tutorial_statistics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "outside-reliance",
   "metadata": {},
   "source": [
    "# Statistics in Python\n",
    "\n",
    "This is a brief overview of statistics in python. In data science we always inspect our data using descriptive statistics and descriptive plots. Such statistics can be used of course for visualisations or dashboards as well. The statistical analysis can be done with a number of tests, depending on the characteristics of the data and the research question to be answered. This notebook refers to the most important ones\n",
    "\n",
    "- Practical: [Descriptive statistics](#0)\n",
    "- Graphical: [Descriptive plots](#1)\n",
    "- Analytical: [Statistical analysis](#2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "dominican-richardson",
   "metadata": {},
   "outputs": [],
   "source": [
    "#import libraries\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "#import scipy \n",
    "#import statsmodels"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "social-patch",
   "metadata": {},
   "source": [
    "<a name='0'></a>\n",
    "## Descriptive statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "collaborative-antigua",
   "metadata": {},
   "source": [
    "Let us create some data for demonstration purpose. We will put the data in a pandas dataframe since pandas has some nice numpy methods built ins, like `mean()`, `sum()`, `max()`,`min()` etc. It can even deliver the descriptive statistics at once with `describe()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "fitting-killing",
   "metadata": {},
   "outputs": [],
   "source": [
    "#series of values with weights\n",
    "x = [8.0, 1, 2.5, 4, 28.0]\n",
    "w = [0.1, 0.2, 0.3, 0.25, 0.15]\n",
    "X = pd.DataFrame({'measurement':x, 'weights':w})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "documentary-scottish",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8.7\n",
      "8.7\n"
     ]
    }
   ],
   "source": [
    "print((1 + 2.5 + 4 + 8 + 28) / 5 )\n",
    "print(X.measurement.mean())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "bacterial-assumption",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "43.5\n",
      "43.5\n"
     ]
    }
   ],
   "source": [
    "print((1 + 2.5 + 4 + 8 + 28))\n",
    "print(X.measurement.sum())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "liked-orlando",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.measurement.min()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "sunrise-directive",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "28.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.measurement.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "corresponding-shock",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.099549540409285"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.measurement.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "postal-holocaust",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.measurement.median()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "amateur-nicholas",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.95\n",
