2a:null 2b:["$","title",null,{"children":"Central limit theorem | Mattias Villani | Observable"}] 31:T97e,viewof params = { var form; if (dist_type == 'binomial'){ form = Inputs.form([ Inputs.range([1, 30], {value: 10, step: 1, label: tex`\text{number of trials }n`}), Inputs.range([Number.EPSILON, 1 - Number.EPSILON], {value: 0.3, step: 0.01, label: tex`\text{success prob }p`}) ]) } if (dist_type == 'poisson'){ form = Inputs.form([ Inputs.range([0, 10], {value: 1, step: 0.1, label: tex`\text{mean }\lambda`}) ]) } if (dist_type == 'beta'){ form = Inputs.form([ Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\text{shape }\alpha`}), Inputs.range([0.1, 20], {value: 5, step: 0.1, label: tex`\text{shape }\beta`}) ]) } if (dist_type == 'cauchy'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{location }m`}), Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\text{scale }\gamma`}), ]) } else if (dist_type == 'uniform'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{lower limit }a`}), Inputs.range([-5, 5], {value: 1, step: 0.1, label: tex`\text{upper limit }b`}) ]) } else if (dist_type == 'studentt'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}), Inputs.range([1, 50], {value: 4, step: 1, label: tex`\text{degrees of freedom }\nu`}) ]) } else if (dist_type == 'chi2'){ form = Inputs.form([ Inputs.range([1, 50], {value: 3, step: 1, label: tex`\text{degrees of freedom }\nu`}) ]) } else if (dist_type == 'lognormal'){ form = Inputs.form([ Inputs.range([-5,5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}) ]) } else if (dist_type == 'bimodal'){ form = Inputs.form([ Inputs.range([-5,5], {value: -2, step: 0.1, label: tex`\text{location }\mu_1`}), Inputs.range([-5,5], {value: 2, step: 0.1, label: tex`\text{location }\mu_2`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma_1`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma_2`}), Inputs.range([0,1], {value: 0.5, step: 0.01, label: tex`\text{weight }w_1`}) ]) } return form; }32:T63c,data = { const size = 10000; if (dist_type == 'binomial'){ const n = params[0]; const p = params[1]; return d3.range(size).map(d => ({x: binomialSample(n, p)})) } if (dist_type == 'poisson'){ const lambda = params[0]; return d3.range(size).map(d => ({x: jStat.poisson.sample(lambda)})) } if (dist_type == 'beta'){ const alpha = params[0]; const beta = params[1]; return d3.range(size).map(d => ({x: randb(alpha, beta)})) } if (dist_type == 'cauchy'){ const location = params[0]; const scale = params[1]; const df = 1; return d3.range(size).map(d => ({x: rand_studentt(location, scale, df)})) } else if (dist_type == 'uniform'){ const min = params[0]; const max = params[1]; return d3.range(size).map(d => ({x: Math.random() * (max-min) + min})) } else if (dist_type == 'studentt'){ const location = params[0]; const scale = params[1]; const df = params[2]; return d3.range(size).map(d => ({x: rand_studentt(location, scale, df)})) } else if (dist_type == 'chi2'){ const df = params[0]; return d3.range(size).map(d => ({x: rand_chi2(df)})) } else if (dist_type == 'lognormal'){ const mu = params[0]; const sigma = params[1]; return d3.range(size).map(d => ({x: rand_lognormal(mu, sigma)})) } else if (dist_type == 'bimodal'){ const mu1 = params[0]; const mu2 = params[1]; const sigma1 = params[2]; const sigma2 = params[3]; const vikt = params[4]; return d3.range(size).map(d => ({x: rand_bimodal(mu1, mu2, sigma1, sigma1, vikt)})) } }33:[] 2e:{"id":"82a1b81f0bb3c541","slug":"central-limit-theorem","title":"Central limit theorem","thumbnail":"6f30924cee9b9ef1d0c9a30def997d264a69bf0dfa53bfb870735a3c9c057d70","access_level":"public","create_time":"$D2023-07-24T08:28:49.311Z","update_time":"$D2025-04-12T15:09:15.509Z","view_time":null,"first_public_time":"$D2023-07-24T08:28:49.311Z","fork":{"id":"5b5361b51ccd4ad7","slug":"cgs","title":"Centrala gränsvärdessatsen","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":1366,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"subscribed":null,"forks":3,"collection":{"id":"9fdb7c9ba7be6ba8","slug":"bayesian-learning","title":"Bayesian Learning","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Central limit theorem","cells":[{"id":0,"value":"# Central limit theorem\n","pinned":false,"hidden":false,"mode":"md"},{"id":913,"value":"## Theory\nIn statistical inference, it is often important to know the **sampling distribution of the sample mean**\n\n${tex`\\bar X_n = \\frac{\\sum_{i=1}^n X_i}{n}`}\n\nWe use a subscript *n* on ${tex`\\bar X_n`} to explicitly show that the sample mean is based on *n* observations. This is useful here since we will now investigate what happens to the sample mean as we increase ${tex`n`}.","pinned":false,"hidden":false,"mode":"md"},{"id":920,"value":"For independent observations from a **normal** population, we know the distribution for ${tex`\\bar X_n`}:","pinned":false,"hidden":false,"mode":"md"},{"id":925,"value":">

