{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [ "remove_input" ] }, "outputs": [], "source": [ "path_data = '../../data/'\n", "\n", "import numpy as np\n", "import pandas as pd\n", "import math\n", "from scipy import stats\n", "\n", "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "plt.style.use('fivethirtyeight')\n", "\n", "import warnings\n", "warnings.filterwarnings('ignore')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# The SD and the Normal Curve\n", "\n", "We know that the mean is the balance point of the histogram. Unlike the mean, the SD is usually not easy to identify by looking at the histogram. \n", "\n", "However, there is one shape of distribution for which the SD is almost as clearly identifiable as the mean. That is the bell-shaped disribution. This section examines that shape, as it appears frequently in probability histograms and also in some histograms of data. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A Roughly Bell-Shaped Histogram of Data\n", "Let us look at the distribution of heights of mothers in our familiar sample of 1,174 mother-newborn pairs. The mothers' heights have a mean of 64 inches and an SD of 2.5 inches. Unlike the heights of the basketball players, the mothers' heights are distributed fairly symmetrically about the mean in a bell-shaped curve." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "baby = pd.read_csv(path_data + 'baby.csv')" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "64.0" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "heights = baby['Maternal Height']\n", "mean_height = np.round(np.mean(heights), 1)\n", "mean_height" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.5" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sd_height = np.round(np.std(heights), 1)\n", "sd_height" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "positions = np.arange(-3, 3.1, 1)*sd_height + mean_height\n", "\n", "unit = 'inch'\n", "\n", "fig, ax = plt.subplots(figsize=(8,5))\n", "\n", "ax.hist(baby['Maternal Height'], bins=np.arange(55.5, 72.5, 1), \n", " density=True, \n", " color='blue', \n", " alpha=0.8, \n", " ec='white', \n", " zorder=5)\n", "\n", "y_vals = ax.get_yticks()\n", "\n", "y_label = 'Percent per ' + (unit if unit else 'unit')\n", "\n", "x_label = 'Maternal Height (' + (unit if unit else 'unit') +')'\n", "\n", "ax.set_yticklabels(['{:g}'.format(x * 100) for x in y_vals])\n", "\n", "plt.xticks(positions)\n", "\n", "plt.ylabel(y_label)\n", "\n", "plt.xlabel(x_label)\n", "\n", "plt.title('');\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The last two lines of code in the cell above change the labeling of the horizontal axis. Now, the labels correspond to \"average $\\pm$ $z$ SDs\" for $z = 0, \\pm 1, \\pm 2$, and $\\pm 3$. Because of the shape of the distribution, the \"center\" has an unambiguous meaning and is clearly visible at 64." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### How to Spot the SD on a Bell Shaped Curve\n", "To see how the SD is related to the curve, start at the top of the curve and look towards the right. Notice that there is a place where the curve changes from looking like an \"upside-down cup\" to a \"right-way-up cup\"; formally, the curve has a point of inflection at which the curve changes from concavity-up to concavity-down depending upon direction of travel on curve. That point is one SD above average. It is the point $z=1$, which is \"average plus 1 SD\" = 66.5 inches.\n", "\n", "Symmetrically on the left-hand side of the mean, the point of inflection is at $z=-1$, that is, \"average minus 1 SD\" = 61.5 inches. \n", "\n", "In general, **for bell-shaped distributions, the SD is the distance between the mean and the points of inflection on either side.**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The standard normal curve\n", "All the bell-shaped histograms that we have seen look essentially the same apart from the labels on the axes. Indeed, there is really just one basic curve from which all of these curves can be drawn just by relabeling the axes appropriately. \n", "\n", "To draw that basic curve, we will use the units into which we can convert every list: standard units. The resulting curve is therefore called the *standard normal curve*. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The standard normal curve has an impressive equation. But for now, it is best to think of it as a smoothed outline of a histogram of a variable that has been measured in standard units and has a bell-shaped distribution.