2b:null 2c:["$","title",null,{"children":"Centrala gränsvärdessatsen | Mattias Villani | Observable"}] 31:T71e,viewof params = { var form; if (dist_type == 'beta'){ form = Inputs.form([ Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\text{form 1 }(\alpha)`}), Inputs.range([0.1, 20], {value: 5, step: 0.1, label: tex`\text{form 2 }(\beta)`}) ]) } else if (dist_type == 'uniform'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{undre gräns }(a)`}), Inputs.range([-5, 5], {value: 1, step: 0.1, label: tex`\text{övre gräns }(b)`}) ]) } else if (dist_type == 'studentt'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{lägesparameter }(\mu)`}), Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\text{skalparameter }(\sigma)`}), Inputs.range([1, 50], {value: 4, step: 1, label: tex`\text{frihetsgrader }(\nu)`}) ]) } else if (dist_type == 'chi2'){ form = Inputs.form([ Inputs.range([1, 50], {value: 3, step: 1, label: tex`\text{frihetsgrader }(\nu)`}) ]) } else if (dist_type == 'lognormal'){ form = Inputs.form([ Inputs.range([-5,5], {value: 0, step: 0.1, label: tex`\text{lägesparameter }(\mu)`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{skalparameter }(\sigma)`}) ]) } else if (dist_type == 'bimodal'){ form = Inputs.form([ Inputs.range([-5,5], {value: -2, step: 0.1, label: tex`\text{läge 1 }(\mu_1)`}), Inputs.range([-5,5], {value: 2, step: 0.1, label: tex`\text{läge 2 }(\mu_2)`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{skala 1 }(\sigma_1)`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{skala 2 }(\sigma_2)`}), Inputs.range([0,1], {value: 0.5, step: 0.01, label: tex`\text{vikt 1 }(w_1)`}) ]) } return form; }32:T45d,data = { const size = 10000; if (dist_type == 'beta'){ const alpha = params[0]; const beta = params[1]; return d3.range(size).map(d => ({x: randb(alpha, beta)})) } 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:[] 2f:{"id":"5b5361b51ccd4ad7","slug":"cgs","title":"Centrala gränsvärdessatsen","thumbnail":"1b2bf378e0771006028aac02f1af75c65af7b68bed99201cc9c6e9d1b0671c68","access_level":"public","create_time":"$D2023-01-03T13:25:56.681Z","update_time":"$D2023-07-24T09:13:54.262Z","view_time":null,"first_public_time":"$D2023-01-03T13:25:56.681Z","fork":{"id":"c48ea0d808158228","slug":"exploring-the-limits-of-the-central-limit-theorem","title":"Exploring the limits of the Central Limit Theorem","owner":{"id":"0148cd6d75e20b3f","login":"amauboussin","name":"Andrew Mauboussin"}},"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":1176,"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":1,"collection":{"id":"209e13808f44ed11","slug":"sda1","title":"SDA1","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Centrala gränsvärdessatsen","cells":[{"id":0,"value":"# Centrala gränsvärdessatsen\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":913,"value":"## Teori\nI statistisk inferens är det ofta viktigt att känna till **samplingfördelningen för medelvärdet** ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":917,"value":"> ${tex`\\bar X = \\frac{\\sum_{i=1}^n X_i}{n}`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":920,"value":"För oberoende observationer från en normalfördelning vet vi fördelningen för ${tex`\\bar X`}:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":925,"value":">

\nOm ${tex`X_1,X_2,\\ldots,X_n \\overset{\\operatorname{ober}}{\\sim}\\operatorname{N}(\\mu,\\sigma^2)`}\nså gäller att ${tex`\\bar X \\sim\\operatorname{N}\\Big(\\mu,\\frac{\\sigma^2}{n}\\Big)`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":948,"value":"Tyvärr har vi inte alltid liknande trevliga resultat när data kommer från andra fördelningar. Hur går man tillväga då? 😕","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":515,"value":"paragraph1 = md`Centrala gränsvärdessatsen beskriver **samplingfördelningen för medelvärdet** ${tex`\\bar X = \\frac{\\sum_{i=1}^n X_i}{n}`} av ett **stort** antal oberoende likafördelade observationer ${tex`X_1,X_2,\\ldots,X_n`} från en **godtycklig fördelning**^{[[1](#footnote1)]}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":719,"value":"En informell version av centrala gränsvärdessatsen säger:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":737,"value":">

Om ${tex`X_1,X_2,\\ldots,X_n`} är oberoende likafördelade observationer från en godtycklig fördelning^{[[1](#footnote1)]} så \n \n>

${tex`\\bar X \\overset{\\operatorname{approx}}{\\sim}\\operatorname{N}\\Big(\\mu,\\frac{\\sigma^2}{n}\\Big)\\text{ när }n\\text{ är stort}`}

","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":722,"value":"Centrala gränsvärdessatsen är därför ofta räddningen: *oavsett vilken fördelning data kommer från* vet vi att ${tex`\\bar X`} är ungefärligen normalfördelad om vi har tillräckligt mycket data. 😍","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1041,"value":">

