2d:null 2e:["$","title",null,{"children":"Mean Estimation & Confidence Interval | Pablo Ruiz | Observable"}] 33:T7d8,chart1 ={ const histogram0 = histogram(xmin,xmax,data,height) d3.select(histogram0).append("line") .style("stroke-width", 2) .style("stroke", "red") // colour the line .attr("x1",x2(mean1) ) // x position of the first end of the line .attr("y1", 0) // y position of the first end of the line .attr("x2", x2(mean1) ) // x position of the second end of the line .attr("y2", height-30); d3.select(histogram0).append("text") .attr("y", 20)//magic number here .attr("x", function(){ return x2(mean1)+70}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Population Mean") .style('fill', 'red'); d3.select(histogram0).append("line") .style("stroke-width", 2) .style("stroke", "black") // colour the line .attr("x1",x2(median1) ) // x position of the first end of the line .attr("y1", 40) // y position of the first end of the line .attr("x2", x2(median1) ) // x position of the second end of the line .attr("y2", height-30); d3.select(histogram0).append("text") .attr("y", 60)//magic number here .attr("x", function(){ return x2(median1)+30}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Median") .style('fill', 'Black'); d3.select(histogram0).append("line") .style("stroke-width", 2) .style("stroke", "Orange") // colour the line .attr("x1",x2(mode1) ) // x position of the first end of the line .attr("y1", 20) // y position of the first end of the line .attr("x2", x2(mode1) ) // x position of the second end of the line .attr("y2", height-30); d3.select(histogram0).append("text") .attr("y", 40)//magic number here .attr("x", function(){ return x2(mode1)+30}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Mode") .style('fill', 'Orange'); return histogram0}34:T583,{ const histogram2 = histogram(xmin,xmax,data,height) d3.select(histogram2).append("line") .style("stroke-width", 2) .style("stroke", "red") // colour the line .attr("x1",x2(mean1)) // x position of the first end of the line .attr("y1", 0) // y position of the first end of the line .attr("x2", x2(mean1)) // x position of the second end of the line .attr("y2", height-30); d3.select(histogram2).append("text") .attr("y", 20)//magic number here .attr("x", function(){ return x2(mean1)+70}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Population Mean") .style('fill', 'red'); d3.select(histogram2).append("line") .style("stroke-width", 2) .style("stroke", "Orange") // colour the line .attr("x1",x2(mean1) ) // x position of the first end of the line .attr("y1", (height-30)/4*3) // y position of the first end of the line .attr("x2", x2(mean1+Math.sqrt(variance1)) ) // x position of the second end of the line .attr("y2", (height-30)/4*3); d3.select(histogram2).append("text") .attr("y", 180)//magic number here .attr("x", function(){ return x2(mean1)+70}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Standard Deviation") .style('fill', 'Orange'); return histogram2}35:T534,{ const histogram3 = histogram(xmin,xmax,data2,height) d3.select(histogram3).append("text") .attr("y", 20)//magic number here .attr("x", function(){ return x2(mean1)+70}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Population Mean") .style('fill', 'red'); d3.select(histogram3).append("line") .style("stroke-width", 2) .style("stroke", "red") // colour the line .attr("x1",x2(mean1) ) // x position of the first end of the line .attr("y1", 0) // y position of the first end of the line .attr("x2", x2(mean1) ) // x position of the second end of the line .attr("y2", height-30); d3.select(histogram3).append("line") .style("stroke", "orange") // colour the line .attr("x1",x2(mean2) ) // x position of the first end of the line .attr("y1", 25) // y position of the first end of the line .attr("x2", x2(mean2) ) // x position of the second end of the line .attr("y2", height-30); d3.select(histogram3).append("text") .attr("y", 45)//magic number here .attr("x", function(){ return x2(mean2)+50}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Sample Mean") .style('fill', 'orange'); return histogram3}36:T8e8,{ const histogram4 = histogram(xmin,xmax,data5,height) d3.select(histogram4).append("text") .attr("y", 20)//magic number here .attr("x", function(){ return x2(mean1)+70}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Population Mean") .style('fill', 'red'); d3.select(histogram4).append("line") .style("stroke-width", 2) .style("stroke", "red") // colour the line .attr("x1",x2(mean1) ) // x position of the first end of the line .attr("y1", 0) // y position of the first end of the line .attr("x2", x2(mean1) ) // x position of the second end of the line .attr("y2", height-30); d3.select(histogram4).append("line") .style("stroke", "orange") // colour the line .attr("x1",x2(mean5) ) // x position of the first end of the line .attr("y1", 25) // y position of the first end of the line .attr("x2", x2(mean5) ) // x position of the second end of the line .attr("y2", height-30); d3.select(histogram4).append("path") .datum(datag) .attr("fill", "none") .attr("stroke", "green") .attr("stroke-width", 1.5) .attr("stroke-linejoin", "round") .attr("stroke-linecap", "round") .attr("d", line); d3.select(histogram4).append("text") .attr("y", 45)//magic number here .attr("x", function(){ return x2(mean5)+90}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("Mean of sample Means") .style('fill', 'orange'); d3.select(histogram4).append("line") .style("stroke-width", 3) .style("stroke", "pink") // colour the line .attr("x1",x2(mean1-2*stdlim) ) // x position of the first end of the line .attr("y1", (height-30)/8*7) // y position of the first end of the line .attr("x2", x2(mean1+2*stdlim) ) // x position of the second end of the line .attr("y2", (height-30)/8*7); d3.select(histogram4).append("text") .attr("y", 180)//magic number here .attr("x", function(){ return x2(mean1)}) .attr('text-anchor', 'middle') .attr("class", "myLabel")//easy to style with CSS .text("2σ interval = 95% CI") .style('fill', 'pink'); return histogram4 }37:[] 31:{"id":"b40ea6b5634f9f3e","slug":"mean-estimation-confidence-interval","title":"Mean Estimation & Confidence Interval","thumbnail":"6f131904f4dc011984c327aa873a6ab28ed2fca363b586ec3555e97fbb0b05f8","access_level":"public","create_time":"$D2019-03-03T16:51:00.111Z","update_time":"$D2021-02-19T19:34:26.587Z","view_time":null,"first_public_time":"$D2019-03-09T15:02:15.142Z","has_unpublished_edits":null,"fork":false,"stars":9,"starred":false,"owner":{"type":"team","id":"aff7cf61197c2be2","login":"pabloruizcuevas","name":"Pablo Ruiz","avatar_url":"https://avatars-next.observableusercontent.com/cc5b605be90117aa3a735df3ea9e14548a10fdaaa9fb51d38a85019182d079d2","tier":"starter_2024"},"version":3085,"publish_version":3085,"role":null,"authored":false,"authors":[{"type":"individual","id":"98041eb57d06192b","login":"pabloruizcuevas","name":"Pablo Ruiz","avatar_url":"https://avatars-next.observableusercontent.com/cc5b605be90117aa3a735df3ea9e14548a10fdaaa9fb51d38a85019182d079d2","tier":null}],"members":[{"type":"individual","id":"98041eb57d06192b","login":"pabloruizcuevas","name":"Pablo Ruiz","avatar_url":"https://avatars-next.observableusercontent.com/cc5b605be90117aa3a735df3ea9e14548a10fdaaa9fb51d38a85019182d079d2","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":7,"collection":{"id":"a320db57d5e9f221","slug":"statistics","title":"Statistics","owner":{"id":"aff7cf61197c2be2","login":"pabloruizcuevas","name":"Pablo Ruiz"}},"type":"notebook","body":{"title":"Mean Estimation & Confidence Interval","cells":[{"id":0,"value":"md`# Mean Estimation & Confidence Interval`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":64,"value":"md ` Supouse we are interested in the **height of some population**, let's say there are N=${N} persons and their height is summarized in the next **histogram**. `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":299,"value":"viewof N = number({placeholder: \"10000\", title: \"Population\", submit: true,value:10000})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1335,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":357,"value":"md ` ${tex.block` \\mu = ${Math.round(SS.mean(data)*10)/10} \\,\\,\\,\\,\\,\\,\\,\\,\\, Mode= ${median1}\\,\\,\\,\\,\\,\\,\\,\\,\\, Median= ${mode1}`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1118,"value":"md ` The height in this population follows a lognormal distribution, as many other natural phenomena related with groth. `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1187,"value":"html`\n

The three estatistics given by the histogram (mean,median and mode) give us a measure of the central tendency of the distribution and the most commonly used is the arithmetic mean ${tex` \\mu `} also known as average.

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1216,"value":"md ` Another important property of the distribution is the **dispersion**, as before there are many measures but the most important are the **standard deviation ${tex` \\sigma `}** and the **variance **${tex` \\sigma^2 `} that is the squared standard deviation. Another dispersion indicators are the range and the Mean absolute deviation MED.` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1206,"value":"md ` ${tex.block` \\sigma= {\\sqrt{\\frac{1}{N} \\sum_{i=1}^N (x_i -\\mu)^2}}=${Math.round(Math.sqrt(variance1)*10)/10} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\sigma^2=${Math.round(variance1*10)/10} `}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1221,"value":"md ` ${tex.block` Range= max(x_i)-min(x_i) =${Math.round(SS.max(data)-SS.min(data))}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, Med= \\frac{1}{N} \\sum_{i=1}^N \\left| x_i - \\mu\\right|=${Math.round(med(data)*10)/10} `}` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1376,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1661,"value":"md ` ${tex.block` \\mu = ${Math.round(SS.mean(data)*10)/10}\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\, \\sigma=${Math.round(Math.sqrt(variance1)*10)/10} `}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1270,"value":"md ` The Standard Deviation will give us a notion about \"how likely is that a new value in the distribution is far from the population mean.