aula08-eda.ipynb

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    "# Aula 08 &mdash; Análise exploratória de dados\n",
    "\n",
    "Renato Vimieiro\n",
    "rv2 {em} cin.ufpe.br\n",
    "\n",
    "abril 2017"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introdução\n",
    "\n",
    "Vimos na aula anterior, a segunda etapa para análise de dados que consistia na limpeza, integração e pré-processamento dos dados coletados. Terminada essa etapa, seguimos para uma exploração inicial dos dados para nos familiarizarmos com eles. Essa etapa é importante para detectarmos falhas na coleta, validar premissas, avaliar possíveis relações entre os atributos e determinar a direção (escolha dos modelos) que utilizaremos na análise. Essa etapa é conhecida como Análise Exploratória dos Dados, tendo sido proposta pelo estatístico John Tukey em 1977 com a publicação do livro de mesmo nome (em inglês *Exploratory Data Analysis*).\n",
    "\n",
    "John Tukey convenceu os estatísticos da época que essa etapa deveria preceder o emprego de qualquer modelo estatístico para análise dos dados. A principal característica da análise exploratória de dados é o uso de estatísticas descritivas e gráficos para entender relações ocultas presentes nos dados.\n",
    "\n",
    "Como discutimos anteriormente, o modelo/paradigma científico anterior era baseado em hipóteses. Dados eram coletados a fim de verificar a validade dessas hipóteses. Dessa forma, a compreensão das relações presentes nos dados era valiosa para levantar outras hipóteses a serem testadas. Essa mesma razão é ainda mais pertinente no paradigma atual centrado em dados. A maior parte dos dados são coletados arbitrariamente sem uma clara intenção de validar uma hipótese. Isso faz com que a análise exploratória dos dados seja ainda mais valiosa para justamente possibilitar que hipóteses sejam levantadas, assim como para entender o que foi coletado.\n",
    "\n",
    "Uma das ideias mais defendidas por Tukey para análise exploratória de dados é a do uso de gráficos para facilitar a compreensão. Veremos na aula de hoje os diversos tipos de visualizações disponíveis para cada tipo de atributo, assim como uma revisão de estatísticas descritivas.\n",
    "\n",
    "Usaremos como exemplo as séries históricas sobre a expectativa de vida no nascimento e produto interno bruto dos países coletadas pelo Banco Mudial. Os dados estão disponíveis nos links: [expectativa de vida](http://data.worldbank.org/indicator/SP.DYN.LE00.IN) e [PIB](http://data.worldbank.org/indicator/NY.GDP.MKTP.CD)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "\n",
    "from IPython.display import display\n",
    "\n",
    "% matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>1960</th>\n",
       "      <th>1961</th>\n",
       "      <th>1962</th>\n",
       "      <th>1963</th>\n",
       "      <th>1964</th>\n",
       "      <th>1965</th>\n",
       "      <th>1966</th>\n",
       "      <th>1967</th>\n",
       "      <th>1968</th>\n",
       "      <th>1969</th>\n",
       "      <th>...</th>\n",
       "      <th>2005</th>\n",
       "      <th>2006</th>\n",
       "      <th>2007</th>\n",
       "      <th>2008</th>\n",
       "      <th>2009</th>\n",
       "      <th>2010</th>\n",
       "      <th>2011</th>\n",
       "      <th>2012</th>\n",
       "      <th>2013</th>\n",
       "      <th>2014</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Country Name</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
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       "      <th></th>\n",
       "      <th></th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Aruba</th>\n",
       "      <td>65.569366</td>\n",
       "      <td>65.988024</td>\n",
       "      <td>66.365537</td>\n",
       "      <td>66.713976</td>\n",
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       "      <td>67.699000</td>\n",
       "      <td>68.034683</td>\n",
       "      <td>68.377146</td>\n",
       "      <td>68.728415</td>\n",
       "      <td>...</td>\n",
       "      <td>74.228073</td>\n",
       "      <td>74.376195</td>\n",
       "      <td>74.526756</td>\n",
       "      <td>74.675732</td>\n",
       "      <td>74.818146</td>\n",
       "      <td>74.953537</td>\n",
       "      <td>75.081390</td>\n",
       "      <td>75.205756</td>\n",
       "      <td>75.328659</td>\n",
       "      <td>75.451098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Afghanistan</th>\n",
       "      <td>32.328512</td>\n",
       "      <td>32.777439</td>\n",
       "      <td>33.219902</td>\n",
       "      <td>33.657878</td>\n",
       "      <td>34.092878</td>\n",
       "      <td>34.525390</td>\n",
       "      <td>34.957415</td>\n",
       "      <td>35.389415</td>\n",
       "      <td>35.822415</td>\n",
       "      <td>36.260390</td>\n",
       "      <td>...</td>\n",
       "      <td>57.027244</td>\n",
       "      <td>57.432561</td>\n",
       "      <td>57.833829</td>\n",
       "      <td>58.225024</td>\n",
       "      <td>58.603683</td>\n",
       "      <td>58.970829</td>\n",
       "      <td>59.327951</td>\n",
       "      <td>59.679610</td>\n",
       "      <td>60.028268</td>\n",
       "      <td>60.374463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Angola</th>\n",
       "      <td>32.984829</td>\n",
       "      <td>33.386220</td>\n",
       "      <td>33.787585</td>\n",
       "      <td>34.188463</td>\n",
       "      <td>34.590341</td>\n",
       "      <td>34.992220</td>\n",
       "      <td>35.395098</td>\n",
       "      <td>35.799976</td>\n",
       "      <td>36.205854</td>\n",
       "      <td>36.610244</td>\n",
       "      <td>...</td>\n",
       "      <td>48.538805</td>\n",
       "      <td>49.007049</td>\n",
       "      <td>49.435732</td>\n",
       "      <td>49.847390</td>\n",
       "      <td>50.251024</td>\n",
       "      <td>50.654171</td>\n",
       "      <td>51.059317</td>\n",
       "      <td>51.464000</td>\n",
       "      <td>51.866171</td>\n",
       "      <td>52.266878</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Albania</th>\n",
       "      <td>62.254366</td>\n",
       "      <td>63.273463</td>\n",
       "      <td>64.162854</td>\n",
       "      <td>64.887098</td>\n",
       "      <td>65.438195</td>\n",
       "      <td>65.827390</td>\n",
       "      <td>66.089317</td>\n",
       "      <td>66.287220</td>\n",
       "      <td>66.474951</td>\n",
       "      <td>66.681610</td>\n",
       "      <td>...</td>\n",
       "      <td>76.081610</td>\n",
       "      <td>76.286122</td>\n",
       "      <td>76.470293</td>\n",
       "      <td>76.652073</td>\n",
       "      <td>76.840366</td>\n",
       "      <td>77.036951</td>\n",
       "      <td>77.240585</td>\n",
       "      <td>77.443976</td>\n",
       "      <td>77.640463</td>\n",
       "      <td>77.830463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Andorra</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
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       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 55 columns</p>\n",
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       "                   1960       1961       1962       1963       1964  \\\n",
       "Country Name                                                          \n",
       "Aruba         65.569366  65.988024  66.365537  66.713976  67.044293   \n",
       "Afghanistan   32.328512  32.777439  33.219902  33.657878  34.092878   \n",
       "Angola        32.984829  33.386220  33.787585  34.188463  34.590341   \n",
       "Albania       62.254366  63.273463  64.162854  64.887098  65.438195   \n",
       "Andorra             NaN        NaN        NaN        NaN        NaN   \n",
       "\n",
       "                   1965       1966       1967       1968       1969  \\\n",
       "Country Name                                                          \n",
       "Aruba         67.369756  67.699000  68.034683  68.377146  68.728415   \n",
       "Afghanistan   34.525390  34.957415  35.389415  35.822415  36.260390   \n",
       "Angola        34.992220  35.395098  35.799976  36.205854  36.610244   \n",
       "Albania       65.827390  66.089317  66.287220  66.474951  66.681610   \n",
       "Andorra             NaN        NaN        NaN        NaN        NaN   \n",
       "\n",
       "                ...           2005       2006       2007       2008  \\\n",
       "Country Name    ...                                                   \n",
       "Aruba           ...      74.228073  74.376195  74.526756  74.675732   \n",
       "Afghanistan     ...      57.027244  57.432561  57.833829  58.225024   \n",
       "Angola          ...      48.538805  49.007049  49.435732  49.847390   \n",
       "Albania         ...      76.081610  76.286122  76.470293  76.652073   \n",
       "Andorra         ...            NaN        NaN        NaN        NaN   \n",
       "\n",
       "                   2009       2010       2011       2012       2013       2014  \n",
       "Country Name                                                                    \n",
       "Aruba         74.818146  74.953537  75.081390  75.205756  75.328659  75.451098  \n",
       "Afghanistan   58.603683  58.970829  59.327951  59.679610  60.028268  60.374463  \n",
       "Angola        50.251024  50.654171  51.059317  51.464000  51.866171  52.266878  \n",
       "Albania       76.840366  77.036951  77.240585  77.443976  77.640463  77.830463  \n",
       "Andorra             NaN        NaN        NaN        NaN        NaN        NaN  \n",
       "\n",
       "[5 rows x 55 columns]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
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   ],
   "source": [
    "expvida = pd.read_csv(\"../2017.1/data/banco_mudial/life_expectancy.csv\",index_col=0,\n",
    "                      skiprows=[0,1,2],)\n",
    "expvida = expvida.drop(['Indicator Name', 'Indicator Code', \n",
    "                        'Country Code'],axis=1).dropna(axis=1,how='all')\n",
    "expvida.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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       "      <th></th>\n",
       "      <th>1960</th>\n",
       "      <th>1961</th>\n",
       "      <th>1962</th>\n",
       "      <th>1963</th>\n",
       "      <th>1964</th>\n",
       "      <th>1965</th>\n",
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       "      <th>Aruba</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>2.331006e+09</td>\n",
       "      <td>2.421475e+09</td>\n",
       "      <td>2.623726e+09</td>\n",
       "      <td>2.791961e+09</td>\n",
       "      <td>2.498933e+09</td>\n",
       "      <td>2.467704e+09</td>\n",
       "      <td>2.584464e+09</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
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       "      <th>Afghanistan</th>\n",
       "      <td>5.377778e+08</td>\n",
       "      <td>5.488889e+08</td>\n",
       "      <td>5.466667e+08</td>\n",
       "      <td>7.511112e+08</td>\n",
       "      <td>8.000000e+08</td>\n",
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       "      <td>1.400000e+09</td>\n",
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       "      <td>1.373333e+09</td>\n",
       "      <td>1.408889e+09</td>\n",
       "      <td>...</td>\n",
       "      <td>6.275074e+09</td>\n",
       "      <td>7.057598e+09</td>\n",
       "      <td>9.843842e+09</td>\n",
       "      <td>1.019053e+10</td>\n",
       "      <td>1.248694e+10</td>\n",
       "      <td>1.593680e+10</td>\n",
       "      <td>1.793024e+10</td>\n",
       "      <td>2.053654e+10</td>\n",
       "      <td>2.004633e+10</td>\n",
       "      <td>2.005019e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Angola</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>2.823371e+10</td>\n",
       "      <td>4.178948e+10</td>\n",
       "      <td>6.044892e+10</td>\n",
       "      <td>8.417803e+10</td>\n",
       "      <td>7.549238e+10</td>\n",
       "      <td>8.247091e+10</td>\n",
       "      <td>1.041159e+11</td>\n",
       "      <td>1.153984e+11</td>\n",
       "      <td>1.249121e+11</td>\n",
       "      <td>1.267769e+11</td>\n",
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       "    <tr>\n",
       "      <th>Albania</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>8.158549e+09</td>\n",
       "      <td>8.992642e+09</td>\n",
       "      <td>1.070101e+10</td>\n",
       "      <td>1.288135e+10</td>\n",
       "      <td>1.204421e+10</td>\n",
       "      <td>1.192695e+10</td>\n",
       "      <td>1.289087e+10</td>\n",
       "      <td>1.231978e+10</td>\n",
       "      <td>1.278103e+10</td>\n",
       "      <td>1.321986e+10</td>\n",
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       "    <tr>\n",
       "      <th>Andorra</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>3.248215e+09</td>\n",
       "      <td>3.536633e+09</td>\n",
       "      <td>4.010991e+09</td>\n",
       "      <td>4.001201e+09</td>\n",
       "      <td>3.650083e+09</td>\n",
       "      <td>3.346517e+09</td>\n",
       "      <td>3.427023e+09</td>\n",
       "      <td>3.146152e+09</td>\n",
       "      <td>3.248925e+09</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
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      "text/plain": [
       "                      1960          1961          1962          1963  \\\n",
       "Country Name                                                           \n",
       "Aruba                  NaN           NaN           NaN           NaN   \n",
       "Afghanistan   5.377778e+08  5.488889e+08  5.466667e+08  7.511112e+08   \n",
       "Angola                 NaN           NaN           NaN           NaN   \n",
       "Albania                NaN           NaN           NaN           NaN   \n",
       "Andorra                NaN           NaN           NaN           NaN   \n",
       "\n",
       "                      1964          1965          1966          1967  \\\n",
       "Country Name                                                           \n",
       "Aruba                  NaN           NaN           NaN           NaN   \n",
       "Afghanistan   8.000000e+08  1.006667e+09  1.400000e+09  1.673333e+09   \n",
       "Angola                 NaN           NaN           NaN           NaN   \n",
       "Albania                NaN           NaN           NaN           NaN   \n",
       "Andorra                NaN           NaN           NaN           NaN   \n",
       "\n",
       "                      1968          1969      ...               2005  \\\n",
       "Country Name                                  ...                      \n",
       "Aruba                  NaN           NaN      ...       2.331006e+09   \n",
       "Afghanistan   1.373333e+09  1.408889e+09      ...       6.275074e+09   \n",
       "Angola                 NaN           NaN      ...       2.823371e+10   \n",
       "Albania                NaN           NaN      ...       8.158549e+09   \n",
       "Andorra                NaN           NaN      ...       3.248215e+09   \n",
       "\n",
       "                      2006          2007          2008          2009  \\\n",
       "Country Name                                                           \n",
       "Aruba         2.421475e+09  2.623726e+09  2.791961e+09  2.498933e+09   \n",
       "Afghanistan   7.057598e+09  9.843842e+09  1.019053e+10  1.248694e+10   \n",
       "Angola        4.178948e+10  6.044892e+10  8.417803e+10  7.549238e+10   \n",
       "Albania       8.992642e+09  1.070101e+10  1.288135e+10  1.204421e+10   \n",
       "Andorra       3.536633e+09  4.010991e+09  4.001201e+09  3.650083e+09   \n",
       "\n",
       "                      2010          2011          2012          2013  \\\n",
       "Country Name                                                           \n",
       "Aruba         2.467704e+09  2.584464e+09           NaN           NaN   \n",
       "Afghanistan   1.593680e+10  1.793024e+10  2.053654e+10  2.004633e+10   \n",
       "Angola        8.247091e+10  1.041159e+11  1.153984e+11  1.249121e+11   \n",
       "Albania       1.192695e+10  1.289087e+10  1.231978e+10  1.278103e+10   \n",
       "Andorra       3.346517e+09  3.427023e+09  3.146152e+09  3.248925e+09   \n",
       "\n",
       "                      2014  \n",
       "Country Name                \n",
       "Aruba                  NaN  \n",
       "Afghanistan   2.005019e+10  \n",
       "Angola        1.267769e+11  \n",
       "Albania       1.321986e+10  \n",
       "Andorra                NaN  \n",
       "\n",
       "[5 rows x 55 columns]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gdp = pd.read_csv(\"../2017.1/data/banco_mudial/gdp.csv\",index_col=0,\n",
    "                      skiprows=[0,1,2],)\n",
    "gdp = gdp.drop(['Indicator Name', 'Indicator Code', \n",
    "                'Country Code', '2015'],axis=1).dropna(axis=1,how='all')\n",
    "gdp.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Formato de dados: longo e amplo\n",
    "\n",
    "Até o momento, usamos uma representação dos dados em um data frame em que cada variável era armazenada em uma coluna. Esse tipo de representação é chamada de formato amplo de dados. Esse formato é bastante intuitivo e faz com que a visualização/compreensão da tabela seja fácil. Contudo, esse formato pode ser inconveniente para o processamento dos dados. No formato longo, os dados são dispostos, em geral, em três colunas: uma com o índice/rótulo dos exemplos; uma com o rótulo da variável; e a última com o valor dessa variável para o respectivo exemplo. Como os exemplos às vezes são identificados por uma combinação de vários rótulos, podemos ter várias colunas para o índice.\n",
    "\n",
    "O código a seguir mostra como podemos mesclar as tabelas de expectativa de vida e PIB carregadas acima e converter a junção para o formato longo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>variable</th>\n",
       "      <th>value</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>country</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Aruba</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>65.569366</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Afghanistan</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>32.328512</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Angola</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>32.984829</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Albania</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>62.254366</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Andorra</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Arab World</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>46.850022</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>United Arab Emirates</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>52.243220</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Argentina</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>65.215537</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Armenia</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>65.863463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>American Samoa</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Antigua and Barbuda</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>61.782732</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Australia</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>70.817073</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Austria</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>68.585610</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Azerbaijan</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>60.836244</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Burundi</th>\n",
       "      <td>1960_ev</td>\n",
       "      <td>41.236049</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                     variable      value\n",
       "country                                 \n",
       "Aruba                 1960_ev  65.569366\n",
       "Afghanistan           1960_ev  32.328512\n",
       "Angola                1960_ev  32.984829\n",
       "Albania               1960_ev  62.254366\n",
       "Andorra               1960_ev        NaN\n",
       "Arab World            1960_ev  46.850022\n",
       "United Arab Emirates  1960_ev  52.243220\n",
       "Argentina             1960_ev  65.215537\n",
       "Armenia               1960_ev  65.863463\n",
       "American Samoa        1960_ev        NaN\n",
       "Antigua and Barbuda   1960_ev  61.782732\n",
       "Australia             1960_ev  70.817073\n",
       "Austria               1960_ev  68.585610\n",
       "Azerbaijan            1960_ev  60.836244\n",
       "Burundi               1960_ev  41.236049"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expv_pib = pd.merge(expvida,gdp,left_index=True, right_index=True, suffixes=['_ev','_pib'])\n",
    "expv_pib = expv_pib.reset_index()\n",
    "expv_pib  = pd.melt(expv_pib,id_vars='Country Name')\n",
    "expv_pib.rename(columns={'Country Name':'country'}, inplace=True)\n",
    "expv_pib.set_index('country',inplace=True)\n",
    "expv_pib.head(15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Revisão estatísticas descritivas\n",
    "\n",
    "A forma mais usada para descrever e sumarizar um conjunto de dados é através da avaliação da tendência central de um atributo e sua variabilidade ou dispersão. As medidas mais comuns de tendência central são: *média (aritmética)*, *mediana*, *moda* e o ponto médio do intervalo. Já as medidas de dispersão são: desvio padrão, variância, amplitude, quantil, e amplitude/intervalo interquartil.\n",
    "\n",
    "#### Média aritmética\n",
    "\n",
    "A média artimética de uma amostra é dada pela fórmula:\n",
    "\n",
    "$$\n",
    "\\bar{v} = \\frac{\\sum v[i]}{N}\n",
    "$$\n",
    "\n",
    "Em Pandas, como já vimos anteriormente, ela é calculada através do método `mean`. Por exemplo, podemos calcular a expectativa de vida média mundial da seguinte forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Country Name\n",
       "San Marino       81.494746\n",
       "Liechtenstein    80.031243\n",
       "Faroe Islands    78.200944\n",
       "Iceland          77.661399\n",
       "Bermuda          77.622164\n",
       "Japan            77.338235\n",
       "Sweden           77.331827\n",
       "Isle of Man      77.247561\n",
       "Switzerland      77.025885\n",
       "Norway           76.846173\n",
       "dtype: float64"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#expvida.mean(axis=0)\n",
    "expvida.mean(axis=1,skipna=True).sort_values(ascending=False).head(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Mediana\n",
    "\n",
    "Embora a média aritmética seja frequentemente para sumarizar um atributo, ela é muito sensível a valores extremos. A presença de um único valor extremo na amostra pode deslocar bastante a média, fazendo com que ela não seja informativa. Essa situação também ocorre com dados assimétricos (distribuições com uma das caudas mais pesada). Nesse caso, o valor que divide a sequência na metade é mais representativo. Esse valor é a mediana. A mediana, ao contrário da média, pode ser obtida tanto para atributos numéricos quanto para ordinais.\n",
    "\n",
    "A mediana é obtida em Pandas com o método `median`. Podemos obter a mediana do PIB mundial como abaixo. Perceba como a média é deslocada em função do PIB dos países mais ricos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>media</th>\n",
       "      <th>mediana</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1960</th>\n",
       "      <td>7.120347e+10</td>\n",
       "      <td>2.037151e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1961</th>\n",
       "      <td>7.393467e+10</td>\n",
       "      <td>2.088012e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1962</th>\n",
       "      <td>7.839931e+10</td>\n",
       "      <td>2.224868e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1963</th>\n",
       "      <td>8.496941e+10</td>\n",
       "      <td>2.583687e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1964</th>\n",
       "      <td>9.349347e+10</td>\n",
       "      <td>2.782474e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1965</th>\n",
       "      <td>9.774862e+10</td>\n",
       "      <td>2.793218e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1966</th>\n",
       "      <td>1.058608e+11</td>\n",
       "      <td>2.876396e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1967</th>\n",
       "      <td>1.103924e+11</td>\n",
       "      <td>2.869407e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1968</th>\n",
       "      <td>1.155847e+11</td>\n",
       "      <td>3.201439e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1969</th>\n",
       "      <td>1.273173e+11</td>\n",
       "      <td>3.529741e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970</th>\n",
       "      <td>1.334579e+11</td>\n",
       "      <td>3.120019e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1971</th>\n",
       "      <td>1.443843e+11</td>\n",
       "      <td>3.872858e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1972</th>\n",
       "      <td>1.670410e+11</td>\n",
       "      <td>4.468150e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1973</th>\n",
       "      <td>2.048221e+11</td>\n",
       "      <td>5.281699e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1974</th>\n",
       "      <td>2.375791e+11</td>\n",
       "      <td>7.680104e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1975</th>\n",
       "      <td>2.605385e+11</td>\n",
       "      <td>7.841458e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1976</th>\n",
       "      <td>2.793974e+11</td>\n",
       "      <td>9.030262e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1977</th>\n",
       "      <td>3.098103e+11</td>\n",
       "      <td>9.355692e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1978</th>\n",
       "      <td>3.645926e+11</td>\n",
       "      <td>1.034550e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1979</th>\n",
       "      <td>4.221517e+11</td>\n",
       "      <td>1.041705e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1980</th>\n",
       "      <td>4.483871e+11</td>\n",
       "      <td>1.015396e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1981</th>\n",
       "      <td>4.478113e+11</td>\n",
       "      <td>8.661264e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1982</th>\n",
       "      <td>4.458401e+11</td>\n",
       "      <td>9.240000e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1983</th>\n",
       "      <td>4.558556e+11</td>\n",
       "      <td>8.836000e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1984</th>\n",
       "      <td>4.656820e+11</td>\n",
       "      <td>9.701357e+09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1985</th>\n",
       "      <td>4.800454e+11</td>\n",
       "      <td>1.000550e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1986</th>\n",
       "      <td>5.618757e+11</td>\n",
       "      <td>1.023488e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1987</th>\n",
       "      <td>6.271937e+11</td>\n",
       "      <td>1.052734e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1988</th>\n",
       "      <td>6.890933e+11</td>\n",
       "      <td>1.041611e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1989</th>\n",
       "      <td>7.233168e+11</td>\n",
       "      <td>1.036228e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990</th>\n",
       "      <td>7.622509e+11</td>\n",
       "      <td>1.229057e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1991</th>\n",
       "      <td>8.060123e+11</td>\n",
       "      <td>1.134148e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1992</th>\n",
       "      <td>8.443227e+11</td>\n",
       "      <td>1.139631e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1993</th>\n",
       "      <td>8.442709e+11</td>\n",
       "      <td>1.318095e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1994</th>\n",
       "      <td>8.918799e+11</td>\n",
       "      <td>1.289916e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995</th>\n",
       "      <td>9.615238e+11</td>\n",
       "      <td>1.335047e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1996</th>\n",
       "      <td>9.860830e+11</td>\n",
       "      <td>1.384365e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1997</th>\n",
       "      <td>9.892436e+11</td>\n",
       "      <td>1.474460e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1998</th>\n",
       "      <td>9.780026e+11</td>\n",
       "      <td>1.498897e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1999</th>\n",
       "      <td>1.000029e+12</td>\n",
       "      <td>1.492598e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2000</th>\n",
       "      <td>1.011896e+12</td>\n",
       "      <td>1.376037e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001</th>\n",
       "      <td>1.009242e+12</td>\n",
       "      <td>1.381270e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2002</th>\n",
       "      <td>1.048959e+12</td>\n",
       "      <td>1.480319e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2003</th>\n",
       "      <td>1.186172e+12</td>\n",
       "      <td>1.782544e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004</th>\n",
