32:T44c,# How the mean and median are affected by extreme observations The **mean** ${tex`\bar x = \frac{\sum_{i=1}^n x_i}{n}`} and the **median** (the middle value in the dataset after arranging observations in order of magnitude) are two common **measures of central tendency** for a dataset, i.e., they attempt to summarize the "middle" of the distribution of the dataset. The mean of a dataset can be sensitive to extreme values in the dataset, whereas the median is not affected at all by the most extreme values. For instance, when summarizing the income distribution in a country, one must consider if one really want the ultra-rich individuals to influence the measure of "average" income. You can explore the difference between mean and median by adjusting the value of the largest observation in the dataset. Drag the slider below, and the largest value in the dataset (marked with an orange circle) will be multiplied by the number you've selected with the slider. Also, investigate whether the effect depends on the size of the dataset, i.e., how many observations are included in the dataset.33:[] 30:{"title":"How the mean and median are affected by extreme observations","cells":[{"id":38,"value":"$32","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof N = Inputs.range([2, 100], {value: 10, step: 1, label: \"number of data points\"})\n ","pinned":false,"hidden":false,"mode":"ojs"},{"id":1251,"value":"viewof maxscalefact = Inputs.range([1, 3], {value: 1, label: \"Multiply the largest data point with\", step: 0.01})","pinned":false,"hidden":false,"mode":"ojs"},{"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 == \"data points\" || d.type == \"max value\", x: \"data\", stroke: \"type\", strokeWidth: \"width\", symbol: \"symbol\", r: \"radius\"}),\n Plot.tickX(datascaled, {filter: d => d.type == \"mean\" || d.type == \"median\", x: \"data\", stroke: \"type\", strokeWidth: \"width\", symbol: \"symbol\"}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some details","pinned":false,"hidden":false,"mode":"md"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":1128,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":888,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"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"},{"id":1325,"value":"datascaled = {\n var tmp;\n tmp = pop.map(d => ({data: d, type: \"data points\", width: 1, symbol: \"circle\", radius: 1}))\n tmp[tmp.length - 1] = {data: d3.max(pop)*maxscalefact, type: \"max value\", 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: \"mean\", 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"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}