{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": [
     "remove-input"
    ]
   },
   "outputs": [],
   "source": [
    "from datascience import *\n",
    "%matplotlib inline\n",
    "path_data = 'https://raw.githubusercontent.com/ChemeketaCS/datasci-textbook/main/assets/data/'\n",
    "import matplotlib.pyplot as plots\n",
    "plots.style.use('fivethirtyeight')\n",
    "import math\n",
    "import numpy as np\n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": [
     "remove-input"
    ]
   },
   "outputs": [],
   "source": [
    "colors = Table.read_table(path_data + 'roulette_wheel.csv').column('Color')\n",
    "pockets = make_array('0','00')\n",
    "for i in np.arange(1, 37):\n",
    "    pockets = np.append(pockets, str(i)) \n",
    "\n",
    "wheel = Table().with_columns(\n",
    "    'Pocket', pockets,\n",
    "    'Color', colors\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Central Limit Theorem\n",
    "Very few of the data histograms that we have seen in this course have been bell shaped. When we have come across a bell shaped distribution, it has almost invariably been an empirical histogram of a statistic based on a random sample.\n",
    "\n",
    "The examples below show two very different situations in which an approximate bell shape appears in such histograms."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Net Gain in Roulette\n",
    "In an earlier section, the bell appeared as the rough shape of the total amount of money we would make if we placed the same bet repeatedly on different spins of a roulette wheel. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table border=\"1\" class=\"dataframe\">\n",
       "    <thead>\n",
       "        <tr>\n",
       "            <th>Pocket</th> <th>Color</th>\n",
       "        </tr>\n",
       "    </thead>\n",
       "    <tbody>\n",
       "        <tr>\n",
       "            <td>0     </td> <td>green</td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>00    </td> <td>green</td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>1     </td> <td>red  </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>2     </td> <td>black</td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>3     </td> <td>red  </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>4     </td> <td>black</td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>5     </td> <td>red  </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>6     </td> <td>black</td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>7     </td> <td>red  </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>8     </td> <td>black</td>\n",
       "        </tr>\n",
       "    </tbody>\n",
       "</table>\n",
       "<p>... (28 rows omitted)</p>"
      ],
      "text/plain": [
       "Pocket | Color\n",
       "0      | green\n",
       "00     | green\n",
       "1      | red\n",
       "2      | black\n",
       "3      | red\n",
       "4      | black\n",
       "5      | red\n",
       "6      | black\n",
       "7      | red\n",
       "8      | black\n",
       "... (28 rows omitted)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wheel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that the bet on red pays even money, 1 to 1. We defined the function `red_winnings` that returns the net winnings on one \\$1 bet on red. Specifically, the function takes a color as its argument and returns 1 if the color is red. For all other colors it returns -1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def red_winnings(color):\n",