       "      <td>1.341986e+12</td>\n",
       "      <td>2.120635e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2005</th>\n",
       "      <td>1.468775e+12</td>\n",
       "      <td>2.618879e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2006</th>\n",
       "      <td>1.604473e+12</td>\n",
       "      <td>3.021606e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2007</th>\n",
       "      <td>1.837086e+12</td>\n",
       "      <td>3.411310e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2008</th>\n",
       "      <td>2.056086e+12</td>\n",
       "      <td>4.278520e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2009</th>\n",
       "      <td>1.957496e+12</td>\n",
       "      <td>4.206622e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2010</th>\n",
       "      <td>2.177742e+12</td>\n",
       "      <td>4.405093e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2011</th>\n",
       "      <td>2.446024e+12</td>\n",
       "      <td>4.765470e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2012</th>\n",
       "      <td>2.547354e+12</td>\n",
       "      <td>5.182157e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013</th>\n",
       "      <td>2.628796e+12</td>\n",
       "      <td>5.664499e+10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2014</th>\n",
       "      <td>2.762845e+12</td>\n",
       "      <td>5.797893e+10</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             media       mediana\n",
       "1960  7.120347e+10  2.037151e+09\n",
       "1961  7.393467e+10  2.088012e+09\n",
       "1962  7.839931e+10  2.224868e+09\n",
       "1963  8.496941e+10  2.583687e+09\n",
       "1964  9.349347e+10  2.782474e+09\n",
       "1965  9.774862e+10  2.793218e+09\n",
       "1966  1.058608e+11  2.876396e+09\n",
       "1967  1.103924e+11  2.869407e+09\n",
       "1968  1.155847e+11  3.201439e+09\n",
       "1969  1.273173e+11  3.529741e+09\n",
       "1970  1.334579e+11  3.120019e+09\n",
       "1971  1.443843e+11  3.872858e+09\n",
       "1972  1.670410e+11  4.468150e+09\n",
       "1973  2.048221e+11  5.281699e+09\n",
       "1974  2.375791e+11  7.680104e+09\n",
       "1975  2.605385e+11  7.841458e+09\n",
       "1976  2.793974e+11  9.030262e+09\n",
       "1977  3.098103e+11  9.355692e+09\n",
       "1978  3.645926e+11  1.034550e+10\n",
       "1979  4.221517e+11  1.041705e+10\n",
       "1980  4.483871e+11  1.015396e+10\n",
       "1981  4.478113e+11  8.661264e+09\n",
       "1982  4.458401e+11  9.240000e+09\n",
       "1983  4.558556e+11  8.836000e+09\n",
       "1984  4.656820e+11  9.701357e+09\n",
       "1985  4.800454e+11  1.000550e+10\n",
       "1986  5.618757e+11  1.023488e+10\n",
       "1987  6.271937e+11  1.052734e+10\n",
       "1988  6.890933e+11  1.041611e+10\n",
       "1989  7.233168e+11  1.036228e+10\n",
       "1990  7.622509e+11  1.229057e+10\n",
       "1991  8.060123e+11  1.134148e+10\n",
       "1992  8.443227e+11  1.139631e+10\n",
       "1993  8.442709e+11  1.318095e+10\n",
       "1994  8.918799e+11  1.289916e+10\n",
       "1995  9.615238e+11  1.335047e+10\n",
       "1996  9.860830e+11  1.384365e+10\n",
       "1997  9.892436e+11  1.474460e+10\n",
       "1998  9.780026e+11  1.498897e+10\n",
       "1999  1.000029e+12  1.492598e+10\n",
       "2000  1.011896e+12  1.376037e+10\n",
       "2001  1.009242e+12  1.381270e+10\n",
       "2002  1.048959e+12  1.480319e+10\n",
       "2003  1.186172e+12  1.782544e+10\n",
       "2004  1.341986e+12  2.120635e+10\n",
       "2005  1.468775e+12  2.618879e+10\n",
       "2006  1.604473e+12  3.021606e+10\n",
       "2007  1.837086e+12  3.411310e+10\n",
       "2008  2.056086e+12  4.278520e+10\n",
       "2009  1.957496e+12  4.206622e+10\n",
       "2010  2.177742e+12  4.405093e+10\n",
       "2011  2.446024e+12  4.765470e+10\n",
       "2012  2.547354e+12  5.182157e+10\n",
       "2013  2.628796e+12  5.664499e+10\n",
       "2014  2.762845e+12  5.797893e+10"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gdp.apply(lambda x: pd.Series({'mediana': x.median(), 'media': x.mean()})).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Moda\n",
    "\n",
    "A moda é o valor mais frequente em um atributo. Ela pode ser calculada tanto para atributos numéricos quanto categóricos. Note que um atributo pode ter mais de um valor mais frequente. Nesse caso, dizemos que o atributo é **bimodal** (duas) ou **multimodal** (várias modas). Quando todos os valores têm a mesma frequência, então o atributo não possui moda.\n",
    "\n",
    "Em Pandas ela é calcula pela função `mode`. Para exemplificar, vamos discretizar a expectativa de vida em *baixa* [0,50), *média* [50,70) e *alta* [70,inf), e, em seguida, calcular a moda."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1960</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1961</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1962</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1963</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1964</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1965</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1966</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1967</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1968</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1969</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1971</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1972</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1973</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1974</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1975</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1976</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1977</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1978</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1979</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1980</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1981</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1982</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1983</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1984</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1985</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1986</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1987</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1988</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1989</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1991</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1992</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1993</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1994</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1996</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1997</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1998</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1999</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2000</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2002</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2003</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2005</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2006</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2007</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2008</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2009</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2010</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2011</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2012</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2014</th>\n",
       "      <td>[70.0, inf)</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 0\n",
       "1960  [50.0, 70.0)\n",
       "1961  [50.0, 70.0)\n",
       "1962  [50.0, 70.0)\n",
       "1963  [50.0, 70.0)\n",
       "1964  [50.0, 70.0)\n",
       "1965  [50.0, 70.0)\n",
       "1966  [50.0, 70.0)\n",
       "1967  [50.0, 70.0)\n",
       "1968  [50.0, 70.0)\n",
       "1969  [50.0, 70.0)\n",
       "1970  [50.0, 70.0)\n",
       "1971  [50.0, 70.0)\n",
       "1972  [50.0, 70.0)\n",
       "1973  [50.0, 70.0)\n",
       "1974  [50.0, 70.0)\n",
       "1975  [50.0, 70.0)\n",
       "1976  [50.0, 70.0)\n",
       "1977  [50.0, 70.0)\n",
       "1978  [50.0, 70.0)\n",
       "1979  [50.0, 70.0)\n",
       "1980  [50.0, 70.0)\n",
       "1981  [50.0, 70.0)\n",
       "1982  [50.0, 70.0)\n",
       "1983  [50.0, 70.0)\n",
       "1984  [50.0, 70.0)\n",
       "1985  [50.0, 70.0)\n",
       "1986  [50.0, 70.0)\n",
       "1987  [50.0, 70.0)\n",
       "1988  [50.0, 70.0)\n",
       "1989  [50.0, 70.0)\n",
       "1990  [50.0, 70.0)\n",
       "1991  [50.0, 70.0)\n",
       "1992  [50.0, 70.0)\n",
       "1993  [50.0, 70.0)\n",
       "1994  [50.0, 70.0)\n",
       "1995  [50.0, 70.0)\n",
       "1996  [50.0, 70.0)\n",
       "1997  [50.0, 70.0)\n",
       "1998  [50.0, 70.0)\n",
       "1999   [70.0, inf)\n",
       "2000   [70.0, inf)\n",
       "2001   [70.0, inf)\n",
       "2002   [70.0, inf)\n",
       "2003   [70.0, inf)\n",
       "2004   [70.0, inf)\n",
       "2005   [70.0, inf)\n",
       "2006   [70.0, inf)\n",
       "2007   [70.0, inf)\n",
       "2008   [70.0, inf)\n",
       "2009   [70.0, inf)\n",
       "2010   [70.0, inf)\n",
       "2011   [70.0, inf)\n",
       "2012   [70.0, inf)\n",
       "2013   [70.0, inf)\n",
       "2014   [70.0, inf)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expvida.apply(lambda x: pd.cut(x,bins=[0,50,70,np.infty],right=False)).mode().T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Variância e desvio padrão\n",
    "\n",
    "A variância e desvio padrão indicam quão espalhada é a distribuição dos dados. Desvios padrão mais baixos indicam que os dados estão mais concentrados perto da média, enquanto valores mais altos indicam maior amplitude de valores.\n",
    "\n",
    "O desvio padrão de uma amostra é calculado por:\n",
    "\n",
    "$$\n",
    "\\sigma_v = \\sqrt{\\frac{1}{N}\\sum (v[i] - \\bar{v})^2}\n",
    "$$\n",
    "\n",
    "Em Pandas, como já vimos, ele é obtido por `std`. Como exemplo, vamos calcular a média e desvio padrão da expectativa de vida dos países."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>media</th>\n",
       "      <th>desvp</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Country Name</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Aruba</th>\n",
       "      <td>72.009732</td>\n",
       "      <td>2.738927</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Afghanistan</th>\n",
       "      <td>46.871411</td>\n",
       "      <td>9.148672</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Angola</th>\n",
       "      <td>41.851346</td>\n",
       "      <td>5.308439</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Albania</th>\n",
       "      <td>71.246746</td>\n",
       "      <td>4.093708</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Andorra</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Arab World</th>\n",
       "      <td>61.024869</td>\n",
       "      <td>7.497234</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>United Arab Emirates</th>\n",
       "      <td>68.820820</td>\n",
       "      <td>7.074375</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Argentina</th>\n",
       "      <td>70.749239</td>\n",
       "      <td>3.504606</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Armenia</th>\n",
       "      <td>70.316793</td>\n",
       "      <td>2.325032</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>American Samoa</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Antigua and Barbuda</th>\n",
       "      <td>69.923561</td>\n",
       "      <td>4.161397</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Australia</th>\n",
       "      <td>76.063947</td>\n",
       "      <td>3.947829</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Austria</th>\n",
       "      <td>74.732993</td>\n",
       "      <td>4.010781</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Azerbaijan</th>\n",
       "      <td>65.756648</td>\n",
       "      <td>2.538131</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Burundi</th>\n",
       "      <td>48.290789</td>\n",
       "      <td>4.200675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Belgium</th>\n",
       "      <td>75.067503</td>\n",
       "      <td>3.466472</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Benin</th>\n",
       "      <td>49.982449</td>\n",
       "      <td>6.900854</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Burkina Faso</th>\n",
       "      <td>46.979641</td>\n",
       "      <td>6.875524</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bangladesh</th>\n",
       "      <td>57.660148</td>\n",
       "      <td>8.480861</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bulgaria</th>\n",
       "      <td>71.624084</td>\n",
       "      <td>1.180425</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bahrain</th>\n",
       "      <td>69.590544</td>\n",
       "      <td>6.626039</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bahamas, The</th>\n",
       "      <td>69.615921</td>\n",
       "      <td>3.593087</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bosnia and Herzegovina</th>\n",
       "      <td>70.376880</td>\n",
       "      <td>4.425167</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Belarus</th>\n",
       "      <td>69.819016</td>\n",
       "      <td>1.122167</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Belize</th>\n",
       "      <td>68.104451</td>\n",
       "      <td>3.042333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bermuda</th>\n",
       "      <td>77.622164</td>\n",
       "      <td>3.600314</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bolivia</th>\n",
       "      <td>54.181093</td>\n",
       "      <td>8.058512</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Brazil</th>\n",
       "      <td>64.807168</td>\n",
       "      <td>5.919131</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Barbados</th>\n",
       "      <td>69.676153</td>\n",
       "      <td>4.248293</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Brunei Darussalam</th>\n",
       "      <td>71.674379</td>\n",
       "      <td>4.526383</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Latin America &amp; the Caribbean (IDA &amp; IBRD countries)</th>\n",
       "      <td>66.193773</td>\n",
       "      <td>5.810838</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Timor-Leste</th>\n",
       "      <td>48.144183</td>\n",
       "      <td>12.378711</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Middle East &amp; North Africa (IDA &amp; IBRD countries)</th>\n",
       "      <td>61.587495</td>\n",
