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Title	Unbiased Variance Visualization
ID	1167453164
Created	13 years ago, 1353696556458
Updated	13 years ago, 1353988919667
Code	/****************************************************** * Hi all! Here's my try at the Unbiased Variance Challenge. * * I made a short YouTube video describing the program. The * link is in the "Tips & Comments" section below, but if you * prefer text, read on! * * I used a flat probabilistic distribution from 0 to 200 for * my population. Use the buttons to sample this population * (50 at a time for now, but go ahead and change it if you * like). As the sample grows, the variance is plotted on the * left for three different ways of calculating (biased, * unbiased n-1 correction, and a custom correction currently * set to n-2). The solid line is the true variance of the * population, and the black dots in each graph show the * variance calculated using the population mean instead of * the sample mean. By looking at these graphs, you can see * that the variance calculated with the n-1 correction tends * to approach the true population variance at large sample * sizes, meaning it is unbiased. * * The plots on the right side help show why the uncorrected * calculation is biased. A point is generated for each * sampling group (again, 50 samples in my version). The * horizontal axis plots the difference between the sample * group mean and the population mean. The vertical axis * plots the difference between the variance calculated using * the sample group's mean and the variance for the sample * that would be calculated using the true population mean. * As you'll see, the uncorrected variance calculated with * the sample mean is always* lower than the variance * calculated with the true population mean (they are equal, when the sample mean happens to be the same as the * population mean) meaning that it will always be an * underestimation of the true variance. The n-1 correction * shifts this curve upward, meaning the higher and lower * portions can balance each other out on average. Some more * interesting questions to think about include: What type of * curve is the red one on the upper right, and why? Why do * the curves below look a bit mushier? * * Beside the n and n-1 methods, a third customizable method * (currently n-2 and changeable in the seventh line of code) * is on the bottom in green. Play around with it and see if * you become convinced that the n-1 method is THE unbiased * method. ****************************************************/ var sampleSize = 50; var newSample = []; var newSampleMean = 0; var showSample = 0; var singleSample = 0; var multiSample = 0; // /////////////////////////////To make the sample variance unbiased, we need to divide by (n - 1) instead of n. Try subracting other numbers from the denominator and see how the variance changes. (make sure the correction factor is less than the sample size to avoid dividing by zero!) var customCorrectionFactor = 2; /////////////////////////////// var numberOfSamples = 0; //(Initially) var sampleMeans = []; var sampleVariancesTrueMean = []; var biasedVariances = []; var unbiasedVariances = []; var customCorrectionVariances = []; var cumulativeSampleVariancesTrueMean = []; var cumulativeBiasedVariances = []; var cumulativeUnbiasedVariances = []; var cumulativeCustomCorrectionVariances = []; // For comparison, these are the true values for mean and variance of the popualtion used to generate random samples. In my code, the total population is a flat distribution of numbers from 0 to 200. If you change the total popualtion, be sure to recalculate these values too. var truePopulationMean = 100; var truePopulationVariance = 3366.7; var generateNewSample = function() { for (var i = 0; i < sampleSize; i++){ newSample[i] = (floor(random() 201)); } numberOfSamples++; }; var computeNewSampleMeanAndVariances = function() { var sampleTotal = 0; for (var