    "    if color == 'red':\n",
    "        return 1\n",
    "    else:\n",
    "        return -1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The table `red` shows each pocket's winnings on red."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table border=\"1\" class=\"dataframe\">\n",
       "    <thead>\n",
       "        <tr>\n",
       "            <th>Pocket</th> <th>Color</th> <th>Winnings: Red</th>\n",
       "        </tr>\n",
       "    </thead>\n",
       "    <tbody>\n",
       "        <tr>\n",
       "            <td>0     </td> <td>green</td> <td>-1           </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>00    </td> <td>green</td> <td>-1           </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>1     </td> <td>red  </td> <td>1            </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>2     </td> <td>black</td> <td>-1           </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>3     </td> <td>red  </td> <td>1            </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>4     </td> <td>black</td> <td>-1           </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>5     </td> <td>red  </td> <td>1            </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>6     </td> <td>black</td> <td>-1           </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>7     </td> <td>red  </td> <td>1            </td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>8     </td> <td>black</td> <td>-1           </td>\n",
       "        </tr>\n",
       "    </tbody>\n",
       "</table>\n",
       "<p>... (28 rows omitted)</p>"
      ],
      "text/plain": [
       "Pocket | Color | Winnings: Red\n",
       "0      | green | -1\n",
       "00     | green | -1\n",
       "1      | red   | 1\n",
       "2      | black | -1\n",
       "3      | red   | 1\n",
       "4      | black | -1\n",
       "5      | red   | 1\n",
       "6      | black | -1\n",
       "7      | red   | 1\n",
       "8      | black | -1\n",
       "... (28 rows omitted)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "red = wheel.with_column(\n",
    "    'Winnings: Red', wheel.apply(red_winnings, 'Color')\n",
    "    )\n",
    "red"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Your net gain on one bet is one random draw from the `Winnings: Red` column. There is an 18/38 chance making \\$1, and a 20/38 chance of making -\\$1. This probability distribution is shown in the histogram below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "red.select('Winnings: Red').hist(bins=np.arange(-1.5, 1.6, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now suppose you bet many times on red. Your net winnings will be the sum of many draws made at random with replacement from the distribution above.\n",
    "\n",
    "It will take a bit of math to list all the possible values of your net winnings along with all of their chances. We won't do that; instead, we will approximate the probability distribution by simulation, as we have done all along in this course. \n",
    "\n",
