2b:null 2c:["$","title",null,{"children":"Standard error of the mean | bluprince13 | Observable"}] 31:T578,md` The standard deviation helps us understand how much variation there is amongst individual observations. For example, Nancy wants to understand how well her students did in the exams. She would summarize the results of her students as a mean and standard deviation. However, the standard deviation doesn't tell us how 'good' our estimate of the mean is. This is something we need to know when we're comparing the means between samples from two or more populations. For example, if we want to compare the mileage between two different types of cars: A and B, we could measure it on a sample of cars from both populations. We can then calculate a mean and standard deviation for both samples. However, we can't make a meaningful comparison between A and B's populations based on just the mean from two small samples. This is where the standard error of the mean comes in. In this [explorable](https://explorabl.es/), we will develop an intuitive understanding of the standard error of the mean. We first cover the basics: normal distribution, standard deviation before delving into standard error. There are more than a few formulas, but feel free to skip past them if you are not a fan. I've included them here as they are really useful for conveying mathematical ideas concisely. Anyhow, there'll be plenty of interactive plots to keep you informed and entertained. It'll be fun. I promise! `32:T454,{ const mu = 0; const sigma = 1; const numberOfStdDev = numberOfStdDev_0; const { x, y } = getPDF({ mu, sigma, numberOfStdDev: 4 }); const { x: xSigma, y: ySigma, fractionOfPopulation } = getPDF({ mu, sigma, numberOfStdDev }); const data = [ { name: "PDF", x: x, y: y, mode: "lines" }, { name: `${numberOfStdDev} sigma`, x: xSigma, y: ySigma, marker: { color: "blue" }, mode: "lines", fill: "tozeroy" } ]; var layout = { title: `Standard normal distribution`, xaxis: { title: sigmaText }, yaxis: { title: probabilityDensityText }, width: width, annotations: [ { xref: "paper", yref: "paper", x: 0.5, y: 0.5, text: `${fractionOfPopulation}% of the
population
scores are within
${numberOfStdDev} std deviations
of the mean.`, showarrow: false, align: "center" } ] }; var div = DOM.element("div"); Plotly.newPlot(div, data, layout); return div; }33:Ta89,{ const mu = mu_1; const sigma = sigma_1; const n = n_1; const refresh = refresh_1; const randomNormal = getRandomNormal(); const randomNormalMean = math.mean(randomNormal.x); const binNumber = math.sqrt(n); const min = math.min(...randomNormal.x); const max = math.max(...randomNormal.x); const extent = rangeVarMean[1] - rangeVarMean[0]; const binSize = extent / (2 * binNumber); const trace_x = { name: xText, x: randomNormal.x, y: randomNormal.y, type: "scatter", mode: "markers", marker: { color: "black" } }; const trace_histogram = { name: xText, x: randomNormal.x, type: "histogram", xaxis: "x2", yaxis: "y2", marker: { color: "black" }, showlegend: false, xbins: { start: min, end: max, size: binSize, autobinx: false } }; const trace_mean = { name: meanText, x: [randomNormalMean, randomNormalMean], y: rangeVarMean, type: "scatter", mode: "lines", line: { color: "red", width: 2 } }; const trace_mu = { name: muText, x: [mu, mu], y: rangeVarMean, type: "scatter", mode: "lines", line: { color: "green", width: 2 } }; const { x, y } = getPDF({ mu, sigma, numberOfStdDev: 4 }); const trace_normal = { name: "PDF", x: x, y: y.map(probability => probability * n * binSize), mode: "lines", xaxis: "x2", yaxis: "y2" }; const data = [trace_x, trace_histogram, trace_mean, trace_mu, trace_normal]; const layout = { title: `Normal distribution
${getInfo({ mu, sigma, n })}`, xaxis: { title: xText, range: rangeVarMean.slice(), zeroline: true }, yaxis: { title: "", range: rangeVarMean.slice(), ticks: "", showticklabels: false }, xaxis2: { title: xText, range: rangeVarMean.slice(), zeroline: true }, yaxis2: { title: "frequency (f)" }, width: width, grid: { rows: 1, columns: 2, pattern: "independent" }, showlegend: true, bargap: 0.05, shapes: [ { type: "line", x0: randomNormalMean, y0: 0, x1: randomNormalMean, y1: 1, line: { color: "red", width: 2 }, xref: "x2", yref: "paper" }, { type: "line", x0: mu, y0: 0, x1: mu, y1: 1, line: { color: "green", width: 2 }, xref: "x2", yref: "paper" } ] }; const div = DOM.element("div"); Plotly.newPlot(div, data, layout); return div; }34:T660,{ const numberOfSamples = numberOfSamples_2; const mu = muDefault; const sigma = sigmaDefault; const n1 = 3; const n2 = 10; const { trace_x, trace_mean, trace_mu } = getTracesExperiment({ mu, sigma, n: n1, numberOfSamples, meanMode: "markers", showMeanError: false }); const { trace_x: trace_x_2, trace_mean: trace_mean_2, trace_mu: trace_mu_2 } = getTracesExperiment({ mu, sigma, n: n2, numberOfSamples, axis: 2, meanMode: "markers", showMeanError: false }); const data = [ trace_x, trace_mean, trace_mu, trace_x_2, trace_mean_2, trace_mu_2 ]; var layout = { title: `Effect of ${nText}
${getInfo({ mu, sigma, numberOfSamples })}`, xaxis: { title: sampleText }, yaxis: { title: scoreText, range: range }, xaxis2: { title: sampleText, zeroline: true }, yaxis2: { title: scoreText, range: range }, width: width, grid: { rows: 1, columns: 2, pattern: "independent" }, showlegend: true, annotations: [ { x: 0.25, y: 1.1, xref: "paper", yref: "paper", xanchor: "center", text: getInfo({ n: n1 }), showarrow: false }, { x: 0.75, y: 1.1, xref: "paper", yref: "paper", xanchor: "center", text: getInfo({ n: n2 }), showarrow: false } ], margin: { t: 150 } }; var div = DOM.element("div"); Plotly.newPlot(div, data, layout); return div; }35:T66e,{ const numberOfSamples = numberOfSamplesDefault; const mu = muDefault; const sigma = sigma_5; const n1 = 3; const n2 = 20; const { trace_stdErrorsSample, trace_mean, trace_mu } = getTracesExperiment({ mu, sigma, n: n1, numberOfSamples }); const { trace_stdErrorsSample: trace_stdErrorsSample_2, trace_mean: trace_mean_2, trace_mu: trace_mu_2 } = getTracesExperiment({ mu, sigma, n: n2, numberOfSamples, axis: 2 }); const data = [ trace_stdErrorsSample, trace_mean, trace_mu, trace_stdErrorsSample_2, trace_mean_2, trace_mu_2 ]; const layout = { title: `Increasing ${nText} reduces ${stdErrorSText}
${getInfo({ mu, sigma, numberOfSamples })}`, xaxis: { title: sampleText }, yaxis: { title: scoreText, range: range.slice() }, xaxis2: { title: sampleText, zeroline: true }, yaxis2: { title: scoreText, range: range.slice() }, width: width, grid: { rows: 1, columns: 2, pattern: "independent" }, showlegend: true, annotations: [ { x: 0.25, y: 1.1, xref: "paper", yref: "paper", xanchor: "center", text: getInfo({ n: n1 }), showarrow: false }, { x: 0.75, y: 1.1, xref: "paper", yref: "paper", xanchor: "center", text: getInfo({ n: n2 }), showarrow: false } ], margin: { t: 150 } }; const div = DOM.element("div"); Plotly.newPlot(div, data, layout); return div; }36:T612,