      "6.95\n"
     ]
    }
   ],
   "source": [
    "#weighted average\n",
    "print((8 * 0.1 + 1 * 0.2 + 2.5 * 0.3 + 4 * 0.25 + 28 * 0.15))\n",
    "print(np.average(X['measurement'], weights=X['weights']))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "executive-forwarding",
   "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>measurement</th>\n",
       "      <th>weights</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>5.00000</td>\n",
       "      <td>5.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>8.70000</td>\n",
       "      <td>0.200000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>11.09955</td>\n",
       "      <td>0.079057</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>1.00000</td>\n",
       "      <td>0.100000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>2.50000</td>\n",
       "      <td>0.150000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>4.00000</td>\n",
       "      <td>0.200000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>8.00000</td>\n",
       "      <td>0.250000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>28.00000</td>\n",
       "      <td>0.300000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       measurement   weights\n",
       "count      5.00000  5.000000\n",
       "mean       8.70000  0.200000\n",
       "std       11.09955  0.079057\n",
       "min        1.00000  0.100000\n",
       "25%        2.50000  0.150000\n",
       "50%        4.00000  0.200000\n",
       "75%        8.00000  0.250000\n",
       "max       28.00000  0.300000"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "green-stock",
   "metadata": {
    "scrolled": true
   },
   "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>count</th>\n",
       "      <th>mean</th>\n",
       "      <th>std</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>measurement</th>\n",
       "      <td>5.0</td>\n",
       "      <td>8.7</td>\n",
       "      <td>11.10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>weights</th>\n",
       "      <td>5.0</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.08</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             count  mean    std\n",
       "measurement    5.0   8.7  11.10\n",
       "weights        5.0   0.2   0.08"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nice_table = X.describe().T.round(2)\n",