**Tumregel**: vi bör ha ${tex`n\\geq30`} för att approximationen i centrala gränsvärdessatsen ska vara tillräckligt bra.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":729,"value":"Exakt *hur* bra approximationen är för ett givet ${tex`n`} beror på den ursprungliga fördelningen av observationerna ${tex`X_1,X_2,\\ldots,X_n`}. Låt oss undersöka detta för olika fördelningar och olika stickprovsstorlekar ${tex`n`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":745,"value":"## Interaktivt utforskande av teorin\n\nHistogrammet till vänster nedan visar simulerade data från den angivna fördelningen där fördelningens parametrar kan varieras med sliders. Prova genom att byta fördelningstyp och variera parametrarna i den givna fördelningen.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":568,"value":"Histogrammet till höger nedan visar stickprovsfördelningen av medelvärdet ${tex`\\bar X`} när man tar upprepade prover av storlek ${tex`n`} från den fördelning du valde. Effekten av centrala gränsvärdessatsen syns genom att fördelningen för ${tex`\\bar X`} blir mer och mer normalfördelad när ${tex`n`} ökar.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":287,"value":"viewof dist_type = Inputs.select(['uniform','studentt','chi2','lognormal','bimodal','beta'], {value: \"uniform\", label: \"Fördelning för data:\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":792,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":175,"value":"viewof sample_size = Inputs.range([1, 300], {value: 10, step: 1, label: tex`\\text{Stickprovstorlek, }n:`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"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","author":"$undefined","summary":"$undefined"},{"id":964,"value":"💪 Prova att välja student-*t* fördelning för data och ändra sen antal frihetgrader i *t*-fördelningen till 1. Detta grymma specialfall av *t*-fördelningen heter [Cauchyfördelningen](https://en.wikipedia.org/wiki/Cauchy_distribution) 😎 och är ett av de få exempel när centrala gränsvärdessatsen inte gäller 😰 ^{[[1](#footnote1)]}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":883,"value":"## Fotnoter","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":880,"value":"footnote1 = md`1. [^](#paragraph1) Det finns vissa krav på fördelningen, t ex att den inte får ha så mycket sannolikhetsmassa i svansarna att dess väntevärde och varians inte existerar. [Cauchyfördelningen](https://en.wikipedia.org/wiki/Cauchy_distribution) är ett exempel på en fördelning som **inte** uppfyller villkoren för centrala gränsvärdessatsen.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1013,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":765,"value":"### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":770,"value":"jStat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":876,"value":"footnote = (referencingCellName, number, text) => {\n return html`

^ ${text}

`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1128,"value":"ss = require('simple-statistics')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1078,"value":"### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":874,"value":"html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1029,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":760,"value":"### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":685,"value":"params_ = params","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":27,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1125,"value":"means = sample_finite_pop(data, sample_size, 1000)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":572,"value":"### Code","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1123,"value":"function sample_finite_pop(data, n, nRep){\n // Sample nRep samples of size n from the data and return the mean in each sample\n var samplemeans = [];\n for (let i = 0; i < nRep; ++i) {\n samplemeans.push({sample: i+1, mean: d3.mean(ss.sample(data, n).map(d => d.x))})\n }\n return samplemeans;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":183,"value":"gaussian_pdf = function(x, mean, sigma) {\n const gaussian_constant = 1 / Math.sqrt(2 * Math.PI);\n x = (x - mean) / sigma;\n return gaussian_constant * Math.exp(-.5 * x * x) / sigma;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":106,"value":"function randb(alpha, beta) {\n const u = randg(alpha);\n return u / (u + randg(beta));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":853,"value":"// Returns a gamma deviate by the method of Marsaglia and Tsang.\nfunction randg(shape) {\n let oalph = shape, a1, a2, u, v, x, mat;\n if (!shape) shape = 1;\n if (shape < 1) shape += 1;\n a1 = shape - 1 / 3;\n a2 = 1 / Math.sqrt(9 * a1);\n do {\n do {\n x = randn();\n v = 1 + a2 * x;\n } while (v <= 0);\n v = v * v * v;\n u = Math.random();\n } while (\n u > 1 - 0.331 * Math.pow(x, 4) &&\n Math.log(u) > 0.5 * x * x + a1 * (1 - v + Math.log(v))\n );\n if (shape === oalph) return a1 * v; // alpha > 1\n do u = Math.random(); while (u === 0); // alpha < 1\n return Math.pow(u, 1 / oalph) * a1 * v;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":98,"value":"// Returns a normal deviate (mu=0, sigma=1).\nfunction randn() {\n let u, v, x, y, q;\n do {\n u = Math.random();\n v = 1.7156 * (Math.random() - 0.5);\n x = u - 0.449871;\n y = Math.abs(v) + 0.386595;\n q = x * x + y * (0.19600 * y - 0.25472 * x);\n } while (q > 0.27597 && (q > 0.27846 || v * v > -4 * Math.log(u) * u * u));\n return v / u;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":782,"value":"function rand_studentt(location, scale, df){\n return location + scale * jStat.studentt.sample(df)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":842,"value":"function rand_chi2(df){\n return jStat.chisquare.sample(df)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":859,"value":"function rand_lognormal(mu, sigma){\n return jStat.lognormal.sample(mu, sigma)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":978,"value":"function rand_bimodal(mu1,mu2,sigma1,sigma2,vikt){\n const comp = Math.random() < vikt\n return comp ? jStat.normal.sample(mu1,sigma1) : jStat.normal.sample(mu2,sigma2)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":62,"value":"d3 = require(\"https://d3js.org/d3.v5.min.js\")\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":79,"value":"\nheight = 250\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":86,"value":"width = 800","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":72,"value":"margin = ({top: 20, right: 20, bottom: 30, left: 40})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1164,"value":"#### Detached plots","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":346,"value":"pltdata = Plot.plot({\n caption:html`ett simulerat datamaterial från populationen. `,\n marks: [\n Plot.rectY(data, Plot.binX({y: \"count\"}, {x: \"x\", fill: \"steelblue\"})),\n Plot.ruleY([0])\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":124,"value":"pltmeans = Plot.plot({\n caption:html`fördelningen av stickprovsmedelvärdena från dataset med n=${sample_size} observationer. `,\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: \"orange\"})),\n Plot.ruleY([0])\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"