\nIf ${tex`X_1,X_2,\\ldots,X_n \\overset{\\operatorname{iid}}{\\sim}\\operatorname{N}(\\mu,\\sigma^2)`}\nthen ${tex`\\bar X_n \\sim\\operatorname{N}\\Big(\\mu,\\frac{\\sigma^2}{n}\\Big)`}","pinned":false,"hidden":false,"mode":"md"},{"id":948,"value":"Unfortunately, we do not always have such nice results when data come from other distributions. What do we do then? 😕","pinned":false,"hidden":false,"mode":"md"},{"id":515,"value":"paragraph1 = md`The central limit theorem describes the **sampling distribution of the mean** ${tex`\\bar X_n`} for a **large** number of independent identically distributed observations ${tex`X_1,X_2,\\ldots,X_n`} from an **arbitrary distribution**^{[[1](#footnote1)]}.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":719,"value":"An informal version of the central limit theorem says:","pinned":false,"hidden":false,"mode":"md"},{"id":737,"value":">

If ${tex`X_1,X_2,\\ldots,X_n`} are independent identically distributed observations from an arbitrary distribution^{[[1](#footnote1)]} then \n\n>

${tex`\\bar X_n \\overset{\\operatorname{approx}}{\\sim}\\operatorname{N}\\Big(\\mu,\\frac{\\sigma^2}{n}\\Big)\\text{ when }n\\text{ is large}`}

","pinned":false,"hidden":false,"mode":"md"},{"id":722,"value":"Therefore, the central limit theorem is often our rescue: *no matter which distribution the data comes from*, we know that ${tex`\\bar X`} is approximately normally distributed if we have enough data. 😍","pinned":false,"hidden":false,"mode":"md"},{"id":1041,"value":">

**Rule of thumb**: we should have ${tex`n\\geq30`} for the approximation in the central limit theorem to be good enough.","pinned":false,"hidden":false,"mode":"md"},{"id":729,"value":"## Interactive exploration\nExactly *how* good the approximation is for a given ${tex`n`} depends on the original distribution of the observations ${tex`X_1,X_2,\\ldots,X_n`}. Let's investigate this for different distributions and different sample sizes ${tex`n`}.","pinned":false,"hidden":false,"mode":"md"},{"id":745,"value":"The histogram to the left below shows simulated data from the specified distribution where the distribution parameters can be varied with sliders. Try by switching the distribution type and varying the parameters in the given distribution.","pinned":false,"hidden":false,"mode":"md"},{"id":568,"value":"The histogram to the right below shows the sampling distribution of the mean ${tex`\\bar X_n`} when taking repeated samples of size ${tex`n`} from the distribution that you selected. The effect of the central limit theorem is seen by the distribution for ${tex`\\bar X_n`} becoming more and more normally distributed as ${tex`n`} increases.","pinned":false,"hidden":false,"mode":"md"},{"id":287,"value":"viewof dist_type = Inputs.select(['beta', 'bimodal', 'binomial', 'cauchy', 'chi2', 'lognormal', 'poisson', 'studentt', 'uniform',], {value: \"uniform\", label: \"data distribution:\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":792,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":175,"value":"viewof sample_size = Inputs.range([1, 300], {value: 2, step: 1, label: tex`\\text{sample size }n:`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1081,"value":"plt = html`\n \n \n \n

