2c:null 2d:["$","title",null,{"children":"Hur medelvärde och median påverkas av extrema observationer | Mattias Villani | Observable"}] 32:T44e,# Hur medelvärde och median påverkas av extrema observationer **Medelvärdet** ${tex`\bar x = \frac{\sum_{i=1}^n x_i}{n}`} och **medianen** (det mittersta värdet i datamaterialet efter att man ordnat observationer i storleksordning) är två vanliga **lägesmått** för ett datamaterial, dvs de försöker sammanfatta "mitten" av datamaterialets fördelning. Medelvärdet av ett datamaterial kan vara känsligt till extrema värden i datamaterialet, medan medianen inte alls påverkas av de mest extrema värdet. Om man t ex vill sammanfatta inkomstfördelningen i ett land måste fundera på om man verkligen vill att de extremrika personerna ska få påverka ditt mått på "genomsnittlig" inkomst. Du kan undersöka skillnaden mellan medelvärde och median genom att ändra värdet på det största värdet i datamaterialet. Dra i slidern nedan så multipliceras det största värdet i datamaterialet (markerat med en orange cirkel) med det tal du valt med slidern. Undersök också om effekten beror på hur stort datamaterialet är, dvs hur många observationer som ingår i datamaterialet.33:T541,distdata = { if (dist_type == 'symmetrisk'){ const draws = d3.range(size).map(d => jstat.normal.sample( 0, 1 )) const pdfdata = d3.range(jstat.normal.inv( 0.001, 0, 1 ), jstat.normal.inv( 0.999, 0, 1 ) ,0.01).map(d => ({x: d, pdf: jstat.normal.pdf(d, 0, 1)})) return [draws, pdfdata, 0, 0, 0] } else if (dist_type == 'skev'){ const mu = 0; const sigma = 1; const draws = d3.range(size).map(d => jstat.lognormal.sample(mu, sigma)) const pdfdata = d3.range(0, jstat.lognormal.inv( 0.99, mu, sigma), 0.01).map(d => ({x: d, pdf: jstat.lognormal.pdf(d, mu, sigma)})) return [draws, pdfdata, 1/2, Math.log(2)/2, 0] } else if (dist_type == 'tunga svansar'){ const df = 4 const draws = d3.range(size).map(d => jstat.studentt.sample( df )) const pdfdata = d3.range(jstat.studentt.inv( 0.001, df), jstat.studentt.inv( 0.999, df), 0.01).map(d => ({x: d, pdf: jstat.studentt.pdf(d, df)})) return [draws, pdfdata, 0, 0, 0] } else if (dist_type == 'bimodal'){ var draws = d3.range(size/2).map(d => jstat.normal.sample( -1, 0.5)) draws = draws.concat(d3.range(size/2).map(d => jstat.normal.sample( 1, 0.2 ))) const pdfdata = d3.range(-3, 3 ,0.01).map(d => ({x: d, pdf: 0.5*jstat.normal.pdf(d, -1, 0.5) + 0.5*jstat.normal.pdf(d, 1, 0.2)})) return [draws, pdfdata, 0, 0, 0] } }34:[] 30:{"id":"b40caa0d2255e34b","slug":"location-measures-outliers","title":"Hur medelvärde och median påverkas av extrema observationer","thumbnail":"8f58513e8f13f37faa147cabd6d75072ec44f8d614df68eb0dda2dbab6730087","access_level":"public","create_time":"$D2023-08-13T11:17:33.695Z","update_time":"$D2023-12-22T09:32:43.150Z","view_time":null,"first_public_time":"$D2023-08-13T11:17:33.695Z","fork":{"id":"9489415a5c3d1f6e","slug":"samplingfordeling-for-andlig-population","title":"Samplingfördelning - leksakspopulation","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":1690,"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":"Hur medelvärde och median påverkas av extrema observationer","cells":[{"id":38,"value":"$32","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof N = Inputs.range([2, 100], {value: 10, step: 1, label: \"antal datapunkter\"})\n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1251,"value":"viewof maxscalefact = Inputs.range([1, 3], {value: 1, label: \"Multiplicera största datavärdet med\", step: 0.01})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1295,"value":"Plot.plot({\n color: {legend: true},\n x: {\n domain: [0, 300]\n },\n height: 100,\n marks: [\n Plot.dotX(datascaled, {filter: d => d.type == \"datapunkter\" || d.type == \"maxvärde\", x: \"data\", stroke: \"type\", strokeWidth: \"width\", symbol: \"symbol\", r: \"radius\"}),\n Plot.tickX(datascaled, {filter: d => d.type == \"medelvärde\" || d.type == \"median\", x: \"data\", stroke: \"type\", strokeWidth: \"width\", symbol: \"symbol\"}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1371,"value":"Plot.plot({\n x: {\n domain: [0, 300]\n },\n height: 200,\n marks: [\n Plot.rectY(datascaled, Plot.binX({y: \"count\"}, {x: \"data\", fill: d3.schemeTableau10[0]})),\n Plot.ruleX(datascaled.filter(d => d.type == \"maxvärde\").map(d => d.data), {stroke: d3.schemeTableau10[1], strokeWidth: 3}),\n Plot.ruleX(datascaled.filter(d => d.type == \"medelvärde\").map(d => d.data), {stroke: d3.schemeTableau10[2], strokeWidth: 3}),\n Plot.ruleX(datascaled.filter(d => d.type == \"median\").map(d => d.data), {stroke: d3.schemeTableau10[3], strokeWidth: 3}),\n Plot.ruleY([0])\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1668,"value":"## Lägesmått för olika datafördelningar\n\nMedelvärde och median kan bli mer eller mindre olika beroende på hur fördelningen av data ser ut. Prova dessa olika fördelningar nedan:\n- symmetrisk (data följer den klassiska klockkurvan)\n- skev (data är positiva värden, med några stora värden i \"högersvansen\")\n- bimodal (data är uppdelat i två \"pucklar\")\n\nNotera varken medelvärde eller median är en speciell bra sammanfattning av en bimodala data. En sådan fördelning är helt enkelt svår att sammanfatta med ett enda lägesmått.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1456,"value":"viewof param = Inputs.form([\n Inputs.select(['symmetrisk','skev','bimodal'], {value: \"symmetrisk\", label: \"fördelning för data:\"}),\n Inputs.range([2, 100], {value: 30, step: 1, label: \"antal datapunkter\"}),\n Inputs.toggle({label: \"Visa teoretisk fördelning\", value: false})\n])\n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1501,"value":"Plot.plot({\n caption: html` \n medelvärde
\n median
\n `,\n y: {\n transform: f => binwidth*f \n },\n marks: [\n Plot.rectY(draws, Plot.binX({y: \"proportion\"}, {x: \"draw\", fill: \"lightsteelblue\", thresholds: \"sturges\"})),\n Plot.line(pdfdata, {filter: param[2], x: \"x\", y: (d) => d.pdf*binwidth, stroke: \"orange\", strokeWidth: 3}),\n Plot.ruleY([0]),\n Plot.ruleX([d3.mean(draws.map(d => d.draw))], {stroke: d3.schemeTableau10[2], strokeWidth: 3}),\n Plot.ruleX([d3.median(draws.map(d => d.draw))], {stroke: d3.schemeTableau10[3], strokeWidth: 3})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1128,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":888,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1028,"value":"pop = d3.sort(d3.range(N).map(d => Math.round(jstat.normal.sample( 50, 10 ).toFixed(2))))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1325,"value":"datascaled = {\n var tmp;\n tmp = pop.map(d => ({data: d, type: \"datapunkter\", width: 1, symbol: \"circle\", radius: 1}))\n tmp[tmp.length - 1] = {data: d3.max(pop)*maxscalefact, type: \"maxvärde\", width: 3, symbol: \"circle\", radius: 1}\n const meanval = d3.mean(tmp.map(d => d.data))\n const medianval = d3.median(tmp.map(d => d.data))\n tmp.push(({data: meanval, type: \"medelvärde\", width: 3, symbol: \"diamond\", radius: 3}))\n tmp.push(({data: medianval, type: \"median\", width: 3, symbol: \"square\", radius: 3}))\n return tmp\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1487,"value":"dist_type = param[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1489,"value":"size = param[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1464,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1499,"value":"draws = distdata[0].map(d => ({draw: d}))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1497,"value":"pdfdata = distdata[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1585,"value":"distdata[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1555,"value":"binwidth = {\n const histbinsdata = Plot.rectY(\n draws,\n Plot.binX({ y: \"proportion\" }, { x: \"draw\", thresholds: \"sturges\" })\n ).initialize()\n return histbinsdata.channels.x2.value[0] - histbinsdata.channels.x1.value[0]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W34","stdlib":"1","titling":"auto"}} 31:"$30:body"