A test gist to see how to share python notebooks.

ipython notebook test file

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 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Sampling Distributions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A **sampling distribution** is a probability distribution for all possible values of a statistic computed from a sample of size *n*. The **sampling distribution of the sample mean** is the probability distribution of all possible values of the mean computed from a sample size of *n*. A sampling distributions mean is written as $\\bar{x}$. The population mean is written as $\\mu$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "In Class Activity"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load?"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import itertools as it\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "def get_sample_means(samples):\n",
      "    \"\"\"\n",
      "    Returns the mean of each sample.\n",
      "    \"\"\"\n",
      "    return [np.mean(sample) for sample in samples]\n",
      "\n",
      "def get_sampling_mean(samples):\n",
      "    \"\"\"\n",
      "    Returns the sampling distribution of the sample mean for the given samples.\n",
      "    \"\"\"\n",
      "    return np.mean(get_sample_means(samples))\n",
      "\n",
      "def get_samples(population, sample_size):\n",
      "    \"\"\"\n",
      "    Returns all possible samples from the population at sample size n.\n",
      "    \"\"\"\n",
      "    return list(it.combinations(population, sample_size))\n",
      "\n",
      "def draw_hist(sample_means, sample_size):\n",
      "    hist, bins = np.histogram(sample_means, 5)\n",
      "    hist = hist / float(len(sample_means))\n",
      "    width = 0.7*(bins[1]-bins[0])\n",
      "    center = (bins[:-1]+bins[1:])/2\n",
      "    plt.title(\"Sampling distribution at n=%(sample_size)d\" % {\"sample_size\": sample_size})\n",
      "    plt.bar(center, hist, align = 'center', width = width)\n",
      "    plt.show()\n",
      "\n",
      "six_students = [np.random.normal(50, 10) for i in range(6)]\n",
      "\n",
      "for sample_size in range(2, 6):\n",
      "    samples = get_samples(six_students, sample_size)\n",
      "    sample_means = get_sample_means(samples)\n",
      "    draw_hist(sample_means, sample_size)\n",
      "    sampling_mean = get_sampling_mean(sample_means)\n",
      "    print \"Mean:\", np.mean(six_students)\n",
      "    print \"Sampling Mean:\", sampling_mean"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Y+5aWFhw6dAjLli0DcPEUrb+/v8esvbP6Ac9Y/8sdOHAAOp0O4eHhbq//sIb9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\n2BMRSYBhT0QkAYY9EZEEGPZERBL4fzjLYmfq2+p1AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x5c65ef0>"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Mean: 49.804150686\n",
        "Sampling Mean: 49.804150686\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x5abc430>"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Mean: 49.804150686\n",
        "Sampling Mean: 49.804150686\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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fYADQ0dGBiIgIAEBERAQ6Ojq8WZLTmobavXs3Vq5c6fOaKisroVarER8f79Va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\nAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x5aa7cd0>"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Mean: 49.804150686\n",
        "Sampling Mean: 49.804150686\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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kAcLNWltbxfjx48X9998vhBBi8eLFQqPRiPHjx4vx48eLnTt3urukdjUJIcSa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\nAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x5a0f130>"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Mean: 49.804150686\n",
        "Sampling Mean: 49.804150686\n"
       ]
      }
     ],
     "prompt_number": 62
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "I have noted in the above that as *n* increases, the distribution is squished together. So as *n* increases the standard deviation of the sampling distribution decreases."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}