    "nice_table[['count', 'mean', 'std']]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "interior-cartridge",
   "metadata": {},
   "source": [
    "<a name='1'></a>\n",
    "## Decriptive plots\n",
    "\n",
    "We can also use the built in plots for our explatory data analyses. Like `boxplot()`, `hist()`, `plot.kde()` or just `plot()`. Seaborn has some nice plots as well"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "referenced-lesson",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# boxplot for outlier check\n",
    "X.boxplot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "welcome-balloon",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#check distribution of data\n",
    "X['measurement'].hist()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "minus-italic",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:ylabel='Density'>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#check distribution for normality\n",
    "X['measurement'].plot.kde()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "fossil-modeling",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYkAAAEGCAYAAACQO2mwAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjQuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/MnkTPAAAACXBIWXMAAAsTAAALEwEAmpwYAAAlSElEQVR4nO3deZgU1dXH8e8RMaISCC4ICEGR4BYFg4qoCUQMilEhcQOjxg2XmGiiRMlmjPqqIdHEJYMoChjUqCBRoiKuuEdAFBQRFSKrCIqgDNtw3j9ujdPC1HQP09XVPfP7PM883V3dVXWmuqfP3Lp17zF3R0REpDpbpB2AiIgULyUJERGJpSQhIiKxlCRERCSWkoSIiMTaMu0AcrHDDjt4+/bt0w5DRKSkTJkyZam771iXbZREkmjfvj2TJ09OOwwRkZJiZv+r6zZ0uklERGIpSYiISCwlCRERiaUkISIisZQkREQkVklc3SQiIl817vUFDJkwi4XLy2ndvAmDeneib5c2ed+PkoSISIkZ9/oCBo+dTvm6CgAWLC9n8NjpAHlPFDrdJCJSYoZMmPVlgqhUvq6CIRNm5X1fShIiIiVm4fLyWi2vCyUJEZES07p5k1otrwslCRGREjOodyeaNG70lWVNGjdiUO9Oed+XOq5FREpMZee0rm4SEZFq9e3SJpGksDGdbhIRkVhKEiIiEktJQkREYilJiIhILCUJERGJpSQhIiKxlCRERCSWkoSIiMRSkhARkVhKEiIiEktJQkREYilJiIhILCUJERGJlViSMLO2ZvaMmc00s7fM7KJo+R/NbIGZTYt++iQVg4iI1E2SU4WvBy5x96lm1hSYYmYTo+dudPe/JLhvERHJg8SShLsvAhZF91ea2Uwg+cnPRUQkbwrSJ2Fm7YEuwKvRogvN7E0zu9PMvhGzzkAzm2xmkz/++ONChCkiIhtJPEmY2XbAGOBid18BlAEdgM6ElsZfq1vP3Ye5e1d377rjjjsmHaaIiFQj0SRhZo0JCWK0u48FcPeP3L3C3TcAtwMHJhmDiIhsviSvbjJgODDT3W/IWN4q42X9gBlJxSAiInWT5NVNhwCnAtPNbFq07DdAfzPrDDgwFzg3wRhERKQOkry66QXAqnnq0aT2KSIi+aUR1yIiEktJQkREYilJiIhILCUJERGJpSQhIiKxlCRERCSWkoSIiMRSkhARkVhKEiIiEktJQkREYilJiIhILCUJERGJpSQhIiKxlCRERCSWkoSIiMRSkhARkVhKEiIiEktJQkREYilJiIhILCUJERGJpSQhIiKxlCRERCSWkoSIiMRSkhARkVhKEiIiEiuxJGFmbc3sGTObaWZvmdlF0fIWZjbRzGZHt99IKgYREambJFsS64FL3H1PoBvwMzPbC7gceMrdOwJPRY9FRKQIJZYk3H2Ru0+N7q8EZgJtgOOAkdHLRgJ9k4pBRETqpiB9EmbWHugCvAq0dPdFEBIJsFPMOgPNbLKZTf74448LEaaIiGwk8SRhZtsBY4CL3X1Fruu5+zB37+ruXXfcccfkAhQRkVhZk4SZdTCzr0X3e5jZL8yseS4bN7PGhAQx2t3HRos/MrNW0fOtgCWbFbmIiCQul5bEGKDCzHYHhgO7AvdkW8nMLHr9THe/IeOph4HTo/unA/+uVcQiIlIwuSSJDe6+HugH/M3dfwm0ymG9Q4BTge+b2bTopw9wHXCEmc0Gjogei4hIEdoyh9esM7P+hP/6j4mWNc62kru/AFjM04fnFp6IiKQpl5bEGcDBwDXuPsfMdgX+mWxYIiJSDLK2JNz9bTO7DGgXPZ6DThGJiDQIuVzddAwwDXg8etzZzB5OOC4RESkCuZxu+iNwILAcwN2nEa5wEhGRei6XJLHe3T/baJknEYyIiNTC2rUwaVKiu8glScwwswFAIzPraGY3Ay8lGpWIiMSbOxd+8xto2xZ69oR58xLbVS5J4ufA3sAa4F5gBXBxYhGJiMimKipg/Hg4+mjYbTe4/nro1g3+8x9o0yax3eZyddMq4LfRj4iIFNLixTB8OAwbBh9+CK1awe9/D2efHVoSCYtNEmb2CDX0Pbj7sYlEJCLS0LnDM89AWRmMGwfr10OvXnDjjXDMMdA463jmvKmpJfGXgkUhIiLwyScwciQMHQrvvgstWsBFF8G550LHjqmEFJsk3P25QgYiItIgucOrr4bE8K9/werV0L07/O53cMIJsPXWqYZX0+mm+939RDObTjWnndx930QjExGpzz7/HEaPDslh2jTYbjs44ww47zzYt3i+Xms63XRRdPvDQgQiItIgTJ8e+hr++U9YuRL22y8kigEDoGnTtKPbRE2nmxZFdy9w98synzOz64HLNl1LREQ2sXo1PPhgSA4vvRROIZ10Umg1HHQQWNyE2enLZZzEEdUsOyrfgYiI1DuzZ8Oll4ZxDKeeCkuXwg03wIIFMGJEGOdQxAkCau6TOB+4ANjNzN7MeKop8GLSgYmIlKR16+CRR0Kr4cknYcstoV+/0Gro2bPok8LGauqTuAd4DLgWuDxj+Up3/yTRqERESs28eXDHHeFn4UJo1w6uvhrOPDMMgCtRNfVJfAZ8BvQ3s0ZAy+j125nZdu7+YYFiFBEpThs2wBNPhI7nRx4Jl7MedVR43KcPNGqUdoR1lnVaDjO7kDBd+EfAhmixA8VzjZaISCEtWQJ33QW33QZz5sBOO8Fll8HAgdC+fdrR5VUuNa4vBjq5+7KEYxERKV7u8PzzoZXw4IOh76FHD7j22tDnsNVWaUeYiFySxDzCaScRkYbns89g1KiQHN5+G5o1gwsuCFNl7Lln2tElLpck8QHwrJn9hzBdOADufkNiUYmIpG3KlHCF0r33wqpVcOCBcOedYXzDNtukHV3B5JIkPox+top+RETqp1Wr4L77QnKYPDkkgwEDwuWr3/lO2tGlIpd6ElcWIhARkdS8/XbohB45Mpxe2ntvuPnmMACuWbO0o0tVLlc37Qj8mlCd7svpCN39+wnGJSKSrDVr4KGHQl/Dc8+Fjufjj4fzz4dDDim5QW9JyWVajtHAO8CuwJXAXOC1bCuZ2Z1mtsTMZmQs+6OZLTCzadFPn82MW0Rk88yZA4MHh6pu/fuHQXDXXw/z54dZWQ89VAkiQy59Etu7+3AzuyiqMfGcmeVSa2IEcAswaqPlN7q7ChqJSOFUVIRa0EOHwuOPhyRw7LGhr+GII2CLXP5fbphySRLrottFZnY0sBDYJdtK7j7JzNrXITYRkbpZtChMk3H77aHF0Lo1/OEPoT70Llm/xoTcksTVZtYMuAS4Gfg68Ms67PNCMzsNmAxc4u6fVvciMxsIDARo165dHXYnIg3Khg1V9aH//e9QH/qII+Dvfw/1obfM5WtPKpn7JkXn8rfx0JIY7+77RI9bAksJ03pcBbRy9zOzbadr164+efLkxOIUkXpg2bKq+tCzZ8P224dKb+eeC7vvnnZ0qTCzKe7etS7byOXqpruovnxp1i/3atb5KGO7twPja7sNEZEvucMrr1TVh16zJlyZdMUV8OMfp14fuj7Ipd2V+UW+NdCP0C9Ra2bWKqPiXT9gRk2vFxGp1sqVVfWh33gjlP0866zQEf3tb6cdXb2Sy2C6MZmPzexe4Mls60Wv6wHsYGbzgSuAHmbWmdAymQucW+uIRaThevPNqvrQn38OnTuHQXADBsB226UdXb20OT04HYGsPcnu3r+axcM3Y38i0pCVl8MDD4RWw8svh1NIJ58cWg0HHqgxDQnLpU9iJeE/f4tuFwOXJRyXiDR0s2eHxDBiBHzyCXTqBDfeCKedBi1apB1dg5HL6aamhQhERIR16+Dhh8MppaeeCper/uhHodXQo4daDSmoMUmYWRPgFGCvaNFk4EF3X5t0YCLSgMybFwa83XFHGADXrh1cc02oD73zzmlH16DFjkU3s28DM4HDCJ3M/wN6Ay+aWXMzu7ogEYpI/VRRAY89FqbHaN8err4a9t8fxo+HDz6A3/xGCaII1NSSuAk4x90nZi40s16ES1ffSjIwEamnliwJxXtuuw3mzoWWLeHyy0N96G9+M+3oZCM1JYlWGycIAHd/0szWEcY5iIhk5w6TJoWO6DFjQt9Dz55h9tW+fettfej6oKYksYWZfc3d12QuNLOtgXXuvirZ0ESk5C1fXlUfeuZMaN4cfvazMFXGHnukHZ3koKYkMQoYY2YXuvtc+HIuppuAu5MPTURK1uTJVfWhy8vhoIPgrrvgxBMbVH3o+iA2Sbj71WZ2ITDJzCrf1S+Av7j7zQWJTkRKxxdfhKQwdChMmQLbbgs/+Umo9NalS9rRyWaq8RJYd78FuMXMmkaPVxYkKhEpHW+9FRLDqFGwYgXssw/ceiucckqDrw9dH+Q0LYeSg4h8xZo1MHZsSA6TJoWO5xNOCK2G7t016K0eUfUNEcndBx/AsGHhEtaPP4YOHeDPfw51G3bYIe3oJAFKEiJSs/Xrq+pDT5gQ6kFX1ofu1Uv1oeu5XCb424ZQurSdu59jZh2BTu6ugkEi9dnChVX1oefPhzZtQjGfs88O96VByKUlcRcwBTg4ejwfeABVlROpfzZsCBPrDR0a6kNXVEDv3nDzzfDDH6o+dAOUyzvewd1PMrP+AO5ebqZeKZF6ZdmyMI7httvgvfdC/8Ill4SpMjp0SDs6SVEuSWJtNBusA5hZB2BNzauISNFzD0V8yspCUZ81a+DQQ+HKK0N96K99Le0IpQjkkiSuAB4H2prZaOAQ4KdJBiUiCVqxItSHLiuD6dNDfeizzw4d0fvsk3Z0UmRyKTo00cymAt0I1ekucveliUcmIvk1bVroaxg9OtSH7tIlXM7av7/qQ0us2CRhZvtvtGhRdNvOzNq5+9TkwhKRvCgvh/vvD62GV18N9aH79w+thgMO0KA3yaqmlsRfa3jOge/nORYRyZdZs0In9IgR8OmnYcbVv/0t1If+xjfSjk5KSE0T/PUsZCAiUkfr1sG4ceGU0tNPQ+PGVfWhv/c9tRpks+QymG5r4ALgUEIL4nlgqLuvTjg2EcnFhx+GvoXhw2Hx4lDd7f/+L9SHbtky7eikxOVyddMoYCVQOT14f0I9iROSCkpEsqioCFNklJXBo4+Gy1mPPjpMsNe7NzRqlHaEUk/kkiQ6uft+GY+fMbM3kgpIRGrw0UehxTBsGPzvf6GlMHgwnHOO6kNLInKZmet1M+tW+cDMDgJezLaSmd1pZkvMbEbGshZmNtHMZke36kETycYdnn0WTjoJ2raF3/42jIK+/36YNw+uvloJQhKTS5I4CHjJzOaa2VzgZeB7ZjbdzN6sYb0RwJEbLbsceMrdOwJPRY9FpDqffgp//zvstRf07AkTJ8KFF8I774T5lU44IXROiyQol9NNG3/R58TdJ0U1sTMdB/SI7o8EngUu25zti9RL7vDaa+EKpfvuC+McunULl7KeeCI0aZJ2hNLA5DLi+n/RaaG2ma/fzMF0Ld19UbT+IjPbKe6FZjYQGAjQrl27zdiVSAn5/POq+tBTp4b60KedBueeq/rQkqpcLoG9ijBX0/tEk/xRgMF07j4MGAbQtWtXz/JykdI0Y0ZIDHff/dX60D/5CXz962lHJ5LT6aYTCdOFr83D/j4ys1ZRK6IVsCQP2xQpLWvWwIMPhuTwwgthttXK+tAHH6xBb1JUckkSM4Dm5OcL/WHgdOC66PbfedimSGl4//0wVcZdd8HSpeEKpSFD4Kc/VX1oKVq5JIlrCZfBziCjjoS7H1vTSmZ2L6GTegczm0+Ycvw64H4zOwv4EA3Ik/pu/XoYPz4MenviiTDI7bjjwlQZhx+u+tBS9HJJEiOB64HpwIZcN+zu/WOeOjzXbYiUrAULqupDL1gQakJfeSWcdZbqQ0tJySVJLHX3mxKPRKTUbdgATz4ZWg2PPFJVH/rWW8OUGaoPLSUol0/tFDO7ltCfkHm6SfUkRCD0L1TWh37//ar60OeeC7vtlnZ0InWSS5KovEi7W8Yy1ZOQhs0dXnwxXKH0wAOwdi0cdhj86U+qDy31Si6D6VRXQqTSihVhTMPQoWGMw9e/DgMHhlaD6kNLPZTTSVIzOxrYG9i6cpm7/ympoESKzuuvh76Ge+6BL76A/fcPndL9+4fR0SL1VC4jrocC2wA9gTuA44H/JhyXSPpWraqqD/3f/4Z5kzLrQ0usca8vYMiEWSxcXk7r5k0Y1LsTfbvoqq5SlEtLoru772tmb7r7lWb2V2Bs0oGJpOadd8LppJEjYfly2HPPMBvrqaeqPnQOxr2+gMFjp1O+rgKABcvLGTx2OoASRQnKJUmUR7erzKw1sAzYNbmQRFKwdm2oD11WFmo3VNaHPv98+O53NVVGLQyZMOvLBFGpfF0FQybMUpIoQbkkifFm1hwYAkwlXNl0e5JBiRTM3Lmhb2H48FD1rX17uPZaOOMM1YfeTAuXl9dquRS3XK5uuiq6O8bMxgNbu/tnyYYlkqCKCnjssXBK6dFHQyuhsj70D36g+tB11Lp5ExZUkxBaN1ctjFIUO3GMmR1gZjtnPD4NuB+4ysxaFCI4kbxavBiuuSYMcDvmGJgyJZQCnTMHHn4YjjpKCSIPBvXuRJPGXz2OTRo3YlDvTilFJHVRU0viNqAXgJl9lzA538+BzoQ6D8cnHZxInVXWhy4rg4ceChPuHX44/PWvYaI9lf/Mu8p+B13dVD/UlCQaufsn0f2TgGHuPoZw2mla4pGJ1MUnn8CoUeGU0qxZ4aqkX/wiDHr71rfSjq7e69uljZJCPVFjkjCzLd19PWHm1oE5rieSDvcwnqGsDP71L1i9OtSHHjkyFPVRfWiRWqvpy/5e4DkzW0q4DPZ5ADPbHVDHtRSPzz8PI6GHDg0jo7fbLhTyOfdc6Nw57ehESlpsknD3a8zsKaAV8IS7V9aZ3oLQNyGSrunTq+pDr1wJ++4bWhGnnAJNm6YdnUi9UONpI3d/pZpl7yYXjkgWq1dX1Yd+8cUw2+qJJ4bLV7t106A3kTxT34KUhvfeq6oPvWwZdOwYrlA6/XTYfvu0oxOpt5QkpHitXx8qvJWVwcSJYQxD375hgr3vf1/1oUUKQElCis/8+VX1oRcuhF12CcV8zjoLWrdOOzqRBkVJQorDhg2htVBZH9odjjwyPO7TR/WhRVKivzxJ18cfV9WH/uAD2HFH+PWv4ZxzVB9apAgoSUjhucMLL4QrlB58MEzT/b3vhXmV+vVTfWiRIqIkIYXz2WdV9aHfeguaNQsD3s47D/baK+3oRKQaShKSvKlTq+pDr1oFXbuG+g0nnaT60CJFLpUkYWZzgZVABbDe3bumEYckaNWqMH9SWRm89lqYN2nAgNBq6Kq3W6RUpNmS6OnuS1PcvyRh5szQCV1ZH3qvveCmm0J96ObN045ORGpJp5uk7tauDbUahg6tqg99/PGh1XDYYZoqQ6SEpZUkHHjCzBy4zd2HpRSH1MXcuTBsWOhfWLIEdt0Vrrsu1Ifeaae0oxORPEgrSRzi7gvNbCdgopm94+6TMl9gZgOJali0a9cujRilOpX1ocvKwq1ZKAV63nmhPrSmyhCpV1JJEu6+MLpdYmYPAQcCkzZ6zTBCmVS6du3qm2xECmvx4jBVxrBhMG8etGoFv/89nH02tG2bdnQikpCCJwkz2xbYwt1XRvd/APyp0HFIDtzhmWdCq2HcuDDhXq9e8Le/hdaD6kOL1HtptCRaAg9Z6MzcErjH3R9PIQ6J88kn4eqkoUPh3XehRQu46KIw8K1jx7SjKwrjXl/AkAmzWLi8nNbNmzCodyfVdJZ6qeBJwt0/APYr9H4lC3d49dWQGCrrQ3fvDr/7XagPvfXWaUdYNMa9voDBY6dTvq4CgAXLyxk8djqAEoXUO7oEtqFbubKqPvS0aaE+9BlnhI7offdNO7qiNGTCrC8TRKXydRUMmTBLSULqHSWJhurNN0Ni+Oc/Q6LYb7/weMAA1YfOYuHy8lotFyllShINyerV8MADIRm89FI4hXTSSaHVcNBBGvSWo9bNm7CgmoTQunmTFKIRSZYuam8IZs+GSy+FNm3gtNNg6VK44QZYsABGjIBu3ZQgamFQ7040adzoK8uaNG7EoN6dUopIJDlqSdRX69ZV1Yd+8slQ2a1fv9Bq6NlTSaEOKvsddHWTNARKEvXNvHlV9aEXLYJ27eDqq+HMM8MAOMmLvl3aKClIg6AkUR9s2ABPPBFaDePHh8tZjzoqzMbapw80apR9GyIi1VCSKGVLllTVh54zJ0yqd9llMHAgtG+fdnQiUg8oSZQad3j++ar60OvWQY8ecO21oc9hq63SjlBE6hEliVLx2WcwalRIDm+/HQr4XHBBmCpjzz0T2aWmnhARJYliN3lySAz33htKgh54INx5ZxjfsM02ie1WU0+ICChJFKcvvoD77gvJYfLkkAwq60N/5zsFCUFTT4gIKEkUl7ffDolh1KhwemnvveGWW+AnP4FmzQoaiqaeEBFQkkjfmjUwdmxIDpMmhY7n44+H88+HQw5JbdCbpp4QEdC0HOmZMwcGDw5V3QYMgPnz4frrw+3o0XDooamOitbUEyICakkU1vr18OijodXw+OMhCRx7bOhrOOKIoqoPraknRASUJApj4UIYPjzUh54/H1q3hj/8IdSH3mWXtKOLpaknRERJIikbNsDTT4dWw7hxUFERWgs33RTqQ2+pQy8ixU/fVPm2bFmYfvu228IU3dtvD7/6VZgqY/fd045ORKRWSjJJFN1IYHd4+eXQarj//nDF0iGHwBVXwI9/rPrQIlKySi5JFNVI4JUrQ/nPoUNDOdCmTeGss0JH9Le/XdhYREQSUDyX0+SoppHABfPGGyERtG4d5k/aYotwemnhQrj1ViUIEak3Sq4lkdpI4PLyUB+6rAxeeSWcQjr55DDo7YADVOlNROqlkksSBR8J/O674XTSiBHw6afQqRPceGOoFd2iRTL7FBEpEiV3uqkgI4HXrQu1Gg4/PCSFm28Ol68+/TTMnAkXX6wEISINQsm1JBIdCfzhh6E29B13wOLFoT70NdeE+tA771z37YuIlJhUkoSZHQn8HWgE3OHu19Vm/byOBK6ogAkTwiml//wnXM569NGhY/rII1UfWkQatIInCTNrBNwKHAHMB14zs4fd/e2CBvLRR6F4z7BhMHcutGwZJtw75xz45jcLGoqISLFKoyVxIPCeu38AYGb3AccByScJd3juudBqGDs29D307BlmX+3bV/WhRUQ2kkaSaAPMy3g8Hzho4xeZ2UBgIEC7du3qtsdPP62qD/3OO6E+9M9+FupD77FH3bYtIlKPpZEkqhtQ4JsscB8GDAPo2rXrJs9n5R5Kf5aVhVKg5eVw0EFw112hPnQTFc8REckmjSQxH2ib8XgXYGHetv7FF3DPPaHVMHUqbLstnHpq6Iju0iVvuxERaQjSSBKvAR3NbFdgAXAyMKDOW50xIySGu++GFStgn33CFBmnnFLw+tAiIvVFwZOEu683swuBCYRLYO9097c2a2Nr1sCYMeGU0gsvhI7nE08MrYbu3TVVhohIHaUyTsLdHwUerdNGHnoo1GhYuhQ6dIA//xnOOAN22CE/QYqISOmNuP7SrrvCYYeFVkOvXkVVH1pEpL4o3STRuXMY6yAiIonRv98iIhJLSUJERGIpSYiISCwlCRERiaUkISIisZQkREQklpKEiIjEUpIQEZFY5l77WbgLzcw+Br4AlqYdSw52oPjjLIUYoTTiLIUYQXHmUynECCHObd19x7pspCSSBICZTXb3rmnHkU0pxFkKMUJpxFkKMYLizKdSiBHyF6dON4mISCwlCRERiVVKSWJY2gHkqBTiLIUYoTTiLIUYQXHmUynECHmKs2T6JEREpPBKqSUhIiIFpiQhIiKxiipJmNkJZvaWmW0ws9hLt8zsSDObZWbvmdnlGctbmNlEM5sd3X4jgRiz7sPMOpnZtIyfFWZ2cfTcH81sQcZzffIdY65xRq+ba2bTo1gm13b9pGM0s7Zm9oyZzYw+GxdlPJfosYz7nGU8b2Z2U/T8m2a2f67rFjDGU6LY3jSzl8xsv4znqn3vU4qzh5l9lvFe/iHXdQsc56CMGGeYWYWZtYieK8jxNLM7zWyJmc2IeT6/n0t