       "      <td>8.229225</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Tonga</th>\n",
       "      <td>68.260691</td>\n",
       "      <td>3.242705</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>South Asia (IDA &amp; IBRD)</th>\n",
       "      <td>56.415401</td>\n",
       "      <td>7.702862</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sub-Saharan Africa (IDA &amp; IBRD countries)</th>\n",
       "      <td>48.866060</td>\n",
       "      <td>4.580167</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Trinidad and Tobago</th>\n",
       "      <td>67.300077</td>\n",
       "      <td>1.932776</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Tunisia</th>\n",
       "      <td>63.468760</td>\n",
       "      <td>10.678343</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Turkey</th>\n",
       "      <td>62.041620</td>\n",
       "      <td>9.092217</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Tuvalu</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Tanzania</th>\n",
       "      <td>50.965893</td>\n",
       "      <td>5.351550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Uganda</th>\n",
       "      <td>48.614232</td>\n",
       "      <td>3.808850</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Ukraine</th>\n",
       "      <td>69.221696</td>\n",
       "      <td>1.050126</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Upper middle income</th>\n",
       "      <td>65.699634</td>\n",
       "      <td>6.894365</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Uruguay</th>\n",
       "      <td>72.065835</td>\n",
       "      <td>2.971958</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>United States</th>\n",
       "      <td>74.461774</td>\n",
       "      <td>2.895459</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Uzbekistan</th>\n",
       "      <td>65.250732</td>\n",
       "      <td>2.578769</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>St. Vincent and the Grenadines</th>\n",
       "      <td>68.063870</td>\n",
       "      <td>3.822522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Venezuela, RB</th>\n",
       "      <td>68.748199</td>\n",
       "      <td>4.200324</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>British Virgin Islands</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Virgin Islands (U.S.)</th>\n",
       "      <td>73.103758</td>\n",
       "      <td>4.953413</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Vietnam</th>\n",
       "      <td>68.188942</td>\n",
       "      <td>5.672938</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Vanuatu</th>\n",
       "      <td>60.940992</td>\n",
       "      <td>7.713914</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>World</th>\n",
       "      <td>63.722157</td>\n",
       "      <td>5.282044</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Samoa</th>\n",
       "      <td>62.944187</td>\n",
       "      <td>7.300075</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Kosovo</th>\n",
       "      <td>68.304161</td>\n",
       "      <td>1.304467</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Yemen, Rep.</th>\n",
       "      <td>52.435930</td>\n",
       "      <td>9.676421</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>South Africa</th>\n",
       "      <td>55.732976</td>\n",
       "      <td>3.949706</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Zambia</th>\n",
       "      <td>48.558585</td>\n",
       "      <td>4.565202</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Zimbabwe</th>\n",
       "      <td>53.010113</td>\n",
       "      <td>6.525704</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>264 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                                                        media      desvp\n",
       "Country Name                                                            \n",
       "Aruba                                               72.009732   2.738927\n",
       "Afghanistan                                         46.871411   9.148672\n",
       "Angola                                              41.851346   5.308439\n",
       "Albania                                             71.246746   4.093708\n",
       "Andorra                                                   NaN        NaN\n",
       "Arab World                                          61.024869   7.497234\n",
       "United Arab Emirates                                68.820820   7.074375\n",
       "Argentina                                           70.749239   3.504606\n",
       "Armenia                                             70.316793   2.325032\n",
       "American Samoa                                            NaN        NaN\n",
       "Antigua and Barbuda                                 69.923561   4.161397\n",
       "Australia                                           76.063947   3.947829\n",
       "Austria                                             74.732993   4.010781\n",
       "Azerbaijan                                          65.756648   2.538131\n",
       "Burundi                                             48.290789   4.200675\n",
       "Belgium                                             75.067503   3.466472\n",
       "Benin                                               49.982449   6.900854\n",
       "Burkina Faso                                        46.979641   6.875524\n",
       "Bangladesh                                          57.660148   8.480861\n",
       "Bulgaria                                            71.624084   1.180425\n",
       "Bahrain                                             69.590544   6.626039\n",
       "Bahamas, The                                        69.615921   3.593087\n",
       "Bosnia and Herzegovina                              70.376880   4.425167\n",
       "Belarus                                             69.819016   1.122167\n",
       "Belize                                              68.104451   3.042333\n",
       "Bermuda                                             77.622164   3.600314\n",
       "Bolivia                                             54.181093   8.058512\n",
       "Brazil                                              64.807168   5.919131\n",
       "Barbados                                            69.676153   4.248293\n",
       "Brunei Darussalam                                   71.674379   4.526383\n",
       "...                                                       ...        ...\n",
       "Latin America & the Caribbean (IDA & IBRD count...  66.193773   5.810838\n",
       "Timor-Leste                                         48.144183  12.378711\n",
       "Middle East & North Africa (IDA & IBRD countries)   61.587495   8.229225\n",
       "Tonga                                               68.260691   3.242705\n",
       "South Asia (IDA & IBRD)                             56.415401   7.702862\n",
       "Sub-Saharan Africa (IDA & IBRD countries)           48.866060   4.580167\n",
       "Trinidad and Tobago                                 67.300077   1.932776\n",
       "Tunisia                                             63.468760  10.678343\n",
       "Turkey                                              62.041620   9.092217\n",
       "Tuvalu                                                    NaN        NaN\n",
       "Tanzania                                            50.965893   5.351550\n",
       "Uganda                                              48.614232   3.808850\n",
       "Ukraine                                             69.221696   1.050126\n",
       "Upper middle income                                 65.699634   6.894365\n",
       "Uruguay                                             72.065835   2.971958\n",
       "United States                                       74.461774   2.895459\n",
       "Uzbekistan                                          65.250732   2.578769\n",
       "St. Vincent and the Grenadines                      68.063870   3.822522\n",
       "Venezuela, RB                                       68.748199   4.200324\n",
       "British Virgin Islands                                    NaN        NaN\n",
       "Virgin Islands (U.S.)                               73.103758   4.953413\n",
       "Vietnam                                             68.188942   5.672938\n",
       "Vanuatu                                             60.940992   7.713914\n",
       "World                                               63.722157   5.282044\n",
       "Samoa                                               62.944187   7.300075\n",
       "Kosovo                                              68.304161   1.304467\n",
       "Yemen, Rep.                                         52.435930   9.676421\n",
       "South Africa                                        55.732976   3.949706\n",
       "Zambia                                              48.558585   4.565202\n",
       "Zimbabwe                                            53.010113   6.525704\n",
       "\n",
       "[264 rows x 2 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expvida.apply(lambda x: pd.Series({'media':x.mean(), 'desvp':x.std()}, \n",
    "                              index=['media','desvp']),axis=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Covariância\n",
    "\n",
    "A covariância de dois atributos é uma medida para avaliar a variância conjunta de seus valores. Ou seja, ela revela se a variação dos valores de um atributo acompanha a do outro. Valores de covariância positiva indicam que valores altos para um atributo $X$ estão sempre associados a grandes valores para outro atributo $Y$. Analogamente, valores negativos indicam a alternância dos valores; toda vez que $X$ sobe, $Y$ é baixo, ou vice-versa. O zero indica ausência de relação.\n",
    "\n",
    "A covariância (amostral) entre dois atributos $X$ e $Y$ é dada por:\n",
    "$$\n",
    "cov(X,Y) = \\frac{\\sum(X_i-\\bar{X})(Y_i-\\bar{Y})}{N-1}\n",
    "$$\n",
    "\n",
    "A matriz de covariância dos atributos de um data frame pode ser computada por:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>expvida</th>\n",
       "      <th>pib</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>expvida</th>\n",
       "      <td>1.396987e+02</td>\n",
       "      <td>6.379557e+11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>pib</th>\n",
       "      <td>6.379557e+11</td>\n",
       "      <td>4.750014e+22</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              expvida           pib\n",
       "expvida  1.396987e+02  6.379557e+11\n",
       "pib      6.379557e+11  4.750014e+22"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = pd.concat([expvida['1960'],gdp['1960']],keys=['expvida','pib'],axis=1)\n",
    "d.cov()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Correlação\n",
    "\n",
    "A correlação entre duas variáveis indica o grau de interdependência entre elas. Assim como a covariância, essa medida indica se o aumento do valor de uma das variáveis está relacionado ao aumento ou decréscimo do valor da outra variável. \n",
    "\n",
    "A medida de correlação mais usual é o ***coeficiente (ou correlação) de Pearson***. Esse coeficiente é dado por:\n",
    "\n",
    "$$\n",
    "\\rho(X,Y) = \\frac{cov(X,Y)}{\\sigma(X)\\sigma(Y)}\n",
    "$$\n",
    "\n",
    "O coeficiente varia entre -1 e 1, sendo -1 a evidência de uma forte anticorrelação (aumento de um associado à baixa do outro), 0 ausência de relação, e +1 correlação positiva (aumento de um associado ao aumento do outro).\n",
    "\n",
    "O coeficiente de Pearson sempre avalia e quantifica a existência de uma relação **linear** entre as variáveis. Caso haja uma interdependência não-linear entre as variáveis, essa medida não será capaz de evidenciá-la. Uma alternativa robusta a esses casos que é também bastante usada é a ***correlação de Spearman***. A sugestão de Spearman foi avaliar a correlação entre o rank das duas variáveis. Dessa forma, ela é calculada por:\n",
    "\n",
    "$$\n",
    "r(X,Y) = 1 - \\frac{6\\sum d_i^2}{N(N^2-1)}\n",
    "$$\n",
    "\n",
    "Em Python os valores são calculados da seguinte forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>expvida</th>\n",
       "      <th>pib</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>expvida</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.235389</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>pib</th>\n",
       "      <td>0.235389</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          expvida       pib\n",
       "expvida  1.000000  0.235389\n",
       "pib      0.235389  1.000000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>expvida</th>\n",
       "      <th>pib</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>expvida</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.330647</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>pib</th>\n",
       "      <td>0.330647</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          expvida       pib\n",
       "expvida  1.000000  0.330647\n",
       "pib      0.330647  1.000000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(d.corr())\n",
    "display(d.corr(method='spearman'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Amplitude\n",
    "\n",
    "A amplitude de um atributo é a diferença entre o menor e maior valor. Ela pode ser calculada por uma função simples, ou através do método `ptp` (peak to peak) de `pandas.Series`. Por exemplo, podemos avaliar a variação do PIB ao longo dos anos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>amplitude</th>\n",
       "      <th>max</th>\n",
       "      <th>min</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1960</th>\n",
       "      <td>1.352922e+12</td>\n",
       "      <td>1.352934e+12</td>\n",
       "      <td>1.201203e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1961</th>\n",
       "      <td>1.408308e+12</td>\n",
       "      <td>1.408320e+12</td>\n",
       "      <td>1.159202e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1962</th>\n",
       "      <td>1.507685e+12</td>\n",
       "      <td>1.507694e+12</td>\n",
       "      <td>9.122751e+06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1963</th>\n",
       "      <td>1.628576e+12</td>\n",
       "      <td>1.628587e+12</td>\n",
       "      <td>1.084010e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1964</th>\n",
       "      <td>1.786698e+12</td>\n",
       "      <td>1.786711e+12</td>\n",
       "      <td>1.271247e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1965</th>\n",
       "      <td>1.946918e+12</td>\n",
       "      <td>1.946931e+12</td>\n",
       "      <td>1.359393e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1966</th>\n",
