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Title	Estimateur sans biais de la variance sur la population
ID	1169428428
Created	Unknown
Updated	2 years ago, 1671613388855
Code	/************************************************************ * This program generates a population of random variables. * The top graph is the distribution of the population. * * Watch a video about this simulation at: * https://www.khanacademy.org/math/probability/descriptive-statistics/variance_std_deviation/v/simulation-showing-bias-in-sample-variance * * Click on the lower graphs to enlarge them. * Click again to return to the normal view. * If multiple dashed lines appear click anywhere. * * The program continually samples the population with a * sample size between 2 and 10. * The second graph shows sample mean vs. sample variance. * It shows that sample means further from the population mean * tend to have a smaller variance. * These samples tend to be smaller samples (more red). * * The third graph shows sample variance as a function of * sample size. Sample variance is plotted as a percentage * of population variance. * * The variance of a sample of size 2 approaches 1/2 the * population variance. * The variance of a sample of size 3 approaches 2/3 the * population variance. * The variance of a sample of size 4 approaches 3/4 the * population variance. * And so on * * In summary, small samples are more likely to have a mean * that is far from the population mean. Samples with a mean * far from the population mean are more likely to have a low * variance. ************************************************************/ frameRate(30); // Population variables var popMin = 1; var popMax = 20; var freqMax = 50; // Sample variables var sampleSizeMin = 2; // Makes no sense to be <2 var sampleSizeMax = 10; var sampleCount = 0; // Display variables var titleFont = createFont("Times", 16); var labelFont = createFont("Arial", 10); var scatterSize = 3; var dashSize = 4; var backgroundColour = color(245, 245, 255); // Used to only refresh graphs when necessary var refresh = true; var mode = 'standard'; var modeSpeed = 50; var modeN = modeSpeed; // Graph locations var popGraph = { x: 15, y: 50, width: 369, height: 110, xMin: popMin, xMax: popMax, yMax: freqMax, xlabel: "Value of random variable", ylabel: "Counts", stroke: color(0, 150, 0), colour1: color(0, 150, 0, 80) }; var sampleGraph = { x: 218, y: 220, width: 167, height: 165, xMin: sampleSizeMin, xMax: sampleSizeMax, yMax: 100, xlabel: "Sample size", ylabel: "Sample var / Pop. var (%)", stroke: color(200, 200, 200), colour1: color(250, 0, 0, 160), colour2: color(0, 0, 250, 160) }; var scatterGraph = { x: 15, y: 220, width: 172, height: 165, xMin: popMin, xMax: popMax, yMax: (0.5popMax) * (0.5popMax), xlabel: "Sample mean", ylabel: "Sample variance", colour1: color(250, 0, 0, 120), colour2: color(0, 0, 250, 120) }; var borderBottom = 25; var borderLeft = 25; // How many samples of each size have we taken var sampleSizeCount = []; // Total variance of samples for each sample size; var sampleSizeVar = []; // Variance of samples relative to population variance for each sample size; var sampleSizeVarPercent = []; // List of sample, sizes, means and variances var sampleStats = []; for (var i=sampleSizeMin; i<=sampleSizeMax; i++) { sampleSizeCount.push(0); sampleSizeVar.push(0); sampleSizeVarPercent.push(0); } popGraph.midX = popGraph.x + borderLeft + (popGraph.width - borderLeft) / 2; sampleGraph.midX = sampleGraph.x + borderLeft + (sampleGraph.width - borderLeft) / 2; scatterGraph.midX = scatterGraph.x + borderLeft + (scatterGraph.width - borderLeft) / 2; var newPopulation = function() { refresh = true; sampleCount = 0; }; // Create a random distribution for N values var createRandomPopulation = function(N) { newPopulation(); var population = []; for (var i=0; i<N; i++) { population.push(floor(random() freqMax)); } return