i = 0; i < sampleSize; i++){ sampleTotal = sampleTotal + newSample[i]; } newSampleMean = sampleTotal/sampleSize; sampleMeans[numberOfSamples - 1] = newSampleMean; var squaredResidualsTotal = 0; for (i = 0; i < sampleSize; i++) { squaredResidualsTotal = squaredResidualsTotal + pow(newSample[i] - newSampleMean,2); } var squaredResidualsTotalTrueMean = 0; for (i = 0; i < sampleSize; i++) { squaredResidualsTotalTrueMean = squaredResidualsTotalTrueMean + pow(newSample[i] - truePopulationMean,2); } if (sampleSize > 0) { biasedVariances[numberOfSamples - 1] = squaredResidualsTotal/sampleSize; sampleVariancesTrueMean[numberOfSamples - 1] = squaredResidualsTotalTrueMean/sampleSize; } if (sampleSize > 1 ) { unbiasedVariances[numberOfSamples - 1] = squaredResidualsTotal/(sampleSize - 1); } if (sampleSize > customCorrectionFactor ) { customCorrectionVariances[numberOfSamples - 1] = squaredResidualsTotal/(sampleSize - customCorrectionFactor); } }; var computeCumulativeVariances = function() { var totalOfSampleMeans = 0; for (var j = 0; j < numberOfSamples; j++) { totalOfSampleMeans = totalOfSampleMeans + sampleMeans[j]; } var totalOfBiasedVariances = 0; for (j = 0; j < numberOfSamples; j++) { totalOfBiasedVariances = totalOfBiasedVariances + biasedVariances[j]; } cumulativeBiasedVariances[j-1] = totalOfBiasedVariances/numberOfSamples; var totalOfUnbiasedVariances = 0; for (j = 0; j < numberOfSamples; j++) { totalOfUnbiasedVariances = totalOfUnbiasedVariances + unbiasedVariances[j]; } cumulativeUnbiasedVariances[j-1] = totalOfUnbiasedVariances/numberOfSamples; var totalOfCustomCorrectionVariances = 0; for (j = 0; j < numberOfSamples; j++) { totalOfCustomCorrectionVariances = totalOfCustomCorrectionVariances + customCorrectionVariances[j]; } cumulativeCustomCorrectionVariances[j-1] = totalOfCustomCorrectionVariances/numberOfSamples; var totalOfSampleVariancesTrueMean = 0; for (j = 0; j < numberOfSamples; j++) { totalOfSampleVariancesTrueMean = totalOfSampleVariancesTrueMean + sampleVariancesTrueMean[j]; } cumulativeSampleVariancesTrueMean[j-1] = totalOfSampleVariancesTrueMean/numberOfSamples; }; var drawButtons = function() { resetMatrix(); stroke(0, 0, 0); strokeWeight(1.5); fill(173, 105, 56); if (showSample) { fill(224, 177, 119); } rect(10,10,100,45,3); fill(173, 105, 56); if (singleSample) { fill(224, 177, 119); } rect(122,10,100,45,3); fill(173, 105, 56); if (multiSample) { fill(224, 177, 119); } rect(234,10,100,45,3); strokeWeight(1); fill(255, 255, 255); text("Show Last",29,29); text("Group Added",24,46); text("Add " + sampleSize,150,29); text("To Sample",143,45); text("Hold To Sample",240,30); text("Continuously",249,46); }; var displayInfo = function() { resetMatrix(); stroke(0, 0, 0); translate(65,80); fill(0, 0, 0); text("Number of Samples",18,-6); rotate(-90); text("Cumulative",-82,-52); text("Biased",-68,-36); text("Variance",-73,-21); text("(n)",-58,-5); rotate(90); fill(255, 255, 255); rect(0,0,150,100); line(0,50,150,50); fill(0, 0, 0); textSize(9); text("True Variance",3,60); textSize(12); stroke(255, 0, 0); fill(255, 0, 0); for (var i = 0; i < numberOfSamples; i++){ stroke(255, 0, 0); fill(255, 0, 0); ellipse((i+1)/(numberOfSamples+1)150,50 - 50((cumulativeBiasedVariances[i]-truePopulationVariance)sqrt(sampleSize)/3000),2,2); stroke(0, 0, 0); fill(0, 0, 0); ellipse((i+1)/(numberOfSamples+1)150,50 - 50((cumulativeSampleVariancesTrueMean[i]-truePopulationVariance)sqrt(sampleSize)/3000),1,1); } translate(200,0); stroke(0, 0, 0); fill(0, 0, 0); text("Difference From True Mean",-19,-6); rotate(-90); text("Variance",-72,-26); text("Difference",-76,-9); rotate(90); fill(255, 255, 255); rect(0,0,100,100); line(0,50,100,50); line(50,0,50,100); stroke(255, 0, 0); fill(255, 0, 0); for (i = 0; i < numberOfSamples; i++){ ellipse(sqrt(sampleSize)/5(sampleMeans[i]-truePopulationMean)+50,50-(biasedVariances[i]-sampleVariancesTrueMean[i])sampleSize/600,2,2); } resetMatrix(); translate(65,185); stroke(0, 