       "      <td>2.110107e+12</td>\n",
       "      <td>2.110121e+12</td>\n",
       "      <td>1.446908e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1967</th>\n",
       "      <td>2.253148e+12</td>\n",
       "      <td>2.253164e+12</td>\n",
       "      <td>1.583518e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1968</th>\n",
       "      <td>2.429354e+12</td>\n",
       "      <td>2.429368e+12</td>\n",
       "      <td>1.460000e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1969</th>\n",
       "      <td>2.672066e+12</td>\n",
       "      <td>2.672081e+12</td>\n",
       "      <td>1.585000e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970</th>\n",
       "      <td>2.936435e+12</td>\n",
       "      <td>2.936449e+12</td>\n",
       "      <td>1.429528e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1971</th>\n",
       "      <td>3.242203e+12</td>\n",
       "      <td>3.242218e+12</td>\n",
       "      <td>1.527863e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1972</th>\n",
       "      <td>3.748849e+12</td>\n",
       "      <td>3.748868e+12</td>\n",
       "      <td>1.893653e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1973</th>\n",
       "      <td>4.561595e+12</td>\n",
       "      <td>4.561619e+12</td>\n",
       "      <td>2.419602e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1974</th>\n",
       "      <td>5.244792e+12</td>\n",
       "      <td>5.244823e+12</td>\n",
       "      <td>3.151486e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1975</th>\n",
       "      <td>5.838358e+12</td>\n",
       "      <td>5.838391e+12</td>\n",
       "      <td>3.250674e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1976</th>\n",
       "      <td>6.338916e+12</td>\n",
       "      <td>6.338946e+12</td>\n",
       "      <td>3.003642e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1977</th>\n",
       "      <td>7.164536e+12</td>\n",
       "      <td>7.164570e+12</td>\n",
       "      <td>3.413939e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1978</th>\n",
       "      <td>8.438105e+12</td>\n",
       "      <td>8.438147e+12</td>\n",
       "      <td>4.156747e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1979</th>\n",
       "      <td>9.816364e+12</td>\n",
       "      <td>9.816407e+12</td>\n",
       "      <td>4.262017e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1980</th>\n",
       "      <td>1.107237e+13</td>\n",
       "      <td>1.107241e+13</td>\n",
       "      <td>3.871555e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1981</th>\n",
       "      <td>1.133919e+13</td>\n",
       "      <td>1.133923e+13</td>\n",
       "      <td>3.102000e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1982</th>\n",
       "      <td>1.121781e+13</td>\n",
       "      <td>1.121785e+13</td>\n",
       "      <td>3.491800e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1983</th>\n",
       "      <td>1.159202e+13</td>\n",
       "      <td>1.159206e+13</td>\n",
       "      <td>3.783784e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1984</th>\n",
       "      <td>1.201678e+13</td>\n",
       "      <td>1.201682e+13</td>\n",
       "      <td>4.124616e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1985</th>\n",
       "      <td>1.264371e+13</td>\n",
       "      <td>1.264375e+13</td>\n",
       "      <td>3.212515e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1986</th>\n",
       "      <td>1.496806e+13</td>\n",
       "      <td>1.496810e+13</td>\n",
       "      <td>3.208556e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1987</th>\n",
       "      <td>1.703077e+13</td>\n",
       "      <td>1.703081e+13</td>\n",
       "      <td>3.360874e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1988</th>\n",
       "      <td>1.906539e+13</td>\n",
       "      <td>1.906544e+13</td>\n",
       "      <td>4.297211e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1989</th>\n",
       "      <td>2.001072e+13</td>\n",
       "      <td>2.001076e+13</td>\n",
       "      <td>4.111972e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990</th>\n",
       "      <td>2.251651e+13</td>\n",
       "      <td>2.251652e+13</td>\n",
       "      <td>8.824448e+06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1991</th>\n",
       "      <td>2.385173e+13</td>\n",
       "      <td>2.385174e+13</td>\n",
       "      <td>9.365166e+06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1992</th>\n",
       "      <td>2.534292e+13</td>\n",
       "      <td>2.534293e+13</td>\n",
       "      <td>9.742949e+06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1993</th>\n",
       "      <td>2.578579e+13</td>\n",
       "      <td>2.578580e+13</td>\n",
       "      <td>9.630763e+06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1994</th>\n",
       "      <td>2.775009e+13</td>\n",
       "      <td>2.775010e+13</td>\n",
       "      <td>1.088683e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995</th>\n",
       "      <td>3.084779e+13</td>\n",
       "      <td>3.084780e+13</td>\n",
       "      <td>1.102595e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1996</th>\n",
       "      <td>3.152869e+13</td>\n",
       "      <td>3.152870e+13</td>\n",
       "      <td>1.233485e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1997</th>\n",
       "      <td>3.141216e+13</td>\n",
       "      <td>3.141217e+13</td>\n",
       "      <td>1.270091e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1998</th>\n",
       "      <td>3.131576e+13</td>\n",
       "      <td>3.131578e+13</td>\n",
       "      <td>1.275763e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1999</th>\n",
       "      <td>3.248788e+13</td>\n",
       "      <td>3.248790e+13</td>\n",
       "      <td>1.368714e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2000</th>\n",
       "      <td>3.354757e+13</td>\n",
       "      <td>3.354758e+13</td>\n",
       "      <td>1.374206e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001</th>\n",
       "      <td>3.334277e+13</td>\n",
       "      <td>3.334279e+13</td>\n",
       "      <td>1.319654e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2002</th>\n",
       "      <td>3.461742e+13</td>\n",
       "      <td>3.461743e+13</td>\n",
       "      <td>1.545099e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2003</th>\n",
       "      <td>3.887595e+13</td>\n",
       "      <td>3.887597e+13</td>\n",
       "      <td>1.823108e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004</th>\n",
       "      <td>4.377913e+13</td>\n",
       "      <td>4.377915e+13</td>\n",
       "      <td>2.153493e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2005</th>\n",
       "      <td>4.739173e+13</td>\n",
       "      <td>4.739175e+13</td>\n",
       "      <td>2.183910e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2006</th>\n",
       "      <td>5.131057e+13</td>\n",
       "      <td>5.131059e+13</td>\n",
       "      <td>2.290286e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2007</th>\n",
       "      <td>5.775632e+13</td>\n",
       "      <td>5.775634e+13</td>\n",
       "      <td>2.043329e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2008</th>\n",
       "      <td>6.334507e+13</td>\n",
       "      <td>6.334510e+13</td>\n",
       "      <td>3.029022e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2009</th>\n",
       "      <td>6.004395e+13</td>\n",
       "      <td>6.004397e+13</td>\n",
       "      <td>2.710108e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2010</th>\n",
       "      <td>6.585313e+13</td>\n",
       "      <td>6.585316e+13</td>\n",
       "      <td>3.182352e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2011</th>\n",
       "      <td>7.317263e+13</td>\n",
       "      <td>7.317267e+13</td>\n",
       "      <td>3.931202e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2012</th>\n",
       "      <td>7.468176e+13</td>\n",
       "      <td>7.468180e+13</td>\n",
       "      <td>3.987575e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013</th>\n",
       "      <td>7.677607e+13</td>\n",
       "      <td>7.677611e+13</td>\n",
       "      <td>3.832236e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2014</th>\n",
       "      <td>7.863009e+13</td>\n",
       "      <td>7.863013e+13</td>\n",
       "      <td>3.725969e+07</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         amplitude           max           min\n",
       "1960  1.352922e+12  1.352934e+12  1.201203e+07\n",
       "1961  1.408308e+12  1.408320e+12  1.159202e+07\n",
       "1962  1.507685e+12  1.507694e+12  9.122751e+06\n",
       "1963  1.628576e+12  1.628587e+12  1.084010e+07\n",
       "1964  1.786698e+12  1.786711e+12  1.271247e+07\n",
       "1965  1.946918e+12  1.946931e+12  1.359393e+07\n",
       "1966  2.110107e+12  2.110121e+12  1.446908e+07\n",
       "1967  2.253148e+12  2.253164e+12  1.583518e+07\n",
       "1968  2.429354e+12  2.429368e+12  1.460000e+07\n",
       "1969  2.672066e+12  2.672081e+12  1.585000e+07\n",
       "1970  2.936435e+12  2.936449e+12  1.429528e+07\n",
       "1971  3.242203e+12  3.242218e+12  1.527863e+07\n",
       "1972  3.748849e+12  3.748868e+12  1.893653e+07\n",
       "1973  4.561595e+12  4.561619e+12  2.419602e+07\n",
       "1974  5.244792e+12  5.244823e+12  3.151486e+07\n",
       "1975  5.838358e+12  5.838391e+12  3.250674e+07\n",
       "1976  6.338916e+12  6.338946e+12  3.003642e+07\n",
       "1977  7.164536e+12  7.164570e+12  3.413939e+07\n",
       "1978  8.438105e+12  8.438147e+12  4.156747e+07\n",
       "1979  9.816364e+12  9.816407e+12  4.262017e+07\n",
       "1980  1.107237e+13  1.107241e+13  3.871555e+07\n",
       "1981  1.133919e+13  1.133923e+13  3.102000e+07\n",
       "1982  1.121781e+13  1.121785e+13  3.491800e+07\n",
       "1983  1.159202e+13  1.159206e+13  3.783784e+07\n",
       "1984  1.201678e+13  1.201682e+13  4.124616e+07\n",
       "1985  1.264371e+13  1.264375e+13  3.212515e+07\n",
       "1986  1.496806e+13  1.496810e+13  3.208556e+07\n",
       "1987  1.703077e+13  1.703081e+13  3.360874e+07\n",
       "1988  1.906539e+13  1.906544e+13  4.297211e+07\n",
       "1989  2.001072e+13  2.001076e+13  4.111972e+07\n",
       "1990  2.251651e+13  2.251652e+13  8.824448e+06\n",
       "1991  2.385173e+13  2.385174e+13  9.365166e+06\n",
       "1992  2.534292e+13  2.534293e+13  9.742949e+06\n",
       "1993  2.578579e+13  2.578580e+13  9.630763e+06\n",
       "1994  2.775009e+13  2.775010e+13  1.088683e+07\n",
       "1995  3.084779e+13  3.084780e+13  1.102595e+07\n",
       "1996  3.152869e+13  3.152870e+13  1.233485e+07\n",
       "1997  3.141216e+13  3.141217e+13  1.270091e+07\n",
       "1998  3.131576e+13  3.131578e+13  1.275763e+07\n",
       "1999  3.248788e+13  3.248790e+13  1.368714e+07\n",
       "2000  3.354757e+13  3.354758e+13  1.374206e+07\n",
       "2001  3.334277e+13  3.334279e+13  1.319654e+07\n",
       "2002  3.461742e+13  3.461743e+13  1.545099e+07\n",
       "2003  3.887595e+13  3.887597e+13  1.823108e+07\n",
       "2004  4.377913e+13  4.377915e+13  2.153493e+07\n",
       "2005  4.739173e+13  4.739175e+13  2.183910e+07\n",
       "2006  5.131057e+13  5.131059e+13  2.290286e+07\n",
       "2007  5.775632e+13  5.775634e+13  2.043329e+07\n",
       "2008  6.334507e+13  6.334510e+13  3.029022e+07\n",
       "2009  6.004395e+13  6.004397e+13  2.710108e+07\n",
       "2010  6.585313e+13  6.585316e+13  3.182352e+07\n",
       "2011  7.317263e+13  7.317267e+13  3.931202e+07\n",
       "2012  7.468176e+13  7.468180e+13  3.987575e+07\n",
       "2013  7.677607e+13  7.677611e+13  3.832236e+07\n",
       "2014  7.863009e+13  7.863013e+13  3.725969e+07"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gdp.apply(lambda x: pd.Series({'amplitude': x.ptp(), 'min':x.min(), 'max':x.max()})).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Quantil e amplitude interquartil\n",
    "\n",
    "Os quantis são valores retirados da amostra que dividem os valores em intervalos com a mesma frequência (mesma quantidade de elementos). A mediana, por exemplo, divide os dados em dois intervalos iguais. Logo, ela é equivalente a 2-quantil. O 4-quantil, ou **quartil**, são os três valores da amostra que dividem o intervalo em quatro partes com a mesma quantidade de elementos. Finalmente, o 100-quantil é o conjunto de pontos que dividem os dados em 100 intervalos com a mesma quantidade de elementos, sendo chamados **percentis**.\n",
    "\n",
    "Pandas oferece suporte direto para obtermos esses valores. A função `quantile` espera como parâmetro uma lista de percentis. Exemplo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th></th>\n",
       "      <th>0.25</th>\n",
       "      <th>0.5</th>\n",
       "      <th>0.75</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1960</th>\n",
       "      <td>43.568488</td>\n",
       "      <td>52.475280</td>\n",
       "      <td>63.761963</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1961</th>\n",
       "      <td>44.225994</td>\n",
       "      <td>53.385537</td>\n",
       "      <td>64.321970</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1962</th>\n",
       "      <td>44.819985</td>\n",
       "      <td>54.289159</td>\n",
       "      <td>64.731751</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1963</th>\n",
       "      <td>45.321005</td>\n",
       "      <td>54.910195</td>\n",
       "      <td>64.966694</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1964</th>\n",
       "      <td>45.915841</td>\n",
       "      <td>55.387293</td>\n",
       "      <td>65.391231</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1965</th>\n",
       "      <td>46.638945</td>\n",
       "      <td>55.896852</td>\n",
       "      <td>65.708360</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1966</th>\n",
       "      <td>47.253622</td>\n",
       "      <td>56.356311</td>\n",
       "      <td>66.046768</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1967</th>\n",
       "      <td>47.770750</td>\n",
       "      <td>57.067614</td>\n",
       "      <td>66.467811</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1968</th>\n",
       "      <td>48.225049</td>\n",
       "      <td>57.693280</td>\n",
       "      <td>66.967018</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1969</th>\n",
       "      <td>48.563878</td>\n",
       "      <td>58.506294</td>\n",
       "      <td>67.444213</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970</th>\n",
       "      <td>49.016415</td>\n",
       "      <td>59.282634</td>\n",
       "      <td>68.055927</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1971</th>\n",
       "      <td>49.412561</td>\n",