\" `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1278,"value":"md`# Confidence Intervals given σ`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":304,"value":"md ` Since we usually **don't have the information** about the all population and we can not obtain it easily, we need to estimate it. In in our case we can not measure the height of everyone, but we take a little sample of n = ${n} persons randomly chosen. With our little sample **we want to estimate the population average height or population mean**.` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2291,"value":"md ` Also we may also have to estimate the standard deviation whenever is unknown. But we will suppose that inthis case the standad deviation is given and we will consider later the opposite case.` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1284,"value":"md `With our little sample we can make an histogram and in order to **estimate the population mean** ${tex` \\mu `} we can just calculate the average height of our sample: \"**sample Mean**\" ${tex` \\bar{X} `}. But is sample mean the same as the real population mean?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3031,"value":"md `As you can see in the following histogram they are usually close one from the other, but how close?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":308,"value":"viewof n = number({placeholder: \"20\", title: \"Population\", submit: true,value:20})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1385,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":472,"value":"md ` ${tex.block` \\mu = ${Math.round(SS.mean(data)*10)/10}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\bar{X} = ${Math.round(SS.mean(data2)*100)/100}`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":480,"value":"md `In order to stimate how far is the sample mean to the population mean, we can imagnine that we continue taking many samples of ${n} persons and then we plot the distribution of the sample mean \"sampling distribution\", which has a mean ${tex`\\bar{S_n}`} and variance ${tex`\\sigma_{s}^2`}.\n\n${tex.block`E(\\bar{X})=\\mu \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\bar{S_n}= \\frac{\\bar{X}_1+\\cdots+\\bar{X}_n}{n}`} \n\nBut how many samples we require take to obtain a good aproximation of the real mean?` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1581,"value":"md `And here is when we observe one of the consecuences of the [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem), for large enough observations, the sample mean distribution converges to a gaussian distribution centered in the population mean and with standad deviation equals to: ${tex.block` \\sigma_{s}= \\frac{\\sigma}{\\sqrt{n}} = \\frac{ ${Math.round(Math.sqrt(variance1)*10)/10}}{\\sqrt{${n}}}= ${Math.round(stdlim*100)/100}`} `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1769,"value":"md `Therefore the distribution of our sample mean is:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1745,"value":"md ` ${tex.block ` \\bar{X}^{${n}} \\sim \\mathcal{N} ( \\mu ,{\\sigma_{s}}^2) = \\mathcal{N} ( \\mu ,${Math.round(Math.sqrt(variance1)/Math.sqrt(n)*100)/100}^2) `}` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3045,"value":"md`Where ${tex` \\mathcal{N} ( \\mu ,{\\sigma_{s}}^2) `} reprents a gaussian distribution.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2153,"value":"md `We can check the validity of this prediction making a histogram of our sample mean. You can take more observations adding more sample means to the histogram, as a result our distribution will aproach to the normal distribution in green.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1501,"value":"{\n const button = html``\n button.onclick = function(event) {\n mutable data5 = data5.concat(Array.from(new Array(1),(val,index)=>SS.mean(shuffle(data).slice(0, n))))\n }\n return button\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2126,"value":"{\n const button1 = html``\n button1.onclick = function(event) {\n mutable data5 = data5.concat(Array.from(new Array(1000),(val,index)=>SS.mean(shuffle(data).slice(0, n))))\n }\n \n return button1\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1436,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1628,"value":"md ` ${tex.block` \\mu = ${Math.round(SS.mean(data)*100)/100}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\bar{S_n} = ${Math.round(SS.mean(data5)*100)/100} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\sigma_n= ${Math.round(Math.sqrt(SS.variance(data5))*100)/100} `}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1678,"value":"`Collected Sample Means: ${data5}` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1776,"value":"md`Since we know the characteristics of the gaussian distribution we can calculate the confidence interval of our sample mean! exactly what we wanted!