T Distribution (Small Sampling Theory).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Let's learn the T-distribution! Note: Learn about the normal distribution first!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For previous distributions the sample size was assumed large (N>30). For sample sizes that are less than 30, otherwise (N<30). Note: Sometimes the t-distribution is known as the student distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The t-distribution allows for use of small samples, but does so by sacrificing certainty with a margin-of-error trade-off. The t-distribution takes into account the sample size using n-1 degrees of freedom, which means there is a different t-distribution for every different sample size. If we see the t-distribution against a normal distribution, you'll notice the tail ends increase as the peak get 'squished' down. \n",
    "\n",
    "It's important to note, that as n gets larger, the t-distribution converges into a normal distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To further explain degrees of freedom and how it relates tothe t-distribution, you can think of degrees of freedom as an adjustment to the sample size, such as (n-1). This is connected to the idea that we are estimating something of a larger population, in practice it gives a slightly larger margin of error in the estimate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's define a new variable called t, where\n",
    ":\n",
    "$$t=\\frac{\\overline{X}-\\mu}{s}\\sqrt{N-1}=\\frac{\\overline{X}-\\mu}{s/\\sqrt{N}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which is analogous to the z statistic given by $$z=\\frac{\\overline{X}-\\mu}{\\sigma/\\sqrt{N}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sampling distribution for t can be obtained:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "source": [
    "## $$  f(t) = \\frac {\\varGamma(\\frac{v+1}{2})}{\\sqrt{v\\pi}\\varGamma(\\frac{v}{2})} (1+\\frac{t^2}{v})^{-\\frac{v+1}{2}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where the gamma function is: $$\\varGamma(n)=(n-1)!$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And v is the number of degrees of freedom, typically equal to N-1. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similar to a z-score table used with a normal distribution, a t-distribution uses a t-table. Knowing the degrees of freedom and the desired cumulative probability (e.g. P(T >= t) ) we can find the value of t. Here is an example of a lookup table for a t-distribution: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's see how to get the t-distribution in Python using scipy!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x181a3c8c410>]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Import for plots\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "#Import the stats library\n",
    "from scipy.stats import t\n",
    "\n",
    "#import numpy\n",
    "import numpy as np\n",
    "\n",
    "# Create x range\n",
    "x = np.linspace(-5,5,100)\n",
    "\n",
    "# Create the t distribution with scipy\n",
    "rv = t(3)\n",
    "\n",
    "# Plot the PDF versus the x range\n",
    "plt.plot(x, rv.pdf(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Additional resources can be found here:\n",
    "\n",
    "1.) http://en.wikipedia.org/wiki/Student%27s_t-distribution\n",
    "\n",
    "2.) http://mathworld.wolfram.com/Studentst-Distribution.html\n",
    "\n",
    "3.) http://stattrek.com/probability-distributions/t-distribution.aspx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.12.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}