Distribution Drill.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "chi_square = np.random.chisquare(100, 100)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(chi_square, bins=20, color='c')\n",
    "\n",
    "# Add a vertical line at the mean.\n",
    "plt.axvline(chi_square.mean(), color='b', linestyle='solid', linewidth=2)\n",
    "\n",
    "# Add a vertical line at one standard deviation above the mean.\n",
    "plt.axvline(chi_square.mean() + chi_square.std(), color='b', linestyle='dashed', linewidth=2)\n",
    "\n",
    "# Add a vertical line at one standard deviation below the mean.\n",
    "plt.axvline(chi_square.mean()-chi_square.std(), color='b', linestyle='dashed', linewidth=2) \n",
    "\n",
    "# Print the histogram.\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Chi Square Distribution\n",
    "\n",
    "In probability theory and statistics, the chi-squared distribution (also chi-square or χ2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables.\n",
    "\n",
    "The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square. Two of the more common tests using the Chi Square distribution are tests of deviations of differences between theoretically expected and observed frequencies (one-way tables) and the relationship between categorical variables (contingency tables).\n",
    "\n",
    "Since there are k independent identically distributed gaussian random variables the resulting distribution is gaussian in the limit as k -> inf. \n",
    "\n",
    "As a result the mean and standard deviation show similar behavior to that of the normal distribution with the mean representing the center of gravity of the histogram distribution and the range of +/- one standard deviations about the mean containing ~68% of the samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.995463419363796 mean and 0.5036048671880883 standard deviation\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rand1 = np.random.normal(5, .5, 1000)\n",
    "plt.hist(rand1)\n",
    "plt.title = print(\"{} mean and {} standard deviation\".format(rand1.mean(),rand1.std()))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9.942854255732014 mean and 0.9829800471592617 standard deviation\n"
     ]
    },
    {
     "data": {
      "image/png": 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f1bGpW98E3NVjlrNW94FFfwlcXVXP9J1nXCddbHA18K2+soyjqr5RVauraqaqZhj8LrJLqup/Ru04tQ8GP+D8PvArfWdZQOa/ZvDFtAf4NPCCvjONmfs/GXwewSPA5X3nmSfn7Qx+HvKT7h/BDcDLGFyF83i3PLfvnGNkvrZb/zFwBPhS3znHyLyfwa823909/qHvnGPm/nz37/FR4F+ANX3nHJX5pNcPACtHvY930EpSA86GaRxJ0giWvSQ1wLKXpAZY9pLUAMtekhpg2UtSAyx7SWqAZS9JDfh/Lz5u7sg2U3cAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rand2 = np.random.normal(10, 1, 1000)\n",
    "plt.hist(rand2)\n",
    "plt.title = print(\"{} mean and {} standard deviation\".format(rand2.mean(),rand2.std()))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "ename": "SyntaxError",
     "evalue": "invalid syntax (<ipython-input-11-e9bbcc918cd3>, line 6)",
     "output_type": "error",
     "traceback": [
      "\u001b[1;36m  File \u001b[1;32m\"<ipython-input-11-e9bbcc918cd3>\"\u001b[1;36m, line \u001b[1;32m6\u001b[0m\n\u001b[1;33m    print(bin[0][i for i, x in enumerate(bin1) if x])\u001b[0m\n\u001b[1;37m                     ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n"
     ]
    }
   ],
   "source": [
    "rand3 = rand2+rand1\n",
    "bin = plt.hist(rand3)\n",
    "rand3_mean = rand3.mean()\n",
    "rand3_std = rand3.std()\n",
    "bin1 = bin[0]>rand3_mean+rand3_std\n",
    "print(bin[0][i for i, x in enumerate(bin1) if x])\n",
    "#print(bin[0])\n",
    "#print(bin1)\n",
    "#plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "14.93831767509581 mean and 1.1094141217823308 standard deviation\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(rand3, bins=20, color='c')\n",
    "\n",
    "# Add a vertical line at the mean.\n",
    "plt.axvline(rand3_mean, color='b', linestyle='solid', linewidth=2)\n",
    "\n",
    "# Add a vertical line at one standard deviation above the mean.\n",
    "plt.axvline(rand3_mean + rand3_std, color='b', linestyle='dashed', linewidth=2)\n",
    "\n",
    "# Add a vertical line at one standard deviation below the mean.\n",
    "plt.axvline(rand3_mean - rand3_std, color='b', linestyle='dashed', linewidth=2) \n",
    "\n",
    "# Title\n",
    "plt.title = print(\"{} mean and {} standard deviation\".format(rand3.mean(),rand3.std()))\n",
    "\n",
    "# Print the histogram.\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "?np.ndarray"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "celltoolbar": "Tags",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}