2b:null 2c:["$","title",null,{"children":"Exploring the limits of the Central Limit Theorem | Andrew Mauboussin | Observable"}] 32:T41f,md`The central limit theory tells us that when we draw repeated samples from a given population, the means of those samples will be approximately normally distributed. It's one of the most important results in statistics because it forms the basis for the assumptions that make hypothesis testing and confidence intervals calculations possible ([full explanation here](https://www.methodsconsultants.com/tutorial/the-central-limit-theorem-and-its-implications-for-statistical-inference/)). Remarkably, the theorem applies to any population distribution with a finite mean and variance. However, it's usually unclear how big the sample size needs to be for it to take effect. Imagine you have a population distribution with a big right skew. A sample size of one clearly won't work, since the distribution of sample means will be exactly the same as the population distribution and have the same skew. But what about a sample size of 10? or 20? It's not obvious exactly when the central limit theorem is approximately correct and when it breaks down. ` 33:Tc21,viewof params = { var form; if (dist_type.value == 'beta'){ form = html``; form.value = [form.alpha, form.beta].map(i => i.valueAsNumber); form.alpha.oninput = () => form.value[0] = form.alpha.valueAsNumber; form.beta.oninput = () => form.value[1] = form.beta.valueAsNumber; } else if (dist_type.value == 'uniform'){ form = html``; form.value = [form.min, form.max].map(i => i.valueAsNumber); form.min.oninput = () => form.value[0] = form.min.valueAsNumber; form.max.oninput = () => form.value[1] = form.max.valueAsNumber; } else if (dist_type.value == 'bimodal'){ form = html``; form.value = [form.diff, form.prop].map(i => i.valueAsNumber); form.diff.oninput = () => form.value[0] = form.diff.valueAsNumber; form.prop.oninput = () => form.value[1] = form.prop.valueAsNumber; } else if (dist_type.value == 'gumbel'){ form = html``; form.value = [form.mean, form.beta].map(i => i.valueAsNumber); form.mean.oninput = () => form.value[0] = form.mean.valueAsNumber; form.beta.oninput = () => form.value[1] = form.beta.valueAsNumber; } else if (dist_type.value == 'power'){ form = html``; form.value = [form.range, form.exponent].map(i => i.valueAsNumber); form.range.oninput = () => form.value[0] = form.range.valueAsNumber; form.exponent.oninput = () => form.value[1] = form.exponent.valueAsNumber; } else if (dist_type.value == 'cauchy'){ form = html``; form.value = [form.mean, form.variance].map(i => i.valueAsNumber); form.mean.oninput = () => form.value[0] = form.mean.valueAsNumber; form.variance.oninput = () => form.value[1] = form.variance.valueAsNumber; } return form; }34:T41b,data = { const size = 10000; if (dist_type.value == 'beta'){ const alpha = params[0]; const beta = params[1]; return new Array(size).fill().map(() => randb(alpha, beta)) } else if (dist_type.value == 'uniform'){ const min = params[0]; const max = params[1]; return new Array(size).fill().map(() => (Math.random() * (max-min) + min)) } else if (dist_type.value == 'bimodal'){ const diff = params[0]; const prop = params[1]; return new Array(size).fill().map(() => bimodal(diff, prop)) } else if (dist_type.value == 'gumbel'){ const u = params[0]; const beta = params[1]; return new Array(size).fill().map(() => rand_gumbel(u, beta)) } else if (dist_type.value == 'cauchy'){ const mean = params[0]; const variance = params[1]; return new Array(size).fill().map(() => rand_cauchy(mean, variance)) } else if (dist_type.value == 'power'){ const range = params[0]; const exp = params[1]; return new Array(size).fill().map(() => rand_power(range, -exp)) } }35:T533,function bar_chart(data){ const svg = d3.select(DOM.svg(width, height)); const x = d3.scaleLinear().domain(d3.extent(data)).nice().range([margin.left, width - margin.right]) const bins = d3.histogram() .domain(x.domain()) .thresholds(x.ticks(50)) (data) const xAxis = g => g .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x).tickSizeOuter(0)) .call(g => g.append("text") .attr("x", width - margin.right) .attr("y", -4) .attr("fill", "#000") .attr("font-weight", "bold") .attr("text-anchor", "end") .text(data.x)) const y = d3.scaleLinear() .domain([0, d3.max(bins, d => d.length)]).nice() .range([height - margin.bottom, margin.top]) const yAxis = g => g .