{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "# HIDDEN\n", "import numpy as np\n", "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "plt.style.use('fivethirtyeight')\n", "import pandas as pd" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Describing distributions\n", "\n", "We have seen several examples of *distributions*.\n", "\n", "We can describe distributions as having a *center*, and a *spread*.\n", "\n", "In [the mean as predictor](../08/mean_meaning), we saw that the mean is a useful measure of the center of a distribution.\n", "\n", "What measure should we use for the spread?\n", "\n", "## Chronic kidney disease\n", "\n", "We're going to work with a data set that was collected to help doctors diagnose chronic kidney disease (CKD). Each row in the data set represents a single patient who was treated in the past and whose diagnosis is known. For each patient, we have a bunch of measurements from a blood test.\n", "\n", "You will see more of this dataset soon.\n", "\n", "If you are running on your laptop, you should download the\n", "[ckd.csv]({{ site.baseurl }}/data/ckd.csv) file to the same\n", "directory as this notebook." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "lines_to_next_cell": 0 }, "outputs": [ { "data": { "text/html": [ "
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AgeBlood PressureSpecific GravityAlbuminSugarRed Blood CellsPus CellPus Cell clumpsBacteriaBlood Glucose Random...Packed Cell VolumeWhite Blood Cell CountRed Blood Cell CountHypertensionDiabetes MellitusCoronary Artery DiseaseAppetitePedal EdemaAnemiaClass
048701.00540normalabnormalpresentnotpresent117...3267003.9yesnonopooryesyes1
153901.02020abnormalabnormalpresentnotpresent70...29121003.7yesyesnopoornoyes1
263701.01030abnormalabnormalpresentnotpresent380...3245003.8yesyesnopooryesno1
368801.01032normalabnormalpresentpresent157...16110002.6yesyesyespooryesno1
461801.01520abnormalabnormalnotpresentnotpresent173...2492003.2yesyesyespooryesyes1
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5 rows × 25 columns

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" ], "text/plain": [ " Age Blood Pressure Specific Gravity Albumin Sugar Red Blood Cells \\\n", "0 48 70 1.005 4 0 normal \n", "1 53 90 1.020 2 0 abnormal \n", "2 63 70 1.010 3 0 abnormal \n", "3 68 80 1.010 3 2 normal \n", "4 61 80 1.015 2 0 abnormal \n", "\n", " Pus Cell Pus Cell clumps Bacteria Blood Glucose Random ... \\\n", "0 abnormal present notpresent 117 ... \n", "1 abnormal present notpresent 70 ... \n", "2 abnormal present notpresent 380 ... \n", "3 abnormal present present 157 ... \n", "4 abnormal notpresent notpresent 173 ... \n", "\n", " Packed Cell Volume White Blood Cell Count Red Blood Cell Count \\\n", "0 32 6700 3.9 \n", "1 29 12100 3.7 \n", "2 32 4500 3.8 \n", "3 16 11000 2.6 \n", "4 24 9200 3.2 \n", "\n", " Hypertension Diabetes Mellitus Coronary Artery Disease Appetite \\\n", "0 yes no no poor \n", "1 yes yes no poor \n", "2 yes yes no poor \n", "3 yes yes yes poor \n", "4 yes yes yes poor \n", "\n", " Pedal Edema Anemia Class \n", "0 yes yes 1 \n", "1 no yes 1 \n", "2 yes no 1 \n", "3 yes no 1 \n", "4 yes yes 1 \n", "\n", "[5 rows x 25 columns]" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ckd_full = pd.read_csv('ckd.csv')\n", "ckd_full.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will use this dataset to get a couple of variables (columns) and therefore a couple of distributions.\n", "\n", "Let's start with the White Blood Cell Count, usually abbreviated as WBC." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "wbc = ckd_full['White Blood Cell Count']\n", "wbc.hist()\n", "plt.title('White Blood Cell Count');" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "count 158.000000\n", "mean 8475.949367\n", "std 3126.880181\n", "min 3800.000000\n", "25% 6525.000000\n", "50% 7800.000000\n", "75% 9775.000000\n", "max 26400.000000\n", "Name: White Blood Cell Count, dtype: float64" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "wbc.describe()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Compare this to Hemoglobin concentrations:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "hgb = ckd_full['Hemoglobin']\n", "hgb.hist()\n", "plt.title('Hemoglobin');" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "count 158.000000\n", "mean 13.687342\n", "std 2.882204\n", "min 3.100000\n", "25% 12.600000\n", "50% 14.250000\n", "75% 15.775000\n", "max 17.800000\n", "Name: Hemoglobin, dtype: float64" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hgb.describe()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that we can't easily plot these two on the same axes, because their units are so different." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We could try and fix this by subtracting the mean, as a center value, so the values are now *deviations* from the mean." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "wbc_deviations = wbc - np.mean(wbc)\n", "wbc_deviations.hist()\n", "plt.title('White Blood Cell Count deviations');" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "hgb_deviations = hgb - np.mean(hgb)\n", "hgb_deviations.hist()\n", "plt.title('Hemoglobin deviations');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The deviations each have a mean very very close to zero, and therefore, they have the same center:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(-1.8420145858692217e-13, 7.195369476051647e-16)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.mean(wbc_deviations), np.mean(hgb_deviations)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We still cannot plot them on the same axes, because the WBC values have a very different *spread*. They will completely dominate the x axis of the graph. We can't reasonably compare the WBC deviations to the Hemoglobin deviations, because they have such different units." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We would like a measure of the spread of the distribution, so we can set the two distributions to have the same spread." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The standard deviation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the [mean as predictor](../08/mean_meaning) section, we found that mean was the best value to use as a predictor, to minimize the sum of *squared* deviations.\n", "\n", "Maybe we could get an idea of the typical *squared* deviation, as a measure of spread?" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0 -2.487342\n", "1 -4.187342\n", "2 -2.887342\n", "3 -8.087342\n", "4 -5.987342\n", "5 -3.887342\n", "6 -1.187342\n", "7 -3.687342\n", "8 -3.187342\n", "9 -3.887342\n", "Name: Hemoglobin, dtype: float64" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hgb_deviations[:10]" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0 6.186869\n", "1 17.533831\n", "2 8.336743\n", "3 65.405097\n", "4 35.848261\n", "5 15.111426\n", "6 1.409780\n", "7 13.596489\n", "8 10.159148\n", "9 15.111426\n", "Name: Hemoglobin, dtype: float64" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hgb_dev_sq = hgb_deviations ** 2\n", "hgb_dev_sq[:10]" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 1.0, 'HGB squared deviations')" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "hgb_dev_sq.hist()\n", "plt.title('HGB squared deviations')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The center, or typical value, of this distribution, could be the *mean*." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "8.254523313571543" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hgb_dev_sq_mean = np.mean(hgb_dev_sq)\n", "hgb_dev_sq_mean" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is the *mean squared deviation*. This is also called the *variance*. Numpy has a function to calculate that in one shot:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "8.254523313571543" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# The mean squared deviation is the variance\n", "np.var(hgb)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The mean squared deviation is a good indicator of the typical squared deviation. What should we used for some measure of the typical devation?\n", "\n", "We could take the square root of the mean squared deviation, like this:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.873068623192203" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.sqrt(hgb_dev_sq_mean)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is a measure of the spread of the distribution. It is a measure of the typical or average deviation.\n", "\n", "It is also called the *standard deviation*." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.873068623192203" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.std(hgb)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can make our distribution have a standard center *and* a standard spread by dividing our mean-centered distribution, by the standard deviation. Then the distribution will have a standard deviation very close to 1.\n", "\n", "This version of the distribution, with mean 0 and standard deviation of 1, is called the *standardized* distribution." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 1.0, 'Standardized Hemoglobin')" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "standardized_hgb = hgb_deviations / np.std(hgb)\n", "standardized_hgb.hist()\n", "plt.title('Standardized Hemoglobin')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can make a function to do this:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "def standard_units(x):\n", " return (x - np.mean(x))/np.std(x)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 1.0, 'Standardized Hemoglobin, again')" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "std_hgb_again = standard_units(hgb)\n", "std_hgb_again.hist()\n", "plt.title('Standardized Hemoglobin, again')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we do the same to the WBC, we can compare values of the distributions:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 1.0, 'Standardized White Blood Cell Count')" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "std_wbc = standard_units(wbc)\n", "std_wbc.hist()\n", "plt.title('Standardized White Blood Cell Count')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Every value in standardized units gives the deviation of the original value from its mean, in terms of the number of standard deviations." ] } ], "metadata": { "anaconda-cloud": {}, "jupytext": { "formats": "", "notebook_metadata_filter": "all,-language_info" }, "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": "3.7.2" } }, "nbformat": 4, "nbformat_minor": 2 }