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2190,"value":"md`Here you can choose the level of confidence of your first measure of ${n} persons in terms of standard deviations:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2220,"value":"\nviewof confidence = radio({\n title: 'Confidence Interval',\n description: 'Select the width of the confidence interval',\n options: [{ value: 1, label: '1σ = 68% CI (really poor)' },{ value: 2, label: '2σ = 95% CI' },{ value: 3, label: '3σ = 99.7% CI' },{ value: 4, label: '4σ = 99.99% CI' },{ value: 5, label: '5σ = 99.99994% CI (Paticle Physics standard)' }],\n value: 4\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2224,"value":"\nmd`${tex.block` \\mu_{estimated} = \\bar{X} \\pm ${confidence} \\frac{\\sigma}{\\sqrt{n}} = ${Math.round(mean2*10)/10}\\, \\pm ${confidence*Math.round(stdlim*10)/10} \\, cm`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2252,"value":"{if(mean1>=(mean2-confidence*stdlim) && mean1<=(mean2+confidence*stdlim)){return \"The real mean is inside the confidence interval\"}else{return \"The real mean is NOT inside the confidence interval\"}}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2300,"value":"md`But what about the case we don't have the standard deviation?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2839,"value":"md`# Confidence Intervals estimating σ`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1725,"value":"md `We will just estimate it as we made with the mean using the sample mean. But in this case the procedure is slightly diferent because the formula of the standard deviation to estimate the sample standard deviation is biased, and we will better estimate the unbiased variance and then derive the standard deviation: ${tex.block` s^2= {\\frac{1}{n-1} \\sum_{i=1}^n (x_i -\\mu)^2}=${Math.round(SS.sampleVariance(data2)*10)/10} \\,\\,\\,\\,\\,s= ${Math.round(Math.sqrt(SS.sampleVariance(data2))*10)/10} `}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2956,"value":"md`Unfortunatelly it is still a bit biased since we took a square root, but it doesn´t exist an unbiased closed formula to estimate estandard deviation.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2321,"value":"md`From now is more or less the same as before, but the intervals in standard deviations with respect to a stantard distribution we will have this intervals with respect a corrected gaussian that is called t ditribution or **student distribution**. Here in stead of introduce our CI by standard deviations we will introduce it directly using α that is the confidence level and (1-α)% is our CI . `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2759,"value":"\nviewof alpha = radio({\n title: 'Confidence Interval',\n description: 'Select the width of the confidence interval',\n options: [{ value: 0.05, label: '0.05 = 95% CI' },{ value: 0.01, label: '0.01 = 99% CI' },{ value: 0.001, label: '0.001 = 99.9% CI' },{ value: 0.000001, label: '0.000001 = 99.999999% CI' }],\n value: 0.001\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2326,"value":"md`${tex.block` \\mu_{estimated} =\\bar{X} \\pm t_{\\alpha/2,n-1} \\frac{s}{\\sqrt{n}} = ${Math.round(mean2*10)/10} \\pm ${Math.round(Math.abs(tint(alpha/2,n-1))*SS.sampleStandardDeviation(data2)/Math.sqrt(n)*10)/10}\\,cm`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2578,"value":"md`As you can see in this case the CI is greater than before because we have less information and we are estimating it.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2545,"value":"md`Since to evaluate the confidence interval of the t distribution is not easy we would substitute the previous fomula for the next one for large enought n (upper 30):`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2791,"value":"\nviewof confidence2 = radio({\n title: 'Confidence Interval',\n description: 'Select the width of the confidence interval',\n options: [{ value: 1, label: '1σ = 68% CI' },{ value: 2, label: '2σ = 95% CI' },{ value: 3, label: '3σ = 99.7% CI' },{ value: 4, label: '4σ = 99.99% CI' },{ value: 5, label: '5σ = 99.99994% CI (Paticle Physics standard)' }],\n value: 4\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2795,"value":"md`${tex.block` \\mu_{estimated} =\\bar{X} \\pm %\n${confidence2} \\frac{s}{\\sqrt{n}} = ${Math.round(mean2*10)/10} \\pm ${Math.round(confidence2*SS.sampleStandardDeviation(data2)/Math.sqrt(n)*10)/10}\\,cm`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2772,"value":"{if(n<30){return \"since n is less than 30 this is not a good estimation\"} else{return \"Since n is big enought this is a correct estimation\"}}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2627,"value":"md`Where we have aproximated the t distribution by a standard distribution.