\n", "\n", "$$\n", "\\phi(z) = {\\frac{1}{\\sqrt{2 \\pi}}} e^{-\\frac{1}{2}z^2}, ~~ -\\infty < z < \\infty\n", "$$ \n", "\n", "The graph below is the output of the function *'plot_normal_cdf'*, this function utilises the probability density function `pdf` method of the `stats` function from the `scipy` module. \n", "\n", "[Scipy stats](https://docs.scipy.org/doc/scipy/reference/stats.html)\n", "\n", "[Scipy stats norm](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.norm)\n", "\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "tags": [ "remove_input" ] }, "outputs": [], "source": [ "# The standard normal curve, a rather lengthy function\n", "# as with all functions - only the function name and parameter are called \n", "\n", "def plot_normal_cdf(rbound=None, lbound=None, mean=0, sd=1):\n", " \"\"\"Plots a normal curve with specified parameters and area below curve shaded\n", " between ``lbound`` and ``rbound``.\n", " Args:\n", " ``rbound`` (numeric): right boundary of shaded region\n", " ``lbound`` (numeric): left boundary of shaded region; by default is negative infinity\n", " ``mean`` (numeric): mean/expectation of normal distribution\n", " ``sd`` (numeric): standard deviation of normal distribution\n", " \"\"\"\n", " shade = rbound is not None or lbound is not None\n", " shade_left = rbound is not None and lbound is not None\n", " inf = 3.5 * sd\n", " step = 0.1\n", " rlabel = rbound\n", " llabel = lbound\n", " if rbound is None:\n", " rbound = inf + mean\n", " rlabel = r\"$\\infty$\"\n", " if lbound is None:\n", " lbound = -inf + mean\n", " llabel = r\"-$\\infty$\"\n", " pdf_range = np.arange(-inf + mean, inf + mean, step)\n", " plt.plot(pdf_range, stats.norm.pdf(pdf_range, loc=mean, scale=sd), color='k', lw=1)\n", " cdf_range = np.arange(lbound, rbound + step, step)\n", " if shade:\n", " plt.fill_between(cdf_range, stats.norm.pdf(cdf_range, loc=mean, scale=sd), color='gold')\n", " if shade_left:\n", " cdf_range = np.arange(-inf+mean, lbound + step, step)\n", " plt.fill_between(cdf_range, stats.norm.pdf(cdf_range, loc=mean, scale=sd), color='darkblue')\n", " plt.ylim(0, stats.norm.pdf(0, loc=0, scale=sd) * 1.25)\n", " plt.xlabel('z')\n", " plt.ylabel(r'$\\phi$(z)', rotation=90)\n", " plt.title(r\"Normal Curve ~ ($\\mu$ = {0}, $\\sigma$ = {1}) \"\n", " \"{2} < z < {3}\".format(mean, sd, llabel, rlabel), fontsize=16)\n", " plt.show()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_normal_cdf()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 'Mechanics' behind the function\n", "For your interest" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#pdf - probability density function\n", "\n", "#cdf - cumulative distribution function\n", "\n", "#ppf - percent point function (inverse of CDF)\n", "\n", "from scipy.stats import norm\n", "\n", "fig, ax = plt.subplots(1, 1)\n", "\n", "#calculate a few first moments\n", "\n", "mean, var, skew, kurt = norm.stats(moments='mvsk')\n", "\n", "#display the pdf\n", "\n", "x = np.linspace(norm.ppf(0.01), norm.ppf(0.99), 100)\n", "\n", "ax.plot(x, norm.pdf(x), 'r-', lw=5, alpha=0.6, label='norm pdf')\n", "\n", "ax.legend(loc='best', frameon=False)\n", "\n", "rv = norm()\n", "\n", "ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')\n", "\n", "#Check accuracy of cdf and ppf:\n", "\n", "vals = norm.ppf([0.001, 0.5, 0.999])\n", "\n", "np.allclose([0.001, 0.5, 0.999], norm.cdf(vals))\n", "\n", "#Generate random numbers:\n", "\n", "r = norm.rvs(size=1000)\n", "\n", "#Compare the histogram:\n", "\n", "ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)\n", "\n", "ax.legend(bbox_to_anchor=(1.04,1), loc=\"upper left\", frameon=False)\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As always when you examine a new histogram, start by looking at the horizontal axis. On the horizontal axis of the standard normal curve, the values are standard units. \n", "\n", "Here are some properties of the curve. Some are apparent by observation, and others require a considerable amount of mathematics to establish.\n", "\n", "- The total area under the curve is 1. So you can think of it as a histogram drawn to the density scale.\n", "\n", "- The curve is symmetric about 0. So if a variable has this distribution, its mean and median are both 0.\n", "\n", "- The points of inflection of the curve are at -1 and +1. \n", "\n", "- If a variable has this distribution, its SD is 1. The normal curve is one of the very few distributions that has an SD so clearly identifiable on the histogram." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since we are thinking of the curve as a smoothed histogram, we will want to represent proportions of the total amount of data by areas under the curve. \n", "\n", "Areas under smooth curves are often found by calculus, using a method called integration. It is a fact of mathematics, however, that the standard normal curve cannot be integrated in any of the usual ways of calculus. \n", "\n", "Therefore, areas under the curve have to be approximated. That is why almost all statistics textbooks carry tables of areas under the normal curve. It is also why all statistical systems, including a module of Python, include methods that provide excellent approximations to those areas." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "from scipy import stats" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The standard normal \"cdf\"\n", "The fundamental function for finding areas under the normal curve is `stats.norm.cdf`. It takes a numerical argument and returns all the area under the curve to the left of that number. Formally, it is called the \"cumulative distribution function\" of the standard normal curve. That rather unwieldy mouthful is abbreviated as cdf.\n", "\n", "Let us use this function to find the area to the left of $z=1$ under the standard normal curve. " ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "tags": [ "remove_input" ] }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Area under the standard normal curve, below 1\n", "\n", "plot_normal_cdf(1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The numerical value of the shaded area can be found by calling `stats.norm.cdf`.\n", "\n", "[Scipy.stats.norm.cdf( )](https://docs.scipy.org/doc/scipy/reference/tutorial/stats.html#common-methods)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.8413447460685429" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stats.norm.cdf(1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That's about 84%. We can now use the symmetry of the curve and the fact that the total area under the curve is 1 to find other areas. \n", "\n", "The area to the right of $z=1$ is about 100% - 84% = 16%." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "tags": [ "remove_input" ] }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Area under the standard normal curve, above 1\n", "\n", "plot_normal_cdf(lbound=1)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.15865525393145707" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1 - stats.norm.cdf(1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Or** we pass the negative equivalent value" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.15865525393145707" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stats.norm.cdf(-1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The area between $z=-1$ and $z=1$ can be computed in several different ways. It is the gold area under the curve below. " ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "tags": [ "remove_input" ] }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Area under the standard normal curve, between -1 and 1\n", "\n", "plot_normal_cdf(1, lbound=-1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For example, we could calculate the area as \"100% - two equal tails\", which works out to roughly 100% - 2x16% = 68%.\n", "\n", "Or we could note that the area between $z=1$ and $z=-1$ is equal to all the area to the left of $z=1$, minus all the area to the left of $z=-1$." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.6826894921370859" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stats.norm.cdf(1) - stats.norm.cdf(-1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By a similar calculation, we see that the area between $-2$ and 2 is about 95%." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "tags": [ "remove_input" ] }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Area under the standard normal curve, between -2 and 2\n", "\n", "plot_normal_cdf(2, lbound=-2)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9544997361036416" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stats.norm.cdf(2) - stats.norm.cdf(-2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In other words, if a histogram is roughly bell shaped, the proportion of data in the range \"average $\\pm$ 2 SDs\" is about 95%. \n", "\n", "That is quite a bit more than Chebychev's lower bound of 75%. Chebychev's bound is weaker because it has to work for all distributions. If we know that a distribution is normal, we have good approximations to the proportions, not just bounds." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The table below compares what we know about all distributions and about normal distributions. Notice that when $z=1$, Chebychev's bound is correct but not illuminating.\n", "\n", "| Percent in Range | All Distributions: Bound | Normal Distribution: Approximation |\n", "| :--------------- | :---------------- --| :-------------------|\n", "|average $\\pm$ 1 SD | at least 0% | about 68% |\n", "|average $\\pm$ 2 SDs | at least 75% | about 95% |\n", "|average $\\pm$ 3 SDs | at least 88.888...% | about 99.73% |" ] } ], "metadata": { "anaconda-cloud": {}, "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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 1 }