population; }; var getPopulationStatistics = function(pop) { var size = 0; var mean = 0; var variance = 0; for (var i=0; i<pop.length; i++) { var x = i + popMin; size += pop[i]; mean += pop[i] * x; variance += pop[i] * x * x; } mean /= size; variance /= size; variance -= mean * mean; return [size, mean, variance]; }; var getSampleStatistics = function(sample) { var mean = 0; var variance = 0; var sampleSize = sample.length; for (var i=0; i<sampleSize; i++) { mean += sample[i]; variance += sample[i] * sample[i]; } mean /= sampleSize; variance /= sampleSize; variance -= mean * mean; return [mean, variance]; }; // Take a random sample of a given size var samplePopulation = function(population, pSize, sSize) { var sample = []; for (var i = 0; i < sSize; i++) { var x = floor(random() * pSize); for (var j = 0; j <= population.length; j++) { x -= population[j]; if (x <=0) { sample.push(j + popMin); break; } } } return sample; }; // Take a random sample of random size var takeRandomSample = function(population, pSize, pVar) { var s = floor(random() * (sampleSizeMax - sampleSizeMin + 1)); var sampleSize = s + sampleSizeMin; var sample = samplePopulation(population, pSize, sampleSize); var samStats = getSampleStatistics(sample); var mean = samStats[0]; var variance = samStats[1]; sampleSizeCount[s]++; sampleSizeVar[s] += variance; sampleSizeVarPercent[s] = (sampleSizeVar[s] / sampleSizeCount[s]) * 100 / pVar; sampleStats.push([sampleSize, mean, variance]); sampleCount++; }; var population = createRandomPopulation(popMax - popMin + 1); var popStats = getPopulationStatistics(population); var popSize = popStats[0]; var popMean = popStats[1]; var popVar = popStats[2]; for (var i=0; i<100; i++) { takeRandomSample(population, popSize, popVar); } // Scale and translate image for current mode var modeTransform = function() { var p = modeN / modeSpeed; resetMatrix(); if (mode === "scatter") { translate(p * 2, -p * 342); scale(1 + 0.9 * p, 1 + 0.9 * p); } else if (mode === "sample-size") { translate(p * 2, -p * 342); translate(p * (-sampleGraph.x -164), p * (-sampleGraph.y + 218)); scale(1 + 0.9 * p, 1 + 0.9 * p); } if (modeN < modeSpeed) { refresh = true; modeN++; } }; var drawBarGraph = function(data, display, title) { var gx = display.x + borderLeft; var gy = display.y + display.height - borderBottom; var gw = display.width - borderLeft; var gh = display.height - borderBottom; var scaleX = (gw - 4) / (display.xMax - display.xMin + 1); var scaleY = gh / display.yMax; var i; modeTransform(); fill(backgroundColour); noStroke(); rect(display.x-10, display.y-10, display.width+12, display.height+12); // Gridlines stroke(220, 230, 220); strokeWeight(1); var dy = display.yMax / 5; for (i = dy; i <= display.yMax; i += dy) { line(gx-2, gy - i * scaleY, gx + gw, gy - i * scaleY); } // Colours for bars if (display.stroke !== undefined) { stroke(display.stroke); } else { noStroke(); } // Draw bars for (i = 0; i <= data.length; i++) { var height = data[i] * scaleY; if (display.colour2 === undefined){ fill(display.colour1); } else { var n = i / data.length; var c = lerpColor(display.colour1, display.colour2, n); fill(c); } rect(gx + 2 + i * scaleX, gy - height, scaleX, height); } // Axes stroke(10, 10, 20); strokeWeight(2); line(gx, gy+1, gx+gw, gy+1); // Axis labels fill(10, 10, 20); textFont(labelFont, 12); textAlign(CENTER, BASELINE); text(display.xlabel, gx + 0.5 * gw, gy+borderBottom); translate(gx - borderLeft, gy - gh/2); rotate(270); text(display.ylabel, 0, 0); resetMatrix(); modeTransform(); // Axis units textFont(labelFont, 10); for (i=0; i<data.length; i++) { text(i + display.xMin, gx + 2 + (i + 0.5) * scaleX, gy + borderBottom - 12); } textAlign(RIGHT, CENTER); for (i = 0; i <= display.yMax; i += dy) { text(i, gx - 4, gy - i * scaleY); } }; var drawScatterGraph = function(data, display, update) { var gx = display.x + borderLeft; var gy = display.y + display.height - borderBottom; var