    "The code below simulates your net gain if you bet \\$1 on red on 400 different spins of the roulette wheel. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_bets = 400\n",
    "repetitions = 10000\n",
    "\n",
    "net_gain_red = make_array()\n",
    "\n",
    "for i in np.arange(repetitions):\n",
    "    spins = red.sample(num_bets)\n",
    "    new_net_gain_red = spins.column('Winnings: Red').sum()\n",
    "    net_gain_red = np.append(net_gain_red, new_net_gain_red)\n",
    "\n",
    "\n",
    "results = Table().with_column(\n",
    "    'Net Gain on Red', net_gain_red\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "results.hist(bins=np.arange(-80, 50, 6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a roughly bell shaped histogram, even though the distribution we are drawing from is nowhere near bell shaped.\n",
    "\n",
    "**Center.** The distribution is centered near -20 dollars, roughly. To see why, note that your winnings will be \\$1 on about 18/38 of the bets, and -$1 on the remaining 20/38. So your average winnings per dollar bet will be roughly -5.26 cents:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.05263157894736842"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "average_per_bet = 1*(18/38) + (-1)*(20/38)\n",
    "average_per_bet"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So in 400 bets you expect that your net gain will be about -\\$21:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-21.052631578947366"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "400 * average_per_bet"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For confirmation, we can compute the mean of the 10,000 simulated net gains:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-20.9586"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(results.column(0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Spread.** Run your eye along the curve starting at the center and notice that the point of inflection is near 0. On a bell shaped curve, the SD is the distance from the center to a point of inflection. The center is roughly -\\$20, which means that the SD of the distribution is around \\$20.\n",
    "\n",
    "In the next section we will see where the \\$20 comes from. For now, let's confirm our observation by simply calculating the SD of the 10,000 simulated net gains:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20.029115957525438"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(results.column(0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Summary.** The net gain in 400 bets is the sum of the 400 amounts won on each individual bet. The probability distribution of that sum is approximately normal, with an average and an SD that we can approximate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Average Flight Delay\n",
    "The table `united` contains data on departure delays of 13,825 United Airlines domestic flights out of San Francisco airport in the summer of 2015. As we have seen before, the distribution of delays has a long right-hand tail."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "united = Table.read_table(path_data + 'united_summer2015.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "united.select('Delay').hist(bins=np.arange(-20, 300, 10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean delay was about 16.6 minutes and the SD was about 39.5 minutes. Notice how large the SD is, compared to the mean. Those large deviations on the right have an effect, even though they are a very small proportion of the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(16.658155515370705, 39.480199851609314)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean_delay = np.mean(united.column('Delay'))\n",