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2598,"value":"md`# Additional information`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1293,"value":"md `The Standard deviation of the sampling distribution decreases with n because as the sample size increases, the sample means are more often closer to the population mean. ` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2843,"value":" md `When we add more observations we will know better how far would be our estimation from the real popluation mean, but the convergence depends on the initial distribution varianc,`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1705,"value":"md` The sample mean it is an unbiased estimator of the population mean since the distribution of sample means is centered in the population mean ${tex`E(\\bar{X}) = \\mu`} and even more is the unbiased estimator of minimum variance.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1226,"value":"md`The mean is also known to be the first \"moment\" of a distribution, to caracterice completely a distribution we would need all their moments, but the first ones are the most relevant for us. The second moment is the standard deviation. `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2965,"value":"md` To estimate the standard deviation there are better aproximations like the next, that is made for a normal distribution: ${tex.block` s= {\\sqrt{\\frac{1}{n-1.5} \\sum_{i=1}^N (x_i -\\mu)^2}} `} `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2805,"value":"md`# Functions`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"function Generate_lognormal_population(N,mu,sigma){\n return Array.from({length: N}, () => Math.round(stdlib.base.random.lognormal(Math.log(mu),Math.log(sigma))));\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1846,"value":"function Generate_normal_population(N,mu,sigma){\n return Array.from({length: N}, () => Math.round(stdlib.base.random.normal(mu,sigma)));\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1562,"value":"function Generate_uniform_population(N,mu,sigma){\n return Array.from({length: N}, () => Math.round(stdlib.base.random.uniform(mu,sigma)));\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2821,"value":"md`you can change the initial population ditribution`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":161,"value":"data=(Generate_lognormal_population(N,168,1.1))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1573,"value":"//data=(Generate_uniform_population(N,100,200))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2814,"value":"md`you can intoduce here your own real observations`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2817,"value":"//data2=[123,134,174,134,....]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":364,"value":"data2=shuffle(data).slice(0, n);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1137,"value":"mode1 = SS.mode(data)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1142,"value":"median1=SS.median(data)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":973,"value":"variance1=SS.variance(data)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":985,"value":"variance2=Math.round(SS.variance(data2))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2091,"value":"stdlim=Math.sqrt(variance1)/Math.sqrt(n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1070,"value":"med1=med(data)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":610,"value":"mean1=SS.mean(data)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":883,"value":"mean2=SS.mean(data2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1535,"value":"mean5=SS.mean(data5)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1851,"value":"datag=I_HATE_TO_PARSE_DATA(xe,ye)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1299,"value":"mutable data5=[mean2];","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":795,"value":"function Arrays_sum(array1, array2) \n{\n var result = [];\n var ctr = 0;\n var x=0;\n\n if (array1.length === 0) \n return \"array1 is empty\";\n if (array2.length === 0) \n return \"array2 is empty\"; \n\n while (ctr < array1.length && ctr < array2.length) \n {\n result.push(array1[ctr] + array2[ctr]);\n ctr++;\n }\n\n if (ctr === array1.length) \n {\n for (x = ctr; x < array2.length; x++) {\n result.push(array2[x]);\n }\n } \n else\n {\n for (x = ctr; x < array1.length; x++) \n {\n result.push(array1[x]);\n }\n }\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1068,"value":"function med(list){\nvar sum=0;\nvar total= list.reduce(function(a,b){\n return a+b;\n});\n// console.log(total);\n\nvar mean=total/list.length;\n\nfunction var_numerator(value){\n return (Math.abs(value-mean));\n}\nvar variance=list.map(var_numerator);\n\nvariance=variance.reduce(function(a,b){\n return (a+b);\n});\n\nvariance = variance/list.length;\nreturn variance\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"function