       "      <td>59.900915</td>\n",
       "      <td>68.217805</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1972</th>\n",
       "      <td>49.810226</td>\n",
       "      <td>60.500024</td>\n",
       "      <td>68.360854</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1973</th>\n",
       "      <td>50.207293</td>\n",
       "      <td>60.912146</td>\n",
       "      <td>68.883902</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1974</th>\n",
       "      <td>50.512340</td>\n",
       "      <td>61.376783</td>\n",
       "      <td>68.668720</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1975</th>\n",
       "      <td>51.145505</td>\n",
       "      <td>61.550610</td>\n",
       "      <td>68.973878</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1976</th>\n",
       "      <td>51.771101</td>\n",
       "      <td>62.061634</td>\n",
       "      <td>69.160787</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1977</th>\n",
       "      <td>52.418293</td>\n",
       "      <td>62.313817</td>\n",
       "      <td>69.460360</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1978</th>\n",
       "      <td>53.003489</td>\n",
       "      <td>63.062483</td>\n",
       "      <td>69.655854</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1979</th>\n",
       "      <td>53.548600</td>\n",
       "      <td>63.533202</td>\n",
       "      <td>69.841634</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1980</th>\n",
       "      <td>54.092801</td>\n",
       "      <td>63.680736</td>\n",
       "      <td>69.868254</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1981</th>\n",
       "      <td>54.373935</td>\n",
       "      <td>64.099276</td>\n",
       "      <td>70.149140</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1982</th>\n",
       "      <td>54.920197</td>\n",
       "      <td>64.699961</td>\n",
       "      <td>70.475796</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1983</th>\n",
       "      <td>55.317420</td>\n",
       "      <td>65.035447</td>\n",
       "      <td>70.434318</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1984</th>\n",
       "      <td>55.741049</td>\n",
       "      <td>65.444615</td>\n",
       "      <td>70.494866</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1985</th>\n",
       "      <td>56.166672</td>\n",
       "      <td>65.793215</td>\n",
       "      <td>70.770585</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1986</th>\n",
       "      <td>56.683109</td>\n",
       "      <td>66.268307</td>\n",
       "      <td>71.130372</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1987</th>\n",
       "      <td>57.275043</td>\n",
       "      <td>66.855064</td>\n",
       "      <td>71.434341</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1988</th>\n",
       "      <td>57.302926</td>\n",
       "      <td>66.928451</td>\n",
       "      <td>71.471280</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1989</th>\n",
       "      <td>57.799501</td>\n",
       "      <td>67.415832</td>\n",
       "      <td>71.465976</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990</th>\n",
       "      <td>58.373494</td>\n",
       "      <td>67.638617</td>\n",
       "      <td>71.362866</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1991</th>\n",
       "      <td>58.642524</td>\n",
       "      <td>67.897558</td>\n",
       "      <td>71.746768</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1992</th>\n",
       "      <td>58.926579</td>\n",
       "      <td>67.792524</td>\n",
       "      <td>71.821543</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1993</th>\n",
       "      <td>58.920407</td>\n",
       "      <td>67.475488</td>\n",
       "      <td>71.935854</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1994</th>\n",
       "      <td>58.969874</td>\n",
       "      <td>67.763976</td>\n",
       "      <td>72.196500</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995</th>\n",
       "      <td>59.126437</td>\n",
       "      <td>68.015439</td>\n",
       "      <td>72.269024</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1996</th>\n",
       "      <td>59.306841</td>\n",
       "      <td>68.523756</td>\n",
       "      <td>72.596256</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1997</th>\n",
       "      <td>59.657732</td>\n",
       "      <td>68.901854</td>\n",
       "      <td>72.940988</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1998</th>\n",
       "      <td>59.412598</td>\n",
       "      <td>68.966415</td>\n",
       "      <td>73.345768</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1999</th>\n",
       "      <td>59.731884</td>\n",
       "      <td>69.274024</td>\n",
       "      <td>73.577049</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2000</th>\n",
       "      <td>60.235388</td>\n",
       "      <td>69.999341</td>\n",
       "      <td>73.752293</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001</th>\n",
       "      <td>60.290585</td>\n",
       "      <td>70.258537</td>\n",
       "      <td>74.500927</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2002</th>\n",
       "      <td>60.888000</td>\n",
       "      <td>70.690732</td>\n",
       "      <td>74.717317</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2003</th>\n",
       "      <td>61.103268</td>\n",
       "      <td>70.845768</td>\n",
       "      <td>74.609817</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004</th>\n",
       "      <td>61.487896</td>\n",
       "      <td>71.106988</td>\n",
       "      <td>74.931201</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2005</th>\n",
       "      <td>62.386156</td>\n",
       "      <td>71.253659</td>\n",
       "      <td>75.044329</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2006</th>\n",
       "      <td>63.135675</td>\n",
       "      <td>71.460220</td>\n",
       "      <td>75.187671</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2007</th>\n",
       "      <td>63.873743</td>\n",
       "      <td>71.697122</td>\n",
       "      <td>75.378543</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2008</th>\n",
       "      <td>64.262877</td>\n",
       "      <td>71.889073</td>\n",
       "      <td>75.591043</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2009</th>\n",
       "      <td>64.640010</td>\n",
       "      <td>72.139451</td>\n",
       "      <td>75.905006</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2010</th>\n",
       "      <td>64.943427</td>\n",
       "      <td>72.304366</td>\n",
       "      <td>76.108872</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2011</th>\n",
       "      <td>65.173256</td>\n",
       "      <td>72.487195</td>\n",
       "      <td>76.355615</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2012</th>\n",
       "      <td>65.346104</td>\n",
       "      <td>72.606427</td>\n",
       "      <td>76.448665</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013</th>\n",
       "      <td>65.678957</td>\n",
       "      <td>72.765707</td>\n",
       "      <td>76.800967</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2014</th>\n",
       "      <td>65.951683</td>\n",
       "      <td>72.936854</td>\n",
       "      <td>76.986146</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           0.25       0.50       0.75\n",
       "1960  43.568488  52.475280  63.761963\n",
       "1961  44.225994  53.385537  64.321970\n",
       "1962  44.819985  54.289159  64.731751\n",
       "1963  45.321005  54.910195  64.966694\n",
       "1964  45.915841  55.387293  65.391231\n",
       "1965  46.638945  55.896852  65.708360\n",
       "1966  47.253622  56.356311  66.046768\n",
       "1967  47.770750  57.067614  66.467811\n",
       "1968  48.225049  57.693280  66.967018\n",
       "1969  48.563878  58.506294  67.444213\n",
       "1970  49.016415  59.282634  68.055927\n",
       "1971  49.412561  59.900915  68.217805\n",
       "1972  49.810226  60.500024  68.360854\n",
       "1973  50.207293  60.912146  68.883902\n",
       "1974  50.512340  61.376783  68.668720\n",
       "1975  51.145505  61.550610  68.973878\n",
       "1976  51.771101  62.061634  69.160787\n",
       "1977  52.418293  62.313817  69.460360\n",
       "1978  53.003489  63.062483  69.655854\n",
       "1979  53.548600  63.533202  69.841634\n",
       "1980  54.092801  63.680736  69.868254\n",
       "1981  54.373935  64.099276  70.149140\n",
       "1982  54.920197  64.699961  70.475796\n",
       "1983  55.317420  65.035447  70.434318\n",
       "1984  55.741049  65.444615  70.494866\n",
       "1985  56.166672  65.793215  70.770585\n",
       "1986  56.683109  66.268307  71.130372\n",
       "1987  57.275043  66.855064  71.434341\n",
       "1988  57.302926  66.928451  71.471280\n",
       "1989  57.799501  67.415832  71.465976\n",
       "1990  58.373494  67.638617  71.362866\n",
       "1991  58.642524  67.897558  71.746768\n",
       "1992  58.926579  67.792524  71.821543\n",
       "1993  58.920407  67.475488  71.935854\n",
       "1994  58.969874  67.763976  72.196500\n",
       "1995  59.126437  68.015439  72.269024\n",
       "1996  59.306841  68.523756  72.596256\n",
       "1997  59.657732  68.901854  72.940988\n",
       "1998  59.412598  68.966415  73.345768\n",
       "1999  59.731884  69.274024  73.577049\n",
       "2000  60.235388  69.999341  73.752293\n",
       "2001  60.290585  70.258537  74.500927\n",
       "2002  60.888000  70.690732  74.717317\n",
       "2003  61.103268  70.845768  74.609817\n",
       "2004  61.487896  71.106988  74.931201\n",
       "2005  62.386156  71.253659  75.044329\n",
       "2006  63.135675  71.460220  75.187671\n",
       "2007  63.873743  71.697122  75.378543\n",
       "2008  64.262877  71.889073  75.591043\n",
       "2009  64.640010  72.139451  75.905006\n",
       "2010  64.943427  72.304366  76.108872\n",
       "2011  65.173256  72.487195  76.355615\n",
       "2012  65.346104  72.606427  76.448665\n",
       "2013  65.678957  72.765707  76.800967\n",
       "2014  65.951683  72.936854  76.986146"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expvida.quantile([0.25, .5, .75]).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A diferença entre o terceiro e o primeiro quartil é chamada de **amplitude interquartil** (IQR). Como o primeiro e terceiro quartis definem os valores tais que 25% e 75% dos dados estão abaixo deles, a diferença entre os dois fornece a amplitude do intervalo que cobre 50% dos valores. Isso nos dá uma ideia de quão espalhados são os dados. Podemos calcular o IQR aplicando uma função que calcule a diferença dos quartis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1960    20.193476\n",
       "1961    20.095976\n",
       "1962    19.911766\n",
       "1963    19.645689\n",
       "1964    19.475390\n",
       "1965    19.069415\n",
       "1966    18.793146\n",
       "1967    18.697061\n",
       "1968    18.741970\n",
       "1969    18.880335\n",
       "1970    19.039512\n",
       "1971    18.805244\n",
       "1972    18.550628\n",
       "1973    18.676610\n",
       "1974    18.156380\n",
       "1975    17.828373\n",
       "1976    17.389686\n",
       "1977    17.042067\n",
       "1978    16.652364\n",
       "1979    16.293035\n",
       "1980    15.775452\n",
       "1981    15.775205\n",
       "1982    15.555599\n",
       "1983    15.116898\n",
       "1984    14.753817\n",
       "1985    14.603914\n",
       "1986    14.447263\n",
       "1987    14.159299\n",
       "1988    14.168354\n",
       "1989    13.666475\n",
       "1990    12.989372\n",
       "1991    13.104244\n",
       "1992    12.894963\n",
       "1993    13.015446\n",
       "1994    13.226626\n",
       "1995    13.142587\n",
       "1996    13.289415\n",
       "1997    13.283256\n",
       "1998    13.933171\n",
       "1999    13.845165\n",
       "2000    13.516905\n",
       "2001    14.210341\n",
       "2002    13.829317\n",
       "2003    13.506549\n",
       "2004    13.443305\n",
       "2005    12.658173\n",
       "2006    12.051995\n",
       "2007    11.504800\n",
       "2008    11.328165\n",
       "2009    11.264996\n",
       "2010    11.165445\n",
       "2011    11.182359\n",
       "2012    11.102561\n",
       "2013    11.122009\n",
       "2014    11.034463\n",
       "dtype: float64"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expvida.apply(lambda x: x.quantile(0.75) - x.quantile(0.25))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Embora as estatísticas acima forneçam uma ideia da dispersão dos dados, todas podem falhar, sobretudo, com dados assimétricos. Dessa forma, é muito comum o **resumo de cinco números** que contém os valores mínimo e máximo do atributo, além do primeiro, segundo (mediana) e terceiro quartis. Se notarmos, essa é a estatística descritiva retornada pelo método `describe`, o qual ainda acrescenta a média, desvio padrão e a quantidade de valores não-nulos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>count</th>\n",
       "      <th>mean</th>\n",
       "      <th>std</th>\n",
       "      <th>min</th>\n",
       "      <th>25%</th>\n",
       "      <th>50%</th>\n",
       "      <th>75%</th>\n",
       "      <th>max</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1960</th>\n",
       "      <td>235.0</td>\n",
       "      <td>53.316972</td>\n",
       "      <td>11.819422</td>\n",
       "      <td>28.211610</td>\n",
       "      <td>43.568488</td>\n",
       "      <td>52.475280</td>\n",
       "      <td>63.761963</td>\n",
       "      <td>73.549756</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1961</th>\n",
       "      <td>236.0</td>\n",
       "      <td>53.870082</td>\n",
       "      <td>11.828587</td>\n",
       "      <td>28.358732</td>\n",
       "      <td>44.225994</td>\n",
       "      <td>53.385537</td>\n",
       "      <td>64.321970</td>\n",
       "      <td>73.652683</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1962</th>\n",
       "      <td>236.0</td>\n",
       "      <td>54.264033</td>\n",
       "      <td>11.689797</td>\n",
       "      <td>28.548463</td>\n",
       "      <td>44.819985</td>\n",
       "      <td>54.289159</td>\n",
       "      <td>64.731751</td>\n",
       "      <td>73.721951</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1963</th>\n",
       "      <td>235.0</td>\n",
       "      <td>54.641853</td>\n",
       "      <td>11.583371</td>\n",
       "      <td>28.796390</td>\n",
       "      <td>45.321005</td>\n",
       "      <td>54.910195</td>\n",
       "      <td>64.966694</td>\n",
       "      <td>73.555366</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1964</th>\n",
       "      <td>235.0</td>\n",