0, 0); fill(0, 0, 0); rotate(-90); text("Cumulative",-82,-52); text("Unbiased",-77,-36); text("Variance",-73,-21); text("(n - 1)",-66,-5); rotate(90); fill(255, 255, 255); rect(0,0,150,100); fill(0, 0, 0); textSize(9); text("True Variance",3,60); textSize(12); line(0,50,150,50); stroke(0, 0, 255); fill(0, 0, 255); for (i = 0; i < numberOfSamples; i++){ stroke(0, 0, 255); fill(0, 0, 255); ellipse((i+1)/(numberOfSamples+1)150,50 - 50((cumulativeUnbiasedVariances[i]-truePopulationVariance)sqrt(sampleSize)/3000),2,2); stroke(0, 0, 0); fill(0, 0, 0); ellipse((i+1)/(numberOfSamples+1)150,50 - 50((cumulativeSampleVariancesTrueMean[i]-truePopulationVariance)sqrt(sampleSize)/3000),1,1); } translate(200,0); stroke(0, 0, 0); rotate(-90); fill(0, 0, 0); text("Variance",-72,-26); text("Difference",-76,-9); rotate(90); fill(255, 255, 255); rect(0,0,100,100); line(0,50,100,50); line(50,0,50,100); stroke(0, 0, 255); fill(0, 0, 255); for (i = 0; i < numberOfSamples; i++){ ellipse(sqrt(sampleSize)/5(sampleMeans[i]-truePopulationMean)+50,50-(unbiasedVariances[i]-sampleVariancesTrueMean[i])sampleSize/600,2,2); } resetMatrix(); translate(65,290); stroke(0, 0, 0); fill(0, 0, 0); rotate(-90); text("Cumulative",-82,-52); text("Custom Corrected",-99,-36); text("Variance",-73,-21); text("(n - "+ customCorrectionFactor + ")",-66,-5); rotate(90); fill(255, 255, 255); rect(0,0,150,100); fill(0, 0, 0); textSize(9); text("True Variance",3,60); textSize(12); line(0,50,150,50); for (i = 0; i < numberOfSamples; i++){ stroke(7, 173, 12); fill(7, 173, 12); ellipse((i+1)/(numberOfSamples+1)150,50 - 50((cumulativeCustomCorrectionVariances[i]-truePopulationVariance)sqrt(sampleSize)/3000),2,2); stroke(0, 0, 0); fill(0, 0, 0); ellipse((i+1)/(numberOfSamples+1)150,50 - 50((cumulativeSampleVariancesTrueMean[i]-truePopulationVariance)sqrt(sampleSize)/3000),1,1); } translate(200,0); stroke(0, 0, 0); rotate(-90); fill(0, 0, 0); text("Variance",-72,-26); text("Difference",-76,-9); rotate(90); fill(255, 255, 255); rect(0,0,100,100); line(0,50,100,50); line(50,0,50,100); stroke(7, 173, 0); fill(7, 173, 0); for (i = 0; i < numberOfSamples; i++){ ellipse(sqrt(sampleSize)/5(sampleMeans[i]-truePopulationMean)+50,50-(customCorrectionVariances[i]-sampleVariancesTrueMean[i])sampleSize/600,2,2); } }; var mouseHeld = 0; var draw = function() { resetMatrix(); background(250, 234, 185); image(getImage("avatars/leaf-green"),344,8,50,49); if (mouseIsPressed) { if (mouseHeld === 0) { var x = mouseX; var y = mouseY; var onShowSampleButton = (10 <= x && x <= 110 && 10 <= y && y <= 55); var onSingleSampleButton = (122 <= x && x <= 222 && 10 <= y && y <= 55); var onMultipleSampleButton = (234 <= x && x <= 334 && 10 <= y && y <= 55); if (onShowSampleButton) { if (showSample === 0) { showSample = 1; } else if (showSample === 1) { showSample = 0; } mouseHeld = 1; } else if (onSingleSampleButton){ generateNewSample(); singleSample = 1; mouseHeld = 1; } else if (onMultipleSampleButton) { generateNewSample(); multiSample = 1; } computeNewSampleMeanAndVariances(); computeCumulativeVariances(); } } else {mouseHeld = 0;singleSample = 0;multiSample = 0;} drawButtons(); displayInfo(); if (showSample === 1){ resetMatrix(); stroke(0, 0, 0); fill(255, 255, 255); rect(2,61,395,336); fill(0, 0, 0); for (var i = 0; i < sampleSize; i++){ text(newSample[i],10 + 40floor(i/22),77+(i%22)15); } } };
Folds
Fork	false
Key	ag5zfmtoYW4tYWNhZGVteXIUCxIKU2NyYXRjaHBhZBjs19esBAw
Votes	247
Spin-Offs	16
Type	PJS
Width	400
Height	400
User Flagged	false
Origin Scratchpad	{"id":"1164356446","title":"Challenge: Unbiased Estimate of Population Variance","deleted":"false","official":"false"}
Hidden From Hotlist	false
Restricted Posting	false
By Child	false
Author Nick	Justin Helps
Author Name	JustinHelps
Author ID	kaid_1202412063512761325716068
Author Profile Access	COACH

Archive Data

Added	10 months ago, 1739280047081
Updated	10 months ago, 1739280047081
Source Deleted	unknown