vari(list){\nvar sum=0;\nvar total= list.reduce(function(a,b){\n return a+b;\n});\n// console.log(total);\n\nvar mean=total/list.length;\n\nfunction var_numerator(value){\n return ((value-mean)*(value-mean));\n}\nvar variance=list.map(var_numerator);\n\nvariance=variance.reduce(function(a,b){\n return (a+b);\n});\n\nvariance = variance/list.length;\nreturn variance\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2672,"value":"function tint(Alfa,df){ //introduce alpha/2 and n-1 df fot the propuose here\n let a,b,c\n a=-10;\n b=10;\n for(let i = 0; i < 100; i++){\n\n c=(a+b)/2;\n if(Alfa !isNaN(d.value))\n .x(d => x2(d.date))\n .y(d => y(d.value))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1990,"value":"max2=70","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2058,"value":"y = d3.scaleLinear()\n .domain([0, d3.max(datag, d => d.value)]).nice() //.domain([d3.min(datag, d => d.value), d3.max(datag, d => d.value)]).nice()\n .range([height - margin.bottom, margin.top])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1958,"value":"xe=Array.from(Array(2*max2*10), (e,i)=>mean1+i/10-max2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2013,"value":"a=Array.from(xe ,(e,i)=>(e-mean1)/stdlim)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1964,"value":"ye=stats.dnorm(a)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":285,"value":"md ` histogram ` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1322,"value":"import{chart as histogram} from \"b15ca1cc633bcb62\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1490,"value":"height=180","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2890,"value":"xmin=SS.min(data)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2893,"value":"xmax=SS.max(data)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2906,"value":"md `Lines in histograms ` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":260,"value":"x2 = d3.scaleLinear()\n .domain([xmin,xmax]).nice()\n .range([margin.left, width - margin.right])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":272,"value":"margin = ({top: 20, right: 20, bottom: 30, left: 40})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":386,"value":"function shuffle(array) {\n let counter = array.length;\n\n // While there are elements in the array\n while (counter > 0) {\n // Pick a random index\n let index = Math.floor(Math.random() * counter);\n\n // Decrease counter by 1\n counter--;\n\n // And swap the last element with it\n let temp = array[counter];\n array[counter] = array[index];\n array[index] = temp;\n }\n\n return array;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":599,"value":"function I_HATE_TO_PARSE_DATA(x2,y2){\n//x,y my data vectors\n// create a new array that follows the format\nvar data = x2.map(function(d,n) {return { date: d, value: y2[n] };});\nreturn data\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2102,"value":"md ` Imports ` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":79,"value":"stdlib = require( 'https://unpkg.com/@stdlib/stdlib@0.0.55' )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2409,"value":"\nfunction Betinc(X,A,B) {\n var A0=0;\n var B0=1;\n var A1=1;\n var B1=1;\n var M9=0;\n var A2=0;\n var C9;\n while (Math.abs((A1-A2)/A1)>.00001) {\n A2=A1;\n C9=-(A+M9)*(A+B+M9)*X/(A+2*M9)/(A+2*M9+1);\n A0=A1+C9*A0;\n B0=B1+C9*B0;\n M9=M9+1;\n C9=M9*(B-M9)*X/(A+2*M9-1)/(A+2*M9);\n A1=A0+C9*A1;\n B1=B0+C9*B1;\n A0=A0/B1;\n B0=B0/B1;\n A1=A1/B1;\n B1=1;\n }\n return A1/A\n}\n\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2414,"value":"\nfunction tdist(X, df) {\n let tcdf\n if (df<=0) {\n alert(\"Degrees of freedom must be positive\")\n } else {\n let A=df/2;\n let S=A+.5;\n let Z=df/(df+X*X);\n let BT=Math.exp(LogGamma(S)-LogGamma(.5)-LogGamma(A)+A*Math.log(Z)+.5*Math.log(1-Z));\n if (Z<(A+1)/(S+2)) {\n Math.betacdf=BT*Betinc(Z,A,.5)\n } else {\n Math.betacdf=1-BT*Betinc(1-Z,.5,A)\n }\n if (X<0) {\n tcdf=Math.betacdf/2\n } else {\n tcdf=1-Math.betacdf/2\n }\n }\n tcdf= Math.round(tcdf*100000)/100000;\n \n return tcdf;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2420,"value":"\nfunction LogGamma(Z) {\n \n var S=1+76.18009173/Z-86.50532033/(Z+1)+24.01409822/(Z+2)-1.231739516/(Z+3)+.00120858003/(Z+4)-.00000536382/(Z+5);\n var LG= (Z-.5)*Math.log(Z+4.5)-(Z+4.5)+Math.log(S*2.50662827465);\n \n return LG\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":353,"value":"SS = require('simple-statistics')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1924,"value":"stats = require('statdists')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":758,"value":"import{number,radio} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":226,"value":"d3 = require(\"d3@5\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2999,"value":"md`style`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3004,"value":"html`\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W37","stdlib":"1","titling":"auto"}} 32:"$31:body"