       "      <td>55.134160</td>\n",
       "      <td>11.547545</td>\n",
       "      <td>29.111049</td>\n",
       "      <td>45.915841</td>\n",
       "      <td>55.387293</td>\n",
       "      <td>65.391231</td>\n",
       "      <td>73.733171</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1965</th>\n",
       "      <td>236.0</td>\n",
       "      <td>55.621615</td>\n",
       "      <td>11.429417</td>\n",
       "      <td>29.501927</td>\n",
       "      <td>46.638945</td>\n",
       "      <td>55.896852</td>\n",
       "      <td>65.708360</td>\n",
       "      <td>73.868780</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1966</th>\n",
       "      <td>236.0</td>\n",
       "      <td>56.088514</td>\n",
       "      <td>11.343648</td>\n",
       "      <td>29.970976</td>\n",
       "      <td>47.253622</td>\n",
       "      <td>56.356311</td>\n",
       "      <td>66.046768</td>\n",
       "      <td>74.078537</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1967</th>\n",
       "      <td>236.0</td>\n",
       "      <td>56.515384</td>\n",
       "      <td>11.222269</td>\n",
       "      <td>30.505634</td>\n",
       "      <td>47.770750</td>\n",
       "      <td>57.067614</td>\n",
       "      <td>66.467811</td>\n",
       "      <td>74.122439</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1968</th>\n",
       "      <td>236.0</td>\n",
       "      <td>56.913283</td>\n",
       "      <td>11.077837</td>\n",
       "      <td>31.093780</td>\n",
       "      <td>48.225049</td>\n",
       "      <td>57.693280</td>\n",
       "      <td>66.967018</td>\n",
       "      <td>74.005854</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1969</th>\n",
       "      <td>236.0</td>\n",
       "      <td>57.298899</td>\n",
       "      <td>10.938343</td>\n",
       "      <td>31.726415</td>\n",
       "      <td>48.563878</td>\n",
       "      <td>58.506294</td>\n",
       "      <td>67.444213</td>\n",
       "      <td>74.084878</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970</th>\n",
       "      <td>237.0</td>\n",
       "      <td>57.778240</td>\n",
       "      <td>10.874455</td>\n",
       "      <td>32.393000</td>\n",
       "      <td>49.016415</td>\n",
       "      <td>59.282634</td>\n",
       "      <td>68.055927</td>\n",
       "      <td>74.649268</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1971</th>\n",
       "      <td>236.0</td>\n",
       "      <td>58.115587</td>\n",
       "      <td>10.780703</td>\n",
       "      <td>33.081512</td>\n",
       "      <td>49.412561</td>\n",
       "      <td>59.900915</td>\n",
       "      <td>68.217805</td>\n",
       "      <td>74.623902</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1972</th>\n",
       "      <td>236.0</td>\n",
       "      <td>58.511045</td>\n",
       "      <td>10.723404</td>\n",
       "      <td>33.785000</td>\n",
       "      <td>49.810226</td>\n",
       "      <td>60.500024</td>\n",
       "      <td>68.360854</td>\n",
       "      <td>74.718049</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1973</th>\n",
       "      <td>237.0</td>\n",
       "      <td>58.950935</td>\n",
       "      <td>10.693110</td>\n",
       "      <td>32.812878</td>\n",
       "      <td>50.207293</td>\n",
       "      <td>60.912146</td>\n",
       "      <td>68.883902</td>\n",
       "      <td>74.867317</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1974</th>\n",
       "      <td>236.0</td>\n",
       "      <td>59.269638</td>\n",
       "      <td>10.646682</td>\n",
       "      <td>28.258488</td>\n",
       "      <td>50.512340</td>\n",
       "      <td>61.376783</td>\n",
       "      <td>68.668720</td>\n",
       "      <td>74.980488</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1975</th>\n",
       "      <td>236.0</td>\n",
       "      <td>59.636184</td>\n",
       "      <td>10.623407</td>\n",
       "      <td>23.884537</td>\n",
       "      <td>51.145505</td>\n",
       "      <td>61.550610</td>\n",
       "      <td>68.973878</td>\n",
       "      <td>75.578780</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1976</th>\n",
       "      <td>236.0</td>\n",
       "      <td>60.011477</td>\n",
       "      <td>10.597988</td>\n",
       "      <td>20.656512</td>\n",
       "      <td>51.771101</td>\n",
       "      <td>62.061634</td>\n",
       "      <td>69.160787</td>\n",
       "      <td>76.972195</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1977</th>\n",
       "      <td>236.0</td>\n",
       "      <td>60.394363</td>\n",
       "      <td>10.553734</td>\n",
       "      <td>19.265512</td>\n",
       "      <td>52.418293</td>\n",
       "      <td>62.313817</td>\n",
       "      <td>69.460360</td>\n",
       "      <td>76.373659</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1978</th>\n",
       "      <td>238.0</td>\n",
       "      <td>60.827633</td>\n",
       "      <td>10.446621</td>\n",
       "      <td>20.064927</td>\n",
       "      <td>53.003489</td>\n",
       "      <td>63.062483</td>\n",
       "      <td>69.655854</td>\n",
       "      <td>76.649268</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1979</th>\n",
       "      <td>237.0</td>\n",
       "      <td>61.136480</td>\n",
       "      <td>10.293123</td>\n",
       "      <td>23.029098</td>\n",
       "      <td>53.548600</td>\n",
       "      <td>63.533202</td>\n",
       "      <td>69.841634</td>\n",
       "      <td>76.773171</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1980</th>\n",
       "      <td>238.0</td>\n",
       "      <td>61.534375</td>\n",
       "      <td>10.120552</td>\n",
       "      <td>27.738976</td>\n",
       "      <td>54.092801</td>\n",
       "      <td>63.680736</td>\n",
       "      <td>69.868254</td>\n",
       "      <td>76.846585</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1981</th>\n",
       "      <td>238.0</td>\n",
       "      <td>61.895084</td>\n",
       "      <td>9.963285</td>\n",
       "      <td>33.449927</td>\n",
       "      <td>54.373935</td>\n",
       "      <td>64.099276</td>\n",
       "      <td>70.149140</td>\n",
       "      <td>76.521463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1982</th>\n",
       "      <td>242.0</td>\n",
       "      <td>62.386842</td>\n",
       "      <td>9.803920</td>\n",
       "      <td>38.186220</td>\n",
       "      <td>54.920197</td>\n",
       "      <td>64.699961</td>\n",
       "      <td>70.475796</td>\n",
       "      <td>77.037805</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1983</th>\n",
       "      <td>239.0</td>\n",
       "      <td>62.640375</td>\n",
       "      <td>9.690835</td>\n",
       "      <td>39.666488</td>\n",
       "      <td>55.317420</td>\n",
       "      <td>65.035447</td>\n",
       "      <td>70.434318</td>\n",
       "      <td>76.961463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1984</th>\n",
       "      <td>239.0</td>\n",
       "      <td>62.982854</td>\n",
       "      <td>9.614115</td>\n",
       "      <td>39.999537</td>\n",
       "      <td>55.741049</td>\n",
       "      <td>65.444615</td>\n",
       "      <td>70.494866</td>\n",
       "      <td>77.576585</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1985</th>\n",
       "      <td>239.0</td>\n",
       "      <td>63.289699</td>\n",
       "      <td>9.529507</td>\n",
       "      <td>40.324805</td>\n",
       "      <td>56.166672</td>\n",
       "      <td>65.793215</td>\n",
       "      <td>70.770585</td>\n",
       "      <td>77.650488</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1986</th>\n",
       "      <td>240.0</td>\n",
       "      <td>63.674581</td>\n",
       "      <td>9.530650</td>\n",
       "      <td>39.886268</td>\n",
       "      <td>56.683109</td>\n",
       "      <td>66.268307</td>\n",
       "      <td>71.130372</td>\n",
       "      <td>78.064634</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1987</th>\n",
       "      <td>244.0</td>\n",
       "      <td>64.046644</td>\n",
       "      <td>9.477522</td>\n",
       "      <td>39.330659</td>\n",
       "      <td>57.275043</td>\n",
       "      <td>66.855064</td>\n",
       "      <td>71.434341</td>\n",
       "      <td>78.483659</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1988</th>\n",
       "      <td>240.0</td>\n",
       "      <td>64.174422</td>\n",
       "      <td>9.517636</td>\n",
       "      <td>38.694073</td>\n",
       "      <td>57.302926</td>\n",
       "      <td>66.928451</td>\n",
       "      <td>71.471280</td>\n",
       "      <td>78.399268</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1989</th>\n",
       "      <td>240.0</td>\n",
       "      <td>64.398744</td>\n",
       "      <td>9.569034</td>\n",
       "      <td>38.018585</td>\n",
       "      <td>57.799501</td>\n",
       "      <td>67.415832</td>\n",
       "      <td>71.465976</td>\n",
       "      <td>78.818049</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990</th>\n",
       "      <td>242.0</td>\n",
       "      <td>64.623422</td>\n",
       "      <td>9.582743</td>\n",
       "      <td>33.485122</td>\n",
       "      <td>58.373494</td>\n",
       "      <td>67.638617</td>\n",
       "      <td>71.362866</td>\n",
       "      <td>78.836829</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1991</th>\n",
       "      <td>243.0</td>\n",
       "      <td>64.843291</td>\n",
       "      <td>9.645045</td>\n",
       "      <td>29.745829</td>\n",
       "      <td>58.642524</td>\n",
       "      <td>67.897558</td>\n",
       "      <td>71.746768</td>\n",
       "      <td>79.100732</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1992</th>\n",
       "      <td>244.0</td>\n",
       "      <td>65.011648</td>\n",
       "      <td>9.667101</td>\n",
       "      <td>27.483659</td>\n",
       "      <td>58.926579</td>\n",
       "      <td>67.792524</td>\n",
       "      <td>71.821543</td>\n",
       "      <td>79.153902</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1993</th>\n",
       "      <td>241.0</td>\n",
       "      <td>65.061818</td>\n",
       "      <td>9.705718</td>\n",
       "      <td>27.078902</td>\n",
       "      <td>58.920407</td>\n",
       "      <td>67.475488</td>\n",
       "      <td>71.935854</td>\n",
       "      <td>79.293659</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1994</th>\n",
       "      <td>243.0</td>\n",
       "      <td>65.329304</td>\n",
       "      <td>9.744893</td>\n",
       "      <td>28.568122</td>\n",
       "      <td>58.969874</td>\n",
       "      <td>67.763976</td>\n",
       "      <td>72.196500</td>\n",
       "      <td>79.687073</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995</th>\n",
       "      <td>243.0</td>\n",
       "      <td>65.483428</td>\n",
       "      <td>9.684162</td>\n",
       "      <td>31.634512</td>\n",
       "      <td>59.126437</td>\n",
       "      <td>68.015439</td>\n",
       "      <td>72.269024</td>\n",
       "      <td>79.536341</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1996</th>\n",
       "      <td>243.0</td>\n",
       "      <td>65.755696</td>\n",
       "      <td>9.729078</td>\n",
       "      <td>35.659195</td>\n",
       "      <td>59.306841</td>\n",
       "      <td>68.523756</td>\n",
       "      <td>72.596256</td>\n",
       "      <td>80.200244</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1997</th>\n",
       "      <td>247.0</td>\n",
       "      <td>66.085685</td>\n",
       "      <td>9.703275</td>\n",
       "      <td>36.426195</td>\n",
       "      <td>59.657732</td>\n",
       "      <td>68.901854</td>\n",
       "      <td>72.940988</td>\n",
       "      <td>80.424146</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1998</th>\n",
       "      <td>243.0</td>\n",
       "      <td>66.211192</td>\n",
       "      <td>9.779394</td>\n",
       "      <td>37.046805</td>\n",
       "      <td>59.412598</td>\n",
       "      <td>68.966415</td>\n",
       "      <td>73.345768</td>\n",
       "      <td>80.501463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1999</th>\n",
       "      <td>244.0</td>\n",
       "      <td>66.437642</td>\n",
       "      <td>9.785452</td>\n",
       "      <td>37.812537</td>\n",
       "      <td>59.731884</td>\n",
       "      <td>69.274024</td>\n",
       "      <td>73.577049</td>\n",
       "      <td>80.570732</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2000</th>\n",
       "      <td>247.0</td>\n",
       "      <td>66.762516</td>\n",
       "      <td>9.789939</td>\n",
       "      <td>38.690146</td>\n",
       "      <td>60.235388</td>\n",
       "      <td>69.999341</td>\n",
       "      <td>73.752293</td>\n",
       "      <td>81.076098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2001</th>\n",
       "      <td>245.0</td>\n",
       "      <td>67.052200</td>\n",
       "      <td>9.863487</td>\n",
       "      <td>39.645488</td>\n",
       "      <td>60.290585</td>\n",
       "      <td>70.258537</td>\n",
       "      <td>74.500927</td>\n",
       "      <td>81.424390</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2002</th>\n",
       "      <td>249.0</td>\n",
       "      <td>67.429526</td>\n",
       "      <td>9.780320</td>\n",
       "      <td>40.642902</td>\n",
       "      <td>60.888000</td>\n",
       "      <td>70.690732</td>\n",
       "      <td>74.717317</td>\n",
       "      <td>81.563415</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2003</th>\n",
       "      <td>246.0</td>\n",
       "      <td>67.612203</td>\n",
       "      <td>9.700922</td>\n",
       "      <td>40.702488</td>\n",
       "      <td>61.103268</td>\n",
       "      <td>70.845768</td>\n",
       "      <td>74.609817</td>\n",
       "      <td>81.760000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004</th>\n",
       "      <td>246.0</td>\n",
       "      <td>67.997679</td>\n",
       "      <td>9.646001</td>\n",
       "      <td>41.059341</td>\n",
       "      <td>61.487896</td>\n",
       "      <td>71.106988</td>\n",
       "      <td>74.931201</td>\n",
       "      <td>82.030244</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2005</th>\n",
       "      <td>247.0</td>\n",
       "      <td>68.305760</td>\n",
       "      <td>9.452554</td>\n",
       "      <td>41.759610</td>\n",
       "      <td>62.386156</td>\n",
       "      <td>71.253659</td>\n",
       "      <td>75.044329</td>\n",
       "      <td>81.975610</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2006</th>\n",
       "      <td>247.0</td>\n",
       "      <td>68.702235</td>\n",
       "      <td>9.305500</td>\n",
       "      <td>42.810707</td>\n",
       "      <td>63.135675</td>\n",
       "      <td>71.460220</td>\n",
       "      <td>75.187671</td>\n",
