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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Compute the variance and standard deviation from their definitions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statistics\n",
    "import math\n",
    "\n",
    "def stats_of_list(values):\n",
    "    \"\"\"\n",
    "    Compute and return a tuple of the mean, population variance, \n",
    "    and population standarard deviation of a list of values.\n",
    "    @param values the list of values.\n",
    "    @return a tuple of the mean, variance, and standard deviation.\n",
    "    \"\"\"\n",
    "    \n",
    "    sum_of_squares = 0\n",
    "    mean = statistics.mean(values)\n",
    "    \n",
    "    for x in values:\n",
    "        diff = x - mean\n",
    "        sum_of_squares += diff*diff\n",
    "       \n",
    "    variance = sum_of_squares/len(values)\n",
    "    stdev = math.sqrt(variance)\n",
    "    \n",
    "    return (mean, variance, stdev)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test with a sequence of normally distributed random values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "\n",
    "mu    = 50\n",
    "sigma = mu/2\n",
    "count = 1_000_000\n",
    "\n",
    "x_list = [random.gauss(mu, sigma) for _ in range(0, count)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_mean, x_variance, x_stdev = stats_of_list(x_list)\n",
    "\n",
    "print(f'              mean = {x_mean:.2f}')\n",
    "print(f'          variance = {x_variance:.2f}')\n",
    "print(f'standard deviation = {x_stdev:.2f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## statistics.pvariance and statistics.pstdev functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(f'statistics.pvariance(x_list) = {statistics.pvariance(x_list):.2f}')\n",
    "print(f'statistics.pstdev(x_list)    = {statistics.pstdev(x_list):.2f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test the standard deviation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_stdev(values, mean, stdev):\n",
    "    \"\"\"\n",
    "    Test the standard deviation by computing the percentage\n",
    "    of values within one standard deviation of the mean.\n",
    "    @param values the list of values.\n",
    "    @param mean the mean of the values.\n",
    "    @param stdev the standard deviation of the values.\n",
    "    @return the percentage of values.\n",
    "    \"\"\"\n",
    "    \n",
    "    total_count = len(values)\n",
    "    range_count = 0;\n",
    "    \n",
    "    lo_range = mean - stdev\n",
    "    hi_range = mean + stdev\n",
    "    \n",
    "    for x in values:\n",
    "        if (x > lo_range) and (x < hi_range):\n",
    "            range_count += 1\n",
    "            \n",
    "    return 100*range_count/total_count"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pct = test_stdev(x_list, x_mean, x_stdev)\n",
    "print(f'Percentage of values within one standard deviation of the mean: {pct:.2f}%')"
   ]
  }
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