    "sd_delay = np.std(united.column('Delay'))\n",
    "\n",
    "mean_delay, sd_delay"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now suppose we sampled 400 delays at random with replacement. You could sample without replacement if you like, but the results would be very similar to with-replacement sampling. If you sample a few hundred out of 13,825 without replacement, you hardly change the population each time you pull out a value.\n",
    "\n",
    "In the sample, what could the average delay be? We expect it to be around 16 or 17, because that's the population average; but it is likely to be somewhat off. Let's see what we get by sampling. We'll work with the table `delay` that only contains the column of delays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "delay = united.select('Delay')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "15.59"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(delay.sample(400).column('Delay'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sample average varies according to how the sample comes out, so we will simulate the sampling process repeatedly and draw the empirical histogram of the sample average. That will be an approximation to the probability histogram of the sample average."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "sample_size = 400\n",
    "repetitions = 10000\n",
    "\n",
    "means = make_array()\n",
    "\n",
    "for i in np.arange(repetitions):\n",
    "    sample = delay.sample(sample_size)\n",
    "    new_mean = np.mean(sample.column('Delay'))\n",
    "    means = np.append(means, new_mean)\n",
    "\n",
    "results = Table().with_column(\n",
    "    'Sample Mean', means\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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yZAk2bNhQZlx2djbOnz+Pfv36GT3gs8WpUql0lmlKSspzr9MY72EuUs1eCmsknz5nwBKCziMDTyt5XCJ6bEXGAzDZ+0s5OwDk5eeZ9HdSqr/vALOLpVAodM7XWWL5+fnqPSIAyMnJQWlpqcYYQRBQq1YtjBs3DhEREc8RVZOTkxMA4O7du+r7NQJAZmZmmb2zp+n7gfVJSUl57vcwFylnTz59Dsu+3iV6/KwJQ2Fvby9qrLWVteixFRkPwGTvL+XsAOBg7wCFwt2gPGJJ+fed2Y1HZ4lNmjQJkyZNAgC0adMGixYt0npih7G5u7vDyckJR44cQfv27QEAhYWFOHnyJKKioiolAxERWTbRJ3b8+eefRl95Xl4erly5AgAoLS3FjRs38Oeff8LR0RGurq6YOnUqli5dCoVCAU9PTyxZsgT29vYYNmyY0bMQEZH0iC6xJx48eIAbN24gKysLKpWqzHxfX1/R73X69GkEBASoXyuVSiiVSgQHB2P16tWYOXMmHj58iLCwMGRnZ8PHxwfx8fFGPYGEiIikS3SJZWVlITw8HDt37sTjx4/LzH9ywsX9+/dFr7xbt27Izs7WOl8QBERGRiIyMlL0exIRUfUhusTeeecd7NmzB5MmTYKvry9kMpkpcxEREeklusR+/PFHTJkyBR9//LEp8xAREYkm+mJnGxsbeHh4mDILERGRQUSX2ODBg3Ho0CFTZiEiIjKI6BJ76623cOfOHbz55pv49ddfcefOHWRkZJT5Q0REVFlEfyfm4+MDQRBw5swZxMXFaR1nyNmJREREz0N0ic2dO1fnPQuJiIgqm+gS47VaRGQoQ54/xmePUUUYfMcOAHj8+DFycnJQp04dWFtX6C2IqBow5PljfPYYVYRBzxM7deoUhgwZgsaNG8PT0xOJiYkAgHv37mHEiBH45ZdfTBKSiIioPKJL7P/+7/8wYMAAXL16FaNGjdK4b2L9+vWRl5eH//73vyYJSUREVB7RJfbhhx/Cw8MDycnJmD9/fpn53bp1w2+//WbUcERERLqILrFTp05h7NixsLOzK/csxSZNmmg8QJOIiMjURJdYjRo1UKOG9uHp6emoWbOmUUIRERGJIbrE2rVrh/3795c7r7i4GNu2bUOnTp2MFoyIiEgf0SU2e/ZsHD16FDNmzMDZs2cBAHfu3MGPP/6IQYMG4erVq5gzZ47JghIRET1L9EVevXr1wtq1axEWFoZNmzYBAKZOnQqVSoW6deti3bp16Nixo8mCEhERPcugK5WHDRuGAQMG4MiRI7h8+TJKS0vRrFkz9OnTBw4ODqbKSEREVC6Db7dRq1Yt+Pv7myILERGRQUR/J7Zv3z6EhYVpnR8WFqb1xA8iIiJTEF1iK1euREFBgdb5hYWFWL58uVFCERERiSG6xM6fP4927dppnd+2bVtcuHDBKKGIiIjEEF1iJSUlePjwodb5Dx8+RFFRkVFCERERiSG6xFq2bImEhASUlpaWmVdaWoqEhAS0aNHCqOGIiIh0EX124ptvvomQkBAEBwcjMjISXl5eAIC///4bixYtwu+//47Vq1ebLChJS3pGFjKzcg1Ygk8NJyLDiS6xoKAgXL16FUqlEocOHQIACIIAlUoFQRAQHh6OkSNHmiwoSUtmVi6iVsSKHj9jXIAJ0xBRVWXQdWKhoaEYNmwYdu/ejdTUVKhUKjRr1gwBAQFo2rSpiSISERGVT1SJFRUVIT4+Hi+99BJ8fHzw1ltvmToXERGRXqJO7LC1tcXMmTPVN/4lIiKyBKLPTlQoFHzoJRERWRTRJTZ37lx89dVXOHfunCnzEBERiSb6xI6jR49CLpeje/fu6NSpE5o1a1bmSc6CIGDJkiVGD0lERFQe0SW2fv169d+TkpKQlJRUZgxLjIiIKpPoEsvKyjJlDiIiIoOJ/k6MiIjI0hj8UMykpCQcPXoUGRkZmDJlCjw9PZGfn48LFy5AoVCgTp06pshJRERUhugSKy4uxoQJE7Bv3z71raYGDhwIT09PWFlZYdiwYZg+fTpCQ0NNmZeIiEhNdIkplUocOHAAixcvRo8ePdCxY0f1PDs7OwwZMgQ//PADS4yIKkQQgHMX00SPtxYMPpBEVZDo34Jt27bhjTfewMSJE3H//v0y8xUKBRISEowajoiqj5wHBfh8/U7R42dPHGzCNCQVok/syMjIgLe3t9b5tra2yM/PN0ooIiIiMUSXmJOTE1JTU7XO//333+Hu7m6MTERERKKILrFBgwbhm2++waVLl9TTBOHfBxn+8MMP2LZtGwIDA42fkIiISAvRJRYeHg5XV1f06NEDISEhEAQBy5YtQ9++fTFmzBi0a9cOM2fONGVWIiIiDaJLrHbt2jh48CBmz56NjIwM2NnZISkpCfn5+YiMjMTu3bthZ2dnyqxEREQaDDpH1c7ODnPmzMGcOXNMlUeDUqlEdHS0xrSGDRvi4sWLlbJ+IiKybHpLrKioCPv27UNqairq1auHfv36oVGjRpWRDcC/p+7v2bNH/drKyqrS1k1ERJZNZ4mlp6djwIABuHr1KlQqFQCgVq1aiIuLg6+vb+UEtLaGk5NTpayLiIikRed3Yh999BFSU1Mxbdo0bN26FUqlEnZ2dpg7d25l5UNqaiq8vLzQpk0bTJgwQedp/kREVL3o3BM7fPgwgoOD8dFHH6mnNWzYECEhIbh58yaaNGli0nAdOnTAqlWroFAokJmZicWLF8PPzw9JSUmoV69eucukpKQ893qN8R7mYinZ8/KLDb743ZDxJY9LRI83ZGxFxgPMbozxFcluKb/vFcHs4igUCp3z9R5O7Ny5s8a0Ll26QKVS4caNGyYvsVdffVXjdYcOHdCuXTts2rQJM2bMKHcZfT+wPikpKc/9HuZiSdnPXUyDvb29QcsYMt7aylr0eEPGVmQ8wOzGGF+R7Jby+24oS9pWDWVp2XUeTnz8+HGZ0+afvC4sLDRdKi0cHBzQokULXLlypdLXTURElkfv2Ympqan4/fff1a9zc3MB/NvGDg4OZcb7+PgYMZ6mwsJCpKSkoFu3biZbBxERSYfeElMqlVAqlWWmP3tyx5NnjJV3h/uKev/99/Haa6/BxcVF/Z1YQUEBgoODjbYOIiKSLp0lFhMTU1k5ynXr1i2EhITg3r17aNCgATp06IBDhw7Bzc3NrLmIiMgy6Cyx0aNHV1aOcq1fv96s66f/Sc/IQmZWrujxRcXFJkxDRPQvPhqVRMnMykXUiljR42dNGGrCNERE/xJ9A2AiIiJLwxIjIiLJYokREZFkscSIiEiyWGJERCRZLDEiIpIslhgREUkWS4yIiCSLJUZERJLFEiMiIsnibaeISJLsbG1x7mKa6PENHOvASe5owkRkDiwxIpKk3PyH+GJDnOjx898ewxKrgng4kYiIJIslRkREksUSIyIiyWKJERGRZLHEiIhIsnh2YjWVnpGFzKxc0eOLiotNmIaIqGJYYtVUZlYuolbEih4/a8JQE6YhIqoYlhgRVQuCAF4cXQWxxIioWsh5UIDP1+8UPZ4XR0sDT+wgIiLJYokREZFkscSIiEiyWGJERCRZLDEiIpIslhgREUkWS4yIiCSL14kREZWDF0dLA0uMiKgcvDhaGng4kYiIJIslRkREksUSIyIiyeJ3YkRERmDIiSDWAv/pNRZ+kkRERmDIiSCzJw42cZrqgyVWhZTCWvT/BPmkZiKqClhiVUh2XgGWfb1L1Fg+qZmIqgKe2EFERJLFEiMiIsni4UQLlp6RhcysXAOWEEyWhYjIErHELFhmVi6iVsSKHj9jXIAJ0xARWR6WGBFRJbOztTXo5sK17GxRUFgkenx1uhmxJEps3bp1WLFiBdLT09GiRQsolUp07drV3LGIiCokN/8hvtgQJ3r8rAlDeTNiLSz+xI74+HhERERgzpw5OHr0KDp16oThw4fj+vXr5o5GRERmZvF7YjExMRg9ejTGjRsHAFi8eDF++uknrF+/HgsWLDBzOsMYeqIGL0gmItJNyM7OVpk7hDbFxcVwdnbG119/jSFDhqinh4aG4vz589i3b58Z0xERkblZ9OHEe/fu4fHjx5DL5RrT5XI57t69a6ZURERkKSy6xJ4QBM3rn1QqVZlpRERU/Vh0idWvXx9WVlZl9royMzPL7J0REVH1Y9ElZmNjg3bt2uHIkSMa048cOYLOnTubKRUREVkKiz87cfr06ZgyZQp8fHzQuXNnrF+/Hnfu3MH48ePNHY2IiMzMovfEACAwMBBKpRKLFy9Gt27dkJSUhLi4OLi5uVXo/RITEzFq1Ch4eXlBJpMhNlbztk4qlQpKpRItWrRAo0aN4O/vj7///tsYP8pz05X90aNHWLBgAbp27YrGjRujefPmCAkJsZjr6fR97k+bOXMmZDIZVq5cWYkJdROT/9KlSxg7dizc3Nzg7OyM7t27459//jFDWk36sufl5SEsLAwtW7ZEo0aN0KFDB8TExJgp7f8sW7YMvXr1gqurKzw8PDBy5EicP39eY4ylbq/6slv69irms3+aObdZiy8xAAgJCcHZs2dx9+5d/PLLL/D19a3we+Xn56Nly5ZYtGgRatasWWb+8uXLERMTg+joaBw+fBhyuRxDhw7FgwcPnudHMApd2QsKCvDHH38gNDQUv/zyCzZt2oSbN29i2LBhKCkpMVPi/9H3uT+xa9cunDp1Cs7OzpWYTj99+VNTU9GvXz+4u7sjISEBJ0+exPvvvw97e3szpNWkL/t7772HgwcPYs2aNUhOTsacOXOwcOFCbNmyxQxp/+f48eOYOHEiDhw4gISEBFhbW2PIkCHIyspSj7HU7VVfdkvfXsV89k+Ye5u16OvETK1Jkyb49NNPMWbMGAD//q+uRYsWmDRpEkJDQwEADx8+hEKhwIcffmhRhzCfzV6eCxcuoEuXLkhMTESrVq0qMZ1u2rJfu3YN/fr1w/fff49hw4Zh8uTJeOutt8yUUrvy8oeEhEAQBHz11VdmTKZfedlfeeUVBAQE4N1331VPGzBgAFq1aoXFixebI2a58vLy4ObmhtjYWPTv319S2+uz2ctjqdsroD2/JWyzktgTqyxpaWlIT09H79691dNq1qyJrl27Ijk52YzJKubJ/0ZlMpmZk+hXUlKCkJAQhIaGonnz5uaOY5DS0lLs378fzZs3R1BQEDw8PNCrVy/Ex8ebO5ooXbp0wf79+3Hjxg0AQHJyMv766y/06dPHzMk05eXlobS0VP37LKXt9dns5bHk7bW8/JayzbLEnpKeng4AVeLi6uLiYrz//vt47bXX0KRJE3PH0UupVMLR0RETJ040dxSDZWRkIC8vT/09ws6dOxEUFIRJkyZh//795o6nV3R0NLy9vdG6dWs0aNAA/v7++OCDD/Daa6+ZO5qGiIgIeHt7o1OnTgCktb0+m/1Zlr69lpffUrZZiz870RykfnF1SUkJJk+ejJycHGzevNnccfQ6fvw4Nm3ahGPHjpk7SoWUlpYC+PcQ3IwZMwAAbdq0wZkzZ7Bu3TqLK4NnrV27FsnJydi8eTNcXV1x4sQJzJs3D25ubujbt6+54wEA3n33XSQlJWH//v2wsrLSmGfp26uu7IDlb6/l5bekbZZ7Yk9xcnICAElfXF1SUoKJEyfi3Llz2LVrF+rVq2fuSHodO3YMd+7cQfPmzVG/fn3Ur18f169fx4IFC9CyZUtzx9Orfv36sLa2LnNI5aWXXlIforNUDx8+RFRUFBYuXIj+/fujdevWmDx5MgIDAy3m7NDIyEjs2LEDCQkJaNq0qXq6FLZXbdmfsPTtVVt+S9pmWWJPcXd3h5OTk8bF1YWFhTh58qQkLq5+9OgRxo8fj3PnzmH37t3qjdzShYSEIDExEceOHVP/cXZ2xrRp07Br1y5zx9PLxsYG7du3R0pKisb0S5cuwdXV1UypxHn06BEePXpUZg/ByspKvYdpTuHh4di+fTsSEhLw0ksvacyz9O1VV3bA8rdXXfktaZutdocT8/LycOXKFQD/Hga6ceMG/vzzTzg6OsLV1RVTp07F0qVLoVAo4OnpiSVLlsDe3h7Dhg0zc3Ld2Z2dnTFu3DicPn0amzdvhiAI6u8M6tSpo/O09sqg73N/9n/O1tbWcGD5+tEAAAiDSURBVHJygkKhMEfcMvTlf/vttzF+/Hh07doV3bt3x7FjxxAfH6/zerjKoi+7r68vFi5cCHt7e7i6uiIxMRFbtmzBwoULzZo7NDQUW7duxXfffQeZTKb+fba3t4eDgwMEQbDY7VVf9pKSEoveXvXll8vlFrPNVrtT7I8dO4aAgIAy04ODg7F69WqoVCosWrQI3377LbKzs+Hj44MlS5ZYxGEtXdkjIiLQtm3bcpeLiYnReSp+ZdD3uT/L29vbok6xF5M/NjYWy5Ytw82bN/Hiiy9i9uzZZv/HFNCfPT09HQsXLsSRI0eQlZUFV1dXvP7665gxY4ZZv1vSdpZeeHg4IiMjAcBit1d92dPS0ix6exXz2T/LXNtstSsxIiKqOvidGBERSRZLjIiIJIslRkREksUSIyIiyWKJERGRZLHEiIhIslhiVC3s2bMH/fv3h6enJxo1aoTWrVtj9OjR+PHHH80dTaupU6fC29vbaO8nk8kgk8kQFRVVZp5KpULbtm0hk8kwefJko62TyNRYYlTlrVmzBmPHjoWHhwdWrlyJrVu3IiwsDABw9OhRM6erXLVr10ZcXBxUKs3LQ0+cOIFr165ZxEM8iQxR7W47RdXPF198AX9/f3zxxRfqaT169MC4ceMs4v6Alcnf3x9xcXE4fvw4unXrpp6+ZcsW+Pr6Ii0tzYzpiAzHPTGq8rKysrTeXLVGjf9tApmZmZg1axZ8fHzg7OyMVq1aISQkBLdu3dJYRqlUQiaT4eLFiwgMDETjxo3RunVrfPfddwD+LYSOHTuiSZMmGDhwIK5evaqx/JPb82zYsAEvv/wynJyc0L17d1F7hQUFBViwYAHatGkDuVyONm3aYMmSJaLL2MXFBb6+vti6dat6WmFhIXbt2oVRo0aVu8y9e/cwe/ZseHl5oWHDhujYsSO+/fZbjTGGfnaXL1/GiBEj0KRJE7Ru3RrR0dHV7j8UZBwsMary2rdvj82bN2PFihW4dOmS1nFZWVmwtbXF/PnzsX37dkRFReHy5cvo168fCgsLy4x/44034Ofnh9jYWLRt2xYzZsxAVFQU1q9fjwULFiAmJgaXLl1CSEhImWUTExMRExODefPm4euvv4atrS2GDx9e5k74TyspKUFQUBA2btyIN998E9u3b8frr7+OxYsXY968eaI/j1GjRiEhIQEPHz4EAOzduxclJSUYPHhwmbG5ubno168fDh48iIiICMTFxeG1117D7NmzsXbt2gp/dmPHjkW3bt0QGxsLf39/KJVKbNq0SfTPQPQEDydSlffZZ5/h9ddfx/z58zF//nzUq1cPvXr1wpgxYzQeba9QKBAdHa1+/fjxY3Tu3BmtW7fGoUOHytxE96233kJwcDAA4OWXX8b+/fvxzTff4I8//kCdOnUAAHfu3EFERASuXbsGNzc39bJ3797FqVOn1I9q6dGjB7y9vbF48WJ8+eWX5f4c27dvx8mTJ7F37174+vqqlwP+fTrzrFmzRD1Ha/DgwQgLC8O+ffsQFBSELVu2wN/fH7Vr1y4zds2aNbh+/TpOnDgBDw8PAEDPnj2Rk5OD6OhoTJw4EdbW1gZ/dtOnT8fYsWPV73fs2DHs2LFDPY1ILO6JUZXn6emJY8eOYe/evQgNDYW3tzf27NmDwMBALF68WGPs119/DV9fXzRp0gT169dH69atAaDcPbhXX31V/XeZTAa5XI6OHTuqCwyA+jlMN2/e1Fi2Q4cOGs8aq127Nvz8/PDrr79q/Tl++uknuLq6onPnzigpKVH/6d27Nx49eqRz2ac5ODhg4MCB2Lp1K9LT03H48GGthxJ/+ukn+Pj4wN3dXWOdffr0wf3793HhwgX1WEM+u379+mm89vLysvgHiJJl4p4YVQtWVlbw9fVV78Hcvn0bQUFBiI6OxqRJkyCTybB27VqEh4dj+vTp6NOnD2QyGUpLS9G3b99yD4k9+7iKF154odxpAFBUVKQxvWHDhmXer2HDhrh9+7bWnyEjIwPXr19HgwYNyp1///59rcs+a9SoURgxYgRWrVoFuVyOnj17al3nlStX9K7T0M/O0dFR47WNjU2544j0YYlRteTs7IzXX38dERERuHz5Mnx8fBAfH48ePXrg448/Vo9LTU01yfrv3r1b7jRnZ2ety9SrVw/u7u5lTqp44unDlfr07NkTcrkcK1euxPTp08s82fnpdcrlcixatKjc+Z6engBQqZ8d0dNYYlTl3bhxAy4uLmWmPzmJ4sleUUFBQZnvhUz1ZObffvtNI9eDBw9w8OBB+Pn5aV2mT58+SEhIgL29fbmPuzdEjRo1EBYWhh9//FHn91B9+vTBl19+CRcXF53ft1XmZ0f0NJYYVXldu3bFf/7zHwwcOBDu7u548OABDh06hPXr12Po0KHq76b69u2Lzz//HEuXLoWPjw+OHj2KXbt2mSRTw4YNERgYiIiICNjY2GD58uUoKCjA3LlztS4zYsQIxMbGYvDgwZg+fTq8vb1RXFyMq1ev4ocffkBsbCxq1aolOsOECRMwYcIEnWOmTZuGnTt3on///pg2bRo8PT1RUFCAlJQUnDhxAps3bwZQuZ8d0dNYYlTlffDBBzh48CA++eQTZGRkwMrKCh4eHvjggw8wdepU9bi5c+ciJycHq1atQlFREXx9fbFjxw60a9fO6Jl8fX3xn//8B1FRUbh16xaaN2+Obdu2qQ/PleeFF15AfHw8PvvsM2zYsAFpaWmoVasWmjVrBj8/P9jY2Bg9Z926dXHw4EFER0fj888/x+3bt1G3bl0oFAqNMw4r87MjepqQnZ2t0j+MiIzF29sbr7zyitZT6YlIPJ5iT0REksUSIyIiyeLhRCIikizuiRERkWSxxIiISLJYYkREJFksMSIikiyWGBERSRZLjIiIJOv/AZfS2Uo4kQn2AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "results.hist(bins=np.arange(10, 25, 0.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once again, we see a rough bell shape, even though we are drawing from a very skewed distribution. The bell is centered somewhere between 16 ad 17, as we expect."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Central Limit Theorem\n",
    "\n",
    "The reason why the bell shape appears in such settings is a remarkable result of probability theory called the **Central Limit Theorem**. \n",
    "\n",
    "**The Central Limit Theorem says that the probability distribution of the sum or average of a large random sample drawn with replacement will be roughly normal, *regardless of the distribution of the population from which the sample is drawn*.**\n",
    "\n",
    "As we noted when we were studying Chebychev's bounds, results that can be applied to random samples *regardless of the distribution of the population* are very powerful, because in data science we rarely know the distribution of the population.\n",
    "\n",
    "The Central Limit Theorem makes it possible to make inferences with very little knowledge about the population, provided we have a large random sample. That is why it is central to the field of statistical inference."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Proportion of Purple Flowers\n",
    "Recall Mendel's probability model for the colors of the flowers of a species of pea plant. The model says that the flower colors of the plants are like draws made at random with replacement from {Purple, Purple, Purple, White}.\n",
    "\n",
    "In a large sample of plants, about what proportion will have purple flowers? We would expect the answer to be about 0.75, the proportion purple in the model. And, because proportions are means, the Central Limit Theorem says that the distribution of the sample proportion of purple plants is roughly normal.\n",
    "\n",
    "We can confirm this by simulation. Let's simulate the proportion of purple-flowered plants in a sample of 200 plants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table border=\"1\" class=\"dataframe\">\n",
       "    <thead>\n",
       "        <tr>\n",