       "      <td>82.375610</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2007</th>\n",
       "      <td>248.0</td>\n",
       "      <td>69.081031</td>\n",
       "      <td>9.100834</td>\n",
       "      <td>44.177756</td>\n",
       "      <td>63.873743</td>\n",
       "      <td>71.697122</td>\n",
       "      <td>75.378543</td>\n",
       "      <td>82.507073</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2008</th>\n",
       "      <td>248.0</td>\n",
       "      <td>69.444249</td>\n",
       "      <td>8.919874</td>\n",
       "      <td>45.739244</td>\n",
       "      <td>64.262877</td>\n",
       "      <td>71.889073</td>\n",
       "      <td>75.591043</td>\n",
       "      <td>82.831707</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2009</th>\n",
       "      <td>248.0</td>\n",
       "      <td>69.813421</td>\n",
       "      <td>8.744318</td>\n",
       "      <td>46.634585</td>\n",
       "      <td>64.640010</td>\n",
       "      <td>72.139451</td>\n",
       "      <td>75.905006</td>\n",
       "      <td>82.995543</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2010</th>\n",
       "      <td>246.0</td>\n",
       "      <td>70.099730</td>\n",
       "      <td>8.604553</td>\n",
       "      <td>47.483415</td>\n",
       "      <td>64.943427</td>\n",
       "      <td>72.304366</td>\n",
       "      <td>76.108872</td>\n",
       "      <td>83.159379</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2011</th>\n",
       "      <td>247.0</td>\n",
       "      <td>70.476308</td>\n",
       "      <td>8.487421</td>\n",
       "      <td>48.219756</td>\n",
       "      <td>65.173256</td>\n",
       "      <td>72.487195</td>\n",
       "      <td>76.355615</td>\n",
       "      <td>83.421951</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2012</th>\n",
       "      <td>246.0</td>\n",
       "      <td>70.696585</td>\n",
       "      <td>8.312338</td>\n",
       "      <td>48.836000</td>\n",
       "      <td>65.346104</td>\n",
       "      <td>72.606427</td>\n",
       "      <td>76.448665</td>\n",
       "      <td>83.480488</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2013</th>\n",
       "      <td>244.0</td>\n",
       "      <td>70.965698</td>\n",
       "      <td>8.237375</td>\n",
       "      <td>48.937927</td>\n",
       "      <td>65.678957</td>\n",
       "      <td>72.765707</td>\n",
       "      <td>76.800967</td>\n",
       "      <td>83.831707</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2014</th>\n",
       "      <td>245.0</td>\n",
       "      <td>71.245815</td>\n",
       "      <td>8.123204</td>\n",
       "      <td>48.934732</td>\n",
       "      <td>65.951683</td>\n",
       "      <td>72.936854</td>\n",
       "      <td>76.986146</td>\n",
       "      <td>83.980488</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      count       mean        std        min        25%        50%        75%  \\\n",
       "1960  235.0  53.316972  11.819422  28.211610  43.568488  52.475280  63.761963   \n",
       "1961  236.0  53.870082  11.828587  28.358732  44.225994  53.385537  64.321970   \n",
       "1962  236.0  54.264033  11.689797  28.548463  44.819985  54.289159  64.731751   \n",
       "1963  235.0  54.641853  11.583371  28.796390  45.321005  54.910195  64.966694   \n",
       "1964  235.0  55.134160  11.547545  29.111049  45.915841  55.387293  65.391231   \n",
       "1965  236.0  55.621615  11.429417  29.501927  46.638945  55.896852  65.708360   \n",
       "1966  236.0  56.088514  11.343648  29.970976  47.253622  56.356311  66.046768   \n",
       "1967  236.0  56.515384  11.222269  30.505634  47.770750  57.067614  66.467811   \n",
       "1968  236.0  56.913283  11.077837  31.093780  48.225049  57.693280  66.967018   \n",
       "1969  236.0  57.298899  10.938343  31.726415  48.563878  58.506294  67.444213   \n",
       "1970  237.0  57.778240  10.874455  32.393000  49.016415  59.282634  68.055927   \n",
       "1971  236.0  58.115587  10.780703  33.081512  49.412561  59.900915  68.217805   \n",
       "1972  236.0  58.511045  10.723404  33.785000  49.810226  60.500024  68.360854   \n",
       "1973  237.0  58.950935  10.693110  32.812878  50.207293  60.912146  68.883902   \n",
       "1974  236.0  59.269638  10.646682  28.258488  50.512340  61.376783  68.668720   \n",
       "1975  236.0  59.636184  10.623407  23.884537  51.145505  61.550610  68.973878   \n",
       "1976  236.0  60.011477  10.597988  20.656512  51.771101  62.061634  69.160787   \n",
       "1977  236.0  60.394363  10.553734  19.265512  52.418293  62.313817  69.460360   \n",
       "1978  238.0  60.827633  10.446621  20.064927  53.003489  63.062483  69.655854   \n",
       "1979  237.0  61.136480  10.293123  23.029098  53.548600  63.533202  69.841634   \n",
       "1980  238.0  61.534375  10.120552  27.738976  54.092801  63.680736  69.868254   \n",
       "1981  238.0  61.895084   9.963285  33.449927  54.373935  64.099276  70.149140   \n",
       "1982  242.0  62.386842   9.803920  38.186220  54.920197  64.699961  70.475796   \n",
       "1983  239.0  62.640375   9.690835  39.666488  55.317420  65.035447  70.434318   \n",
       "1984  239.0  62.982854   9.614115  39.999537  55.741049  65.444615  70.494866   \n",
       "1985  239.0  63.289699   9.529507  40.324805  56.166672  65.793215  70.770585   \n",
       "1986  240.0  63.674581   9.530650  39.886268  56.683109  66.268307  71.130372   \n",
       "1987  244.0  64.046644   9.477522  39.330659  57.275043  66.855064  71.434341   \n",
       "1988  240.0  64.174422   9.517636  38.694073  57.302926  66.928451  71.471280   \n",
       "1989  240.0  64.398744   9.569034  38.018585  57.799501  67.415832  71.465976   \n",
       "1990  242.0  64.623422   9.582743  33.485122  58.373494  67.638617  71.362866   \n",
       "1991  243.0  64.843291   9.645045  29.745829  58.642524  67.897558  71.746768   \n",
       "1992  244.0  65.011648   9.667101  27.483659  58.926579  67.792524  71.821543   \n",
       "1993  241.0  65.061818   9.705718  27.078902  58.920407  67.475488  71.935854   \n",
       "1994  243.0  65.329304   9.744893  28.568122  58.969874  67.763976  72.196500   \n",
       "1995  243.0  65.483428   9.684162  31.634512  59.126437  68.015439  72.269024   \n",
       "1996  243.0  65.755696   9.729078  35.659195  59.306841  68.523756  72.596256   \n",
       "1997  247.0  66.085685   9.703275  36.426195  59.657732  68.901854  72.940988   \n",
       "1998  243.0  66.211192   9.779394  37.046805  59.412598  68.966415  73.345768   \n",
       "1999  244.0  66.437642   9.785452  37.812537  59.731884  69.274024  73.577049   \n",
       "2000  247.0  66.762516   9.789939  38.690146  60.235388  69.999341  73.752293   \n",
       "2001  245.0  67.052200   9.863487  39.645488  60.290585  70.258537  74.500927   \n",
       "2002  249.0  67.429526   9.780320  40.642902  60.888000  70.690732  74.717317   \n",
       "2003  246.0  67.612203   9.700922  40.702488  61.103268  70.845768  74.609817   \n",
       "2004  246.0  67.997679   9.646001  41.059341  61.487896  71.106988  74.931201   \n",
       "2005  247.0  68.305760   9.452554  41.759610  62.386156  71.253659  75.044329   \n",
       "2006  247.0  68.702235   9.305500  42.810707  63.135675  71.460220  75.187671   \n",
       "2007  248.0  69.081031   9.100834  44.177756  63.873743  71.697122  75.378543   \n",
       "2008  248.0  69.444249   8.919874  45.739244  64.262877  71.889073  75.591043   \n",
       "2009  248.0  69.813421   8.744318  46.634585  64.640010  72.139451  75.905006   \n",
       "2010  246.0  70.099730   8.604553  47.483415  64.943427  72.304366  76.108872   \n",
       "2011  247.0  70.476308   8.487421  48.219756  65.173256  72.487195  76.355615   \n",
       "2012  246.0  70.696585   8.312338  48.836000  65.346104  72.606427  76.448665   \n",
       "2013  244.0  70.965698   8.237375  48.937927  65.678957  72.765707  76.800967   \n",
       "2014  245.0  71.245815   8.123204  48.934732  65.951683  72.936854  76.986146   \n",
       "\n",
       "            max  \n",
       "1960  73.549756  \n",
       "1961  73.652683  \n",
       "1962  73.721951  \n",
       "1963  73.555366  \n",
       "1964  73.733171  \n",
       "1965  73.868780  \n",
       "1966  74.078537  \n",
       "1967  74.122439  \n",
       "1968  74.005854  \n",
       "1969  74.084878  \n",
       "1970  74.649268  \n",
       "1971  74.623902  \n",
       "1972  74.718049  \n",
       "1973  74.867317  \n",
       "1974  74.980488  \n",
       "1975  75.578780  \n",
       "1976  76.972195  \n",
       "1977  76.373659  \n",
       "1978  76.649268  \n",
       "1979  76.773171  \n",
       "1980  76.846585  \n",
       "1981  76.521463  \n",
       "1982  77.037805  \n",
       "1983  76.961463  \n",
       "1984  77.576585  \n",
       "1985  77.650488  \n",
       "1986  78.064634  \n",
       "1987  78.483659  \n",
       "1988  78.399268  \n",
       "1989  78.818049  \n",
       "1990  78.836829  \n",
       "1991  79.100732  \n",
       "1992  79.153902  \n",
       "1993  79.293659  \n",
       "1994  79.687073  \n",
       "1995  79.536341  \n",
       "1996  80.200244  \n",
       "1997  80.424146  \n",
       "1998  80.501463  \n",
       "1999  80.570732  \n",
       "2000  81.076098  \n",
       "2001  81.424390  \n",
       "2002  81.563415  \n",
       "2003  81.760000  \n",
       "2004  82.030244  \n",
       "2005  81.975610  \n",
       "2006  82.375610  \n",
       "2007  82.507073  \n",
       "2008  82.831707  \n",
       "2009  82.995543  \n",
       "2010  83.159379  \n",
       "2011  83.421951  \n",
       "2012  83.480488  \n",
       "2013  83.831707  \n",
       "2014  83.980488  "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expvida.describe().T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Sumarização de dados categóricos\n",
    "\n",
    "Vimos que a moda e mediana podem ser calculadas para valores categóricos, mas quais outras estatísticas podemos computar para descrever os dados? O mais comum com dados nominais é contar a frequência das categorias, apresentando-as de forma relativa ou absoluta. Também podemos explorar a relação entre dois atributos categóricos, montando uma tabela (tabela de contigência) entre elas. Por exemplo, imagine que estivéssemos trabalhando com dados discretos para a expectativa de vida e PIB. Poderíamos querer investigar qual a relação (se existe alguma) entre a faixa de valores do PIB e a expectativa de vida. Podemos tabular os dois valores e ver a contagem das ocorrências."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>expvida</th>\n",
       "      <th>pib</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>country</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Aruba</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Afghanistan</th>\n",
       "      <td>[0.0, 50.0)</td>\n",
       "      <td>(0.0002, 0.0673]</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Angola</th>\n",
       "      <td>[0.0, 50.0)</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Albania</th>\n",
       "      <td>[50.0, 70.0)</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Andorra</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                  expvida               pib\n",
       "country                                    \n",
       "Aruba        [50.0, 70.0)               NaN\n",
       "Afghanistan   [0.0, 50.0)  (0.0002, 0.0673]\n",
       "Angola        [0.0, 50.0)               NaN\n",
       "Albania      [50.0, 70.0)               NaN\n",
       "Andorra               NaN               NaN"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.DataFrame({'expvida':pd.cut(expv_pib[expv_pib.variable.str.contains('1960_ev')].value,\n",
    "                                    bins=[0,50,70,np.infty],right=False),\n",
    "                  'pib':pd.qcut(expv_pib[expv_pib.variable.str.contains('1960_pib')].value/10E9,\n",
    "                                q=3)})\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
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       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>pib</th>\n",
       "      <th>(0.0002, 0.0673]</th>\n",
       "      <th>(0.0673, 0.818]</th>\n",
       "      <th>(0.818, 135.293]</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>expvida</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>[0.0, 50.0)</th>\n",
       "      <td>26</td>\n",
       "      <td>16</td>\n",
       "      <td>20</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>[50.0, 70.0)</th>\n",
       "      <td>12</td>\n",
       "      <td>21</td>\n",
       "      <td>16</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>[70.0, inf)</th>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "pib           (0.0002, 0.0673]  (0.0673, 0.818]  (0.818, 135.293]\n",
       "expvida                                                          \n",
       "[0.0, 50.0)                 26               16                20\n",
       "[50.0, 70.0)                12               21                16\n",
       "[70.0, inf)                  1                3                 6"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.crosstab(df.expvida,df.pib)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>pib</th>\n",
       "      <th>(0.00218, 1.283]</th>\n",
       "      <th>(1.283, 23.769]</th>\n",
       "      <th>(23.769, 6585.316]</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>expvida</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>[0.0, 50.0)</th>\n",
       "      <td>6</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>[50.0, 70.0)</th>\n",
       "      <td>40</td>\n",
       "      <td>27</td>\n",
       "      <td>26</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>[70.0, inf)</th>\n",
       "      <td>27</td>\n",
       "      <td>54</td>\n",
       "      <td>56</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "pib           (0.00218, 1.283]  (1.283, 23.769]  (23.769, 6585.316]\n",
       "expvida                                                            \n",
       "[0.0, 50.0)                  6                0                   0\n",
       "[50.0, 70.0)                40               27                  26\n",
       "[70.0, inf)                 27               54                  56"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.DataFrame({'expvida':pd.cut(expv_pib[expv_pib.variable.str.contains('2010_ev')].value,bins=[0,50,70,np.infty],right=False),\\\n",
    "                  'pib':pd.qcut(expv_pib[expv_pib.variable.str.contains('2010_pib')].value/10E9,q=3)})\n",
    "pd.crosstab(df.expvida,df.pib)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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