\n ${pltdata}\n

\n \n

\n ${pltmeans}\n

\n\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":964,"value":"Try selecting the [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution) 😎 as data distribution. This nasty distribution is one of the few examples where the Central Limit Theorem does not apply. 😰 ^{[[1](#footnote1)]}.","pinned":false,"hidden":false,"mode":"md"},{"id":883,"value":"## Footnotes","pinned":false,"hidden":false,"mode":"md"},{"id":880,"value":"footnote1 = md`1. [^](#paragraph1) There are certain requirements on the distribution, such as that it should not have so much probability mass in the tails that its expected value and variance do not exist. The [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution) is an example of a distribution that **does not** satisfy the conditions for the Central Limit Theorem.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1013,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and implementation details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":765,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":1342,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":770,"value":"jStat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":876,"value":"footnote = (referencingCellName, number, text) => {\n return html`

^ ${text}

`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1128,"value":"ss = require('simple-statistics')","pinned":false,"hidden":false,"mode":"ojs"},{"id":1078,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":874,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1029,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":760,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1313,"value":"nRep = 5000","pinned":false,"hidden":false,"mode":"ojs"},{"id":685,"value":"params_ = params","pinned":false,"hidden":false,"mode":"ojs"},{"id":27,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":1125,"value":"means = sample_finite_pop(data, sample_size, nRep)","pinned":false,"hidden":false,"mode":"ojs"},{"id":572,"value":"#### Code","pinned":false,"hidden":false,"mode":"md"},{"id":1245,"value":"discrete = ([\"poisson\", \"binomial\"].includes(dist_type))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1123,"value":"function sample_finite_pop(data, n, nRep){\n // Sample nRep samples of size n from the data and return the mean in each sample\n var samplemeans = [];\n for (let i = 0; i < nRep; ++i) {\n samplemeans.push({sample: i+1, mean: d3.mean(ss.sample(data, n).map(d => d.x))})\n }\n return samplemeans;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1317,"value":"function sample_finite_pop_prealloc(data, n, nRep){\n // Sample nRep samples of size n from the data and return the mean in each sample\n var samplemeans = new Array(nRep);\n for (let i = 0; i < nRep; ++i) {\n samplemeans[i] = ({sample: i+1, mean: d3.mean(ss.sample(data, n).map(d => d.x))})\n }\n return samplemeans;\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":853,"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":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":782,"value":"function rand_studentt(location, scale, df){\n return location + scale * jStat.studentt.sample(df)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":842,"value":"function rand_chi2(df){\n return jStat.chisquare.sample(df)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":859,"value":"function rand_lognormal(mu, sigma){\n return jStat.lognormal.sample(mu, sigma)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":978,"value":"function rand_bimodal(mu1,mu2,sigma1,sigma2,vikt){\n const comp = Math.random() < vikt\n return comp ? jStat.normal.sample(mu1,sigma1) : jStat.normal.sample(mu2,sigma2)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1269,"value":"function binomialSample(n,p){\n return d3.sum(d3.range(n).map(d => Math.random()a single dataset simulated from the data distribution. `,\n marks: [\n Plot.barY(data, Plot.groupX({y: \"count\"}, {x: \"x\", fill: \"colors[0]\"})),\n Plot.ruleY([0])\n ]\n }) \n }\n else {\n \n return Plot.plot({\n caption:html`a single dataset simulated from the data distribution. `,\n marks: [\n Plot.rectY(data.filter(d => Math.abs(d.x) <=50), Plot.binX({y: \"count\"}, {x: \"x\", fill: colors[0]})),\n Plot.ruleY([0])\n ]\n })\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":124,"value":"pltmeans = Plot.plot({\n caption:html`distribution of the sample mean from samples with n=${sample_size} observations. `,\n x: {\n label: \"sample means\"\n },\n y:{\n axis: false\n },\n marks: [\n Plot.rectY(means, Plot.binX({y: \"count\"}, {x: \"mean\", fill: colors[8]})),\n //Plot.areaY(means, Plot.binX({y: \"count\", filter: null}, {x: \"mean\", fillOpacity: 0.2})),\n //Plot.lineY(means, Plot.binX({y: \"count\", filter: null}, {x: \"mean\"})),\n Plot.ruleY([0])\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"