       "            <th>Color</th>\n",
       "        </tr>\n",
       "    </thead>\n",
       "    <tbody>\n",
       "        <tr>\n",
       "            <td>Purple</td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>Purple</td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>Purple</td>\n",
       "        </tr>\n",
       "        <tr>\n",
       "            <td>White </td>\n",
       "        </tr>\n",
       "    </tbody>\n",
       "</table>"
      ],
      "text/plain": [
       "Color\n",
       "Purple\n",
       "Purple\n",
       "Purple\n",
       "White"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "colors = make_array('Purple', 'Purple', 'Purple', 'White')\n",
    "\n",
    "model = Table().with_column('Color', colors)\n",
    "\n",
    "model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "props = make_array()\n",
    "\n",
    "num_plants = 200\n",
    "repetitions = 10000\n",
    "\n",
    "for i in np.arange(repetitions):\n",
    "    sample = model.sample(num_plants)\n",
    "    new_prop = np.count_nonzero(sample.column('Color') == 'Purple')/num_plants\n",
    "    props = np.append(props, new_prop)\n",
    "    \n",
    "results = Table().with_column('Sample Proportion: 200', props)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "results.hist(bins=np.arange(0.65, 0.85, 0.01))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There's that normal curve again, as predicted by the Central Limit Theorem, centered at around 0.75 just as you would expect.\n",
    "\n",
    "How would this distribution change if we increased the sample size? Let's run the code again with a sample size of 800, and collect the results of simulations in the same table in which we collected simulations based on a sample size of 200. We will keep the number of `repetitions` the same as before so that the two columns have the same length."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "props2 = make_array()\n",
    "\n",
    "num_plants = 800\n",
    "\n",
    "for i in np.arange(repetitions):\n",
    "    sample = model.sample(num_plants)\n",
    "    new_prop = np.count_nonzero(sample.column('Color') == 'Purple')/num_plants\n",
    "    props2 = np.append(props2, new_prop)\n",
    "    \n",
    "results = results.with_column('Sample Proportion: 800', props2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "results.hist(bins=np.arange(0.65, 0.85, 0.01))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both distributions are approximately normal but one is narrower than the other. The proportions based on a sample size of 800 are more tightly clustered around 0.75 than those from a sample size of 200. Increasing the sample size has decreased the variability in the sample proportion.\n",
    "\n",
    "This should not be surprising.  We have leaned many times on the intuition that a larger sample size generally reduces the variability of a statistic.  However, in the case of a sample average, we can *quantify* the relationship between sample size and variability.\n",
    "\n",
    "Exactly how does the sample size affect the variability of a sample average or proportion? That is the question we will examine in the next section."
   ]
  }
 ],
 "metadata": {
  "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": "3.7.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}