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)) .call(g => g.select(".domain").remove()) const bar = svg.append("g") .attr("fill", "steelblue") .selectAll("rect") .data(bins) .enter().append("rect") .attr("x", d => x(d.x0) + 1) .attr("width", d => x(d.x1) - x(d.x0) - 1) .attr("y", d => y(d.length)) .attr("height", d => y(0) - y(d.length)); svg.append("g") .call(xAxis); svg.append("g") .call(yAxis); return svg.node(); }36:T8d1,function gaussian_bar_chart(sample_means){ const svg = d3.select(DOM.svg(width, height)); const mean = d3.mean(data) const std = (d3.deviation(data) ** 2. / sample_size) ** .5; const lower_bound = d3.min([mean - 3 * std, d3.min(sample_means)]) - .05 const upper_bound = d3.max([mean + 3 * std, d3.max(sample_means)]) + .05 const x = d3.scaleLinear().domain([lower_bound, upper_bound]).nice().range([margin.left, width - margin.right]) const bins = d3.histogram() .domain(x.domain()) .thresholds(x.ticks(60)) (sample_means) const xAxis = g => g .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x).tickSizeOuter(0)) .call(g => g.append("text") .attr("x", width - margin.right) .attr("y", -4) .attr("fill", "#000") .attr("font-weight", "bold") .attr("text-anchor", "end") .text(sample_means.x)) const y = d3.scaleLinear() .domain([0, d3.max(bins, d => d.length)]).nice() .range([height - margin.bottom, margin.top]) const yAxis = g => g .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)) .call(g => g.select(".domain").remove()) const line_data= new Array(1000).fill().map(function (d, i) { return {'x' : lower_bound + (upper_bound - lower_bound) * i/1000, 'y':sample_means.length * (bins[1].x1 - bins[1].x0) * gaussian_pdf(lower_bound + (upper_bound - lower_bound) * i/1000, mean, std)};}) const line = d3.line() .x(d => x(d.x)) .y(d => y(d.y)) const bar = svg.append("g") .attr("fill", "steelblue") .selectAll("rect") .data(bins) .enter().append("rect") .attr("x", d => x(d.x0) + 1) .attr("width", d => x(d.x1) - x(d.x0) - 1 > 0 ? x(d.x1) - x(d.x0) - 1 : 0) .attr("y", d => y(d.length))// * (d.x1 - d.x0))) .attr("height", d => y(0) - y(d.length)); svg.append("path") .datum(line_data) .attr("fill", "none") .attr("stroke", "red") .attr("stroke-width", 1.5) .attr("stroke-linejoin", "round") .attr("stroke-linecap", "round") .attr("d", line); svg.append("g") .call(xAxis); svg.append("g") .call(yAxis); return svg.node(); }37:[] 2f:{"id":"c48ea0d808158228","slug":"exploring-the-limits-of-the-central-limit-theorem","title":"Exploring the limits of the Central Limit Theorem","thumbnail":"df7370bc2349c4bb2e08c938387b67d1d7c839fa42e471d2d41ff8fe2adb54e9","access_level":"public","create_time":"$D2018-02-25T21:34:19.517Z","update_time":"$D2023-01-23T23:25:09.733Z","view_time":null,"first_public_time":"$D2018-03-04T22:46:14.346Z","fork":false,"stars":3,"starred":false,"owner":{"type":"team","id":"0148cd6d75e20b3f","login":"amauboussin","name":"Andrew Mauboussin","avatar_url":"https://avatars-next.observableusercontent.com/280c5eb6ea02ac5f496252a8f59e62c9267abd76bb6a6153b3f05bbd480f6be2","tier":"starter_2024"},"version":704,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"858238618d174455","login":"amauboussin","name":"Andrew Mauboussin","avatar_url":"https://avatars-next.observableusercontent.com/280c5eb6ea02ac5f496252a8f59e62c9267abd76bb6a6153b3f05bbd480f6be2","tier":null}],"members":[{"type":"individual","id":"858238618d174455","login":"amauboussin","name":"Andrew Mauboussin","avatar_url":"https://avatars-next.observableusercontent.com/280c5eb6ea02ac5f496252a8f59e62c9267abd76bb6a6153b3f05bbd480f6be2","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":6,"collection":null,"type":"notebook","body":{"title":"Exploring the limits of the Central Limit Theorem","cells":[{"id":0,"value":"md`# Exploring the limits of the Central Limit Theorem`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":16,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":515,"value":"md`\nIn this notebook, you can see how well the central limit theorem works with different population distributions and sample sizes. First, select the shape and parameters for the population distribution. \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":287,"value":"viewof dist_type = {\n const form = html``;\n form.value = form.dist;\n\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":280,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":346,"value":"population_distribution = {\n \treturn bar_chart(data)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":568,"value":"md`Then you can adjust the sample size and see how the distribution of sample means changes.\nThe standard rule of thumb given in most textbooks is that the central limit theorem tends to work once you have a sample size of around 30.