simulation/z_score.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.stats as st\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample mean: -0.00; sample var: 1.00\n"
     ]
    },
    {
     "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": [
    "sample = np.random.normal(0, 1, 100000)\n",
    "\n",
    "plt.hist(sample, density=True, bins=100)\n",
    "print(\"sample mean: %.2f; sample var: %.2f\"%(np.mean(sample), np.var(sample)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Look-up z-score by p-value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def z_score(sample, p):\n",
    "    z = np.percentile(sample, p)\n",
    "    return z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "z-score at 1.0%: -2.33\n",
      "z-score at 2.5%: -1.96\n",
      "z-score at 5.0%: -1.65\n",
      "z-score at 95.0%: 1.64\n",
      "z-score at 97.5%: 1.95\n",
      "z-score at 99.0%: 2.33\n"
     ]
    }
   ],
   "source": [
    "p_vals = [1, 2.5, 5, 95, 97.5, 99]\n",
    "for p in p_vals:\n",
    "    z = z_score(sample, p)\n",
    "    print(\"z-score at %.1f%%: %.2f\"%(p, z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Look-up p-valie by z-score"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lookup_p(sample, z):\n",
    "    p = sum(sample <= z) / len(sample)\n",
    "    return p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p(z <= -2.32) = 1.02% (scipy: 1.02%)\n",
      "p(z <= -1.96) = 2.53% (scipy: 2.50%)\n",
      "p(z <= -1.64) = 5.11% (scipy: 5.05%)\n",
      "p(z <= 1.64) = 95.02% (scipy: 94.95%)\n",
      "p(z <= 1.96) = 97.55% (scipy: 97.50%)\n",
      "p(z <= 2.32) = 98.97% (scipy: 98.98%)\n"
     ]
    }
   ],
   "source": [
    "z_scores = [-2.32, -1.96, -1.64, 1.64, 1.96, 2.32]\n",
    "for z in z_scores:\n",
    "    p = lookup_p(sample, z)\n",
    "    \n",
    "    print(\"p(z <= %.2f) = %.2f%% (scipy: %.2f%%)\"\n",
    "          %(z, 100 * p, 100 * st.norm.cdf(z)))"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
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   "pygments_lexer": "ipython3",
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