3L5ofYE+gE/As0DXmNY2A94HdgK2AN4C9ouf+DFwe3b8cuD6BGGu1jyjexcA3o8d/BC4twLHMKU5gLrBDXX/PpGIEWgH7R/ebAu9mvN+JHcuaPmcZr+kDPAYY0A14Ndd1Cxhjd+Ab0f2jKmOs6b1PKc4ewPjNWbeQcW70+mOAp1M4nt8F9gdmxDyf189lUbUk3H2mu8/K8rIDgffc/QN3XwvcBxwXPXccMDK6PxLom0CYtd3H4cD77v6/BGKpSV2PRVEcS3df5O5To/srgZlAmwRi2VhNn7NKxwGjPHgFaG5mrXJctyAxuvtL7v5p9PAVYJcE4simLsejUMdyc/bVH7g3oVhiufsk4JMaXpLXz2VRJYkctQHmZTyeT9WXRkt3XwThywXYKYH913YfJ7PpB+nCqBl4ZxKncSK5xunAE2Y2xcwGbsb6hYgRADNrD3QBXs1YnNSxrOlzlu01uaxbqBgznUX4D7NS3Hufb7nGebCZvWFmj5nZ3rVcNx9y3peZbQMcCYzJWFyo45lNXj+XW+Y1tByY2ZPAztU89Vt3/3cum6hmWV6v460pxlpuZyvgWGBwxuIy4CpCzFcBfwXOTDHOQ9x9oZntBEw0s3ei/1TyIo/HcjvCH+TF7r4iWpy3Y1ndLqtZtvHnLO41iX9Gs+x/0xea9SQkiUMzFif63tcyzqmEU7KfR31L44COOa6bL7XZ1zHAi+6e+R99oY5nNnn9XBY8Sbh7rzpuYj7QNuPxLsDC6P5HZtbK3RdFzasl+Y7RzGqzj6OAqe7+Uca2v7xvZrcD4zcnxnzF6e4Lo9slZvYQoUk6iSI6lmbWmJAgRrv72Ixt5+1YVqOmz1m212yVw7qFihEz2xe4AzjK3ZdVLq/hvS94nBmJH3d/1Mz+YWY75LJuIePMsMkZggIez2zy+rksxdNNrwEdzWzX6D/1k4GHo+ceBk6P7p8O5NIyqa3a7GOTc5bRl2GlfkC1VyjkQdY4zWxbM2taeR/4QUY8RXEszcyA4cBMd79ho+eSPJY1fc4qPQycFl1N0g34LDptlsu6BYnRzNoBY4FT3f3djOU1vfdpxLlz9F5jZgcSvpuW5bJuIeOM4msGfI+Mz2uBj2c2+f1cJt0TX5sfwh/6fGAN8BEwIVreGng043V9CFe5vE84TVW5fHvgKWB2dNsigRir3Uc1MW5D+JA322j9u4HpwJvRG9QqoWOZNU7CVQ5vRD9vFeOxJJwe8eh4TYt++hTiWFb3OQPOA86L7htwa/T8dDKuyIv7jCZwDLPFeAfwacaxm5ztvU8pzgujON4gdLB3L/SxzCXO6PFPgfs2Wq9gx5Pwj+ciYB3h+/KsJD+XmpZDRERileLpJhERKRAlCRERiaUkISIisZQkREQklpKEiIjEUpKQgjGz7a1qBs3FVjWD63Ize7vAsfQ1s70yHv/JzGo90NPM2lv8bJx7m9nTZvaumb1vZleaWd7/5mr6XczsWathRmWRbJQkpGDcfZm7d3b3zsBQ4MbofmdgQ773Z2Y1zSjQF/jyi9Xd/+DuT+Zx300IYzeuc/dvAd8mjMC9qMYVN09fEvxdpGFTkpBi0cjMbrdQM+KJ6EsWM+tgZo9Hk6Y9b2Z7RMu/aWZPWZjc76loZDFmNsLMbjCzZ4Drq1vfzLoT5tQaErVkOkTrHR9t4wALtRfeMLP/mlnTqMXwvJlNjX66Z/l9BhDm9nkCwN1XEQaMDYr28Uczu7TyxRZqE7SP7o+L4n3LMiaKM7PPzeyaKK5XzKxltt8lk5n9wMxejuJ/wMJ8WJjZdWb2dnQs/1L7t07qMyUJKRYdgVvdfW9gOfDjaPkw4Ofu/h3gUuAf0fJbCNMh7wuMBm7K2Na3gF7ufkl167v7S4T/8gdFLZv3K1eMpiv4F3CRu+8H9ALKCfNKHeHu+wMnbbS/6uwNTMlcEO2niZk1z7LumVG8XYFfmNn20fJtgVeiuCYB59T0u2SyMA/S76Ljsj8wGfiVhYI5/YC9o2N5dZbYpIEp+AR/IjHmuPu06P4UoH30n2534IFoWh+Ar0W3BwM/iu7fTShgVOkBd6/Isn6cTsAid38Nqiaei+bjucXMOgMVhERUE6P6GTarm4lzY78ws37R/baEBLoMWEvVJIZTgCNy2FalboRTUi9Gx2Ir4GVgBbAauMPM/kN+J0mUekBJQorFmoz7FUATQkt3edRvkU3mF/IX0W1t1q8U9+X+S8J8YvtF212dZTtvESqIVW3YbDdgqbsvN7P1fLUlv3X0mh6E1svB7r7KzJ6tfA5Y51Xz6FRQu79fAya6e/9NnggT6h1OmPDtQuD7tdiu1HM63SRFK/ovfo6ZnQBf1u6trNH8EuFLDeAU4IVarr+SUA51Y+8Arc3sgGidplEHeDNCC2MDcCqhFGRNRgOHZlxl1IRwiuqK6Pm5hBKUWKhBvGu0vBnwaZQg9iC0ALKJ+10yvQIcYma7R/vcxsy+FbW2mrn7o8DFhIsIRL6kJCHF7hTgLDOrnF2zstziL4AzzOxNwpd23FVDcevfBwwys9fNrEPliz2UdTwJuDlaZyLhP/l/AKeb2SuEU01fUAN3Lyd0KP/WzN4FlhI6skdHLxkDtDCzacD5hJk5AR4Htox+r6sIX+7ZVPu7bBTPx4TZS++Ntv0KsAchuYyPlj1HaDGJfEmzwIoUgJn1BW4Aenrh652LbDYlCRERiaXTTSIiEktJQkREYilJiIhILCUJERGJpSQhIiKxlCRERCTW/wMpyTnHJFRaVwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#check normality with qqplot\n",