gw = display.width - borderLeft; var gh = display.height - borderBottom; var scaleX = gw / (display.xMax - display.xMin); var scaleY = gh / display.yMax; var dy = display.yMax / 5; var i, n, c, x, y; modeTransform(); if (!update) { // Clear background noStroke(); fill(backgroundColour); noStroke(); rect(display.x-10, display.y-10, display.width+20, display.height+12); // Draw all data for (i=0; i<data.length; i++) { n = (data[i][0] - display.xMin) / (display.xMax - display.xMin); c = lerpColor(display.colour1, display.colour2, n); fill(c); ellipse(gx + (data[i][1] - display.xMin) * scaleX, gy - data[i][2] * scaleY, scatterSize, scatterSize); } // Axis labels fill(10, 10, 20); textFont(labelFont, 12); textAlign(CENTER, BASELINE); text(display.xlabel, gx + 0.5 * gw, gy+borderBottom); translate(gx - borderLeft+4, gy - gh/2); rotate(270); text(display.ylabel, 0, 0); resetMatrix(); modeTransform(); // Axis units textFont(labelFont, 10); strokeWeight(1); for (i=1; i<popMax; i+=2) { x = gx + 2 + (i - 0.5) * scaleX + 2; text(i + display.xMin, x, gy + borderBottom - 12); line(x, gy, x, gy + 3); } textAlign(RIGHT, CENTER); for (i = 0; i <= display.yMax; i += dy) { y = gy - i * scaleY; text(i, gx - 4, y); line(gx, y, gx-3, y); } } else { i = data.length - 1; n = (data[i][0] - display.xMin) / (display.xMax - display.xMin); c = lerpColor(display.colour1, display.colour2, n); fill(c); ellipse(gx + (data[i][1] - display.xMin) * scaleX, gy - data[i][2] * scaleY, scatterSize, scatterSize); } // Draw population values stroke(140, 140, 160); strokeWeight(1); // Mean of means x = gx + (popMean - display.xMin) * scaleX; for (i=0; i < gh; i += dashSize * 2) { line(x, gy - i, x, gy - i - dashSize); } // Mean of variance y = gy - popVar * scaleY; for (i=0; i < gw; i += dashSize * 2) { line(gx + i, y, gx + i + dashSize, y); } // Axes stroke(10, 10, 20); strokeWeight(2); line(gx, gy+1, gx+gw, gy+1); line(gx, gy, gx, gy - gh+1); }; var drawTitle =function(display, title, subtitle) { fill(backgroundColour); noStroke(); rect(display.x, display.y-40, display.width+20, 35); fill(10, 10, 20); textFont(titleFont, 15); textAlign(CENTER, BASELINE); text(title, display.midX, display.y - 25); if (subtitle) { textFont(labelFont, 12); text(subtitle, display.midX, display.y -10); } }; var draw = function() { takeRandomSample(population, popSize, popVar); drawScatterGraph(sampleStats,scatterGraph, true); // Only update display every 250 frames. if (frameCount % 250 === 0) { refresh = true; } if (refresh) { background(backgroundColour); drawBarGraph(population, popGraph); drawScatterGraph(sampleStats,scatterGraph, false); refresh = false; } drawBarGraph(sampleSizeVarPercent, sampleGraph); var statString = "N = " + popSize; statString += ", mean = " + round(10 * popMean) / 10; statString += ", variance = " + round(10 * popVar) / 10; drawTitle(scatterGraph, "Sample Statistics", "Lines indicate Pop. statistics"); drawTitle(popGraph, "Population Distribution", statString); drawTitle(sampleGraph, "Sample Size vs. Variance", sampleCount + " samples"); }; var mouseClicked = function(){ if (mode === "scatter" \|\| mode === "sample-size") { mode = "standard"; } else { if (mouseX > scatterGraph.x && mouseX < scatterGraph.x + scatterGraph.width && mouseY > scatterGraph.y) { mode = "scatter"; modeN = 0; } else if (mouseX > sampleGraph.x && mouseX < sampleGraph.x + scatterGraph.width && mouseY > sampleGraph.y) { mode = "sample-size"; modeN = 0; } } refresh = true; }; // By Peter Collingridge
Folds
Fork	false
Key	ag5zfmtoYW4tYWNhZGVteXIUCxIKU2NyYXRjaHBhZBjMn9CtBAw
Votes	1988
Spin-Offs	259
Type
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	true
By Child	false
Author Nick	Peter Collingridge
Author Name	peterwcollingridge
Author ID	kaid_1128810778687926889052316
Author Profile Access	COACH

Archive Data

Added	1 years ago, 1707881094551
Updated	1 years ago, 1707884110070
Source Deleted	unknown