mann_whitney.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "fc2c74d6",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt \n",
    "import seaborn as sns \n",
    "import numpy as np \n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "79359fdf",
   "metadata": {},
   "outputs": [],
   "source": [
    "sns.set(context='talk', style='white', rc={'figure.facecolor':'white'}, font_scale=1)\n",
    "sns.set_style('ticks')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "id": "c8a3a89c",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.random.choice([1, 1, 1, 4, 5], 1000)\n",
    "y = np.random.choice([1, 2, 3, 4, 5], 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "id": "0fcfc1ec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.histplot(x)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "id": "14ed00e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.histplot(y)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "05fda74d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MannwhitneyuResult(statistic=497871.0, pvalue=0.8663900567778527)"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.mannwhitneyu(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "id": "e9c8eab3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ttest_indResult(statistic=-0.17637137892649216, pvalue=0.860020062981373)"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.ttest_ind(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "id": "481400ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Power_divergenceResult(statistic=1275.6666666666665, pvalue=5.636848140435057e-09)"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.chisquare(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "id": "76370bda",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Power_divergenceResult(statistic=666.2821685173092, pvalue=1.0)"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.chisquare(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd2acec6",
   "metadata": {},
   "source": [
    "# contingency table analysis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "56aa0a5e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10,  1],\n",
       "       [ 5, 20]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d1cfb71c",
   "metadata": {},
   "outputs": [],
   "source": []
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