    "from statsmodels.graphics.gofplots import qqplot\n",
    "from matplotlib import pyplot\n",
    "qqplot(X['measurement'], line='s')\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "measured-basis",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot to see time effect\n",
    "X['measurement'].plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "honey-installation",
   "metadata": {},
   "source": [
    "<a name='2'></a>\n",
    "## Analytical statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "contemporary-renewal",
   "metadata": {},
   "source": [
    "\n",
    "### Normality check with Shapiro-Wilk Test"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "expressed-delay",
   "metadata": {},
   "source": [
    "It is good practice to check for normality. The Shapiro-Wilk Test is a good test for checking normality"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "blocked-nutrition",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Statistics=0.754, p=0.032\n",
      "Sample does not look Gaussian (reject H0)\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import shapiro\n",
    "# normality test\n",
    "stat, p = shapiro(X['measurement'])\n",
    "print('Statistics=%.3f, p=%.3f' % (stat, p))\n",
    "# interpret\n",
    "alpha = 0.05\n",
    "if p > alpha:\n",
    "    print('Sample looks Gaussian (fail to reject H0)')\n",
    "else:\n",
    "    print('Sample does not look Gaussian (reject H0)')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "informal-browse",
   "metadata": {},
   "source": [
    "source: https://machinelearningmastery.com/statistical-hypothesis-tests-in-python-cheat-sheet/"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "pharmaceutical-register",
   "metadata": {},
   "source": [
    "### More statistics\n",
    "\n",
    "There are a number of cheatsheets and tutorials on the internet. The next overview is a compact overview of tutorials"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ahead-restaurant",
   "metadata": {},
   "source": [
    "- https://www.kaggle.com/hamelg/python-for-data-21-descriptive-statistics\n",
    "- https://www.kaggle.com/hamelg/python-for-data-22-probability-distributions\n",
    "- https://www.kaggle.com/hamelg/python-for-data-23-confidence-intervals\n",
    "- https://www.kaggle.com/hamelg/python-for-data-24-hypothesis-testing\n",
    "- https://www.kaggle.com/hamelg/python-for-data-25-chi-squared-tests\n",
    "- https://www.kaggle.com/hamelg/python-for-data-26-anova/notebook\n",
    "- https://www.kaggle.com/hamelg/python-for-data-27-linear-regression"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "ITANN",
   "language": "python",
   "name": "itann"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}