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":175,"value":"viewof sample_size_slider = {\n var form = html``\n form.value = [form.n].map(i => i.valueAsNumber);\n form.n.oninput = () => form.value[0] = form.n.valueAsNumber;\n\treturn form\n }","pinned":false,"hidden":false,"mode":"ojs"},{"id":351,"value":"sample_size = Math.ceil(sample_size_slider)","pinned":false,"hidden":false,"mode":"ojs"},{"id":124,"value":"\n\ndistribution_of_sample_means = {\n \treturn gaussian_bar_chart(means)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":449,"value":"md`Next, we take 1000 samples of size \"sample_size\" and plot the sample means. The red line outlines a gaussian distribution with the mean and standard deviation predicted by the central limit theorem (mean = population mean and variance = population variance / sample_size). If the histogram roughly matches the red line, then the sample size is large enough for central limit theorem to work. All of the distributions will eventually converge except the Cauchy, which has infinite variance.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":453,"value":"md`This notebook is adapted from Mike Bostock's [beta distributed random variable notebook](https://beta.observablehq.com/@mbostock/beta-distributed-random-values).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":685,"value":"params_ = params","pinned":false,"hidden":false,"mode":"ojs"},{"id":27,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":277,"value":"population_mean = d3.mean(data)","pinned":false,"hidden":false,"mode":"ojs"},{"id":458,"value":"population_variance = d3.deviation(data) ** 2","pinned":false,"hidden":false,"mode":"ojs"},{"id":128,"value":"means = new Array(1000).fill().map(() => d3.mean(d3.shuffle(data).slice(0, sample_size))) ","pinned":false,"hidden":false,"mode":"ojs"},{"id":572,"value":"md`## Code`","pinned":false,"hidden":false,"mode":"ojs"},{"id":132,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":192,"value":"$36","pinned":false,"hidden":false,"mode":"ojs"},{"id":185,"value":"function bimodal(diff, prop){\n \tif (Math.random() < prop){\n return randn();\n }\n \telse{\n return diff + randn();\n }\n \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":183,"value":"gaussian_pdf = function(x, mean, sigma) {\n const gaussian_constant = 1 / Math.sqrt(2 * Math.PI);\n x = (x - mean) / sigma;\n return gaussian_constant * Math.exp(-.5 * x * x) / sigma;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":106,"value":"function randb(alpha, beta) {\n const u = randg(alpha);\n return u / (u + randg(beta));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":98,"value":"// Returns a normal deviate (mu=0, sigma=1).\nfunction randn() {\n let u, v, x, y, q;\n do {\n u = Math.random();\n v = 1.7156 * (Math.random() - 0.5);\n x = u - 0.449871;\n y = Math.abs(v) + 0.386595;\n q = x * x + y * (0.19600 * y - 0.25472 * x);\n } while (q > 0.27597 && (q > 0.27846 || v * v > -4 * Math.log(u) * u * u));\n return v / u;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"// Returns a gamma deviate by the method of Marsaglia and Tsang.\nfunction randg(shape) {\n let oalph = shape, a1, a2, u, v, x, mat;\n if (!shape) shape = 1;\n if (shape < 1) shape += 1;\n a1 = shape - 1 / 3;\n a2 = 1 / Math.sqrt(9 * a1);\n do {\n do {\n x = randn();\n v = 1 + a2 * x;\n } while (v <= 0);\n v = v * v * v;\n u = Math.random();\n } while (\n u > 1 - 0.331 * Math.pow(x, 4) &&\n Math.log(u) > 0.5 * x * x + a1 * (1 - v + Math.log(v))\n );\n if (shape === oalph) return a1 * v; // alpha > 1\n do u = Math.random(); while (u === 0); // alpha < 1\n return Math.pow(u, 1 / oalph) * a1 * v;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":534,"value":"function rand_gumbel(u, b){\n return u - b * Math.log(-Math.log(Math.random()));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":607,"value":"function rand_power(range, n){\n let x0 = .1;\n let x1 = x0 + range;\n\n return Math.pow((Math.pow(x1, n+1) - Math.pow(x0, n+1) * Math.random() + Math.pow(x0, n+1)), 1/(n + 1.))\n //x = [(x1^(n+1) - x0^(n+1))*y + x0^(n+1)]^(1/(n+1)) \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":582,"value":"function rand_cauchy(mean, variance){\n return mean + variance * Math.tan((Math.random() - .5) * Math.PI)\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":62,"value":"d3 = require(\"https://d3js.org/d3.v5.min.js\")\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":79,"value":"\nheight = 250\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":86,"value":"width = 800","pinned":false,"hidden":false,"mode":"ojs"},{"id":704,"value":"","pinned":true,"hidden":false,"mode":"ojs"},{"id":703,"value":"null","pinned":true,"hidden":false,"mode":"ojs"},{"id":72,"value":"margin = ({top: 20, right: 20, bottom: 30, left: 40})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W37","stdlib":"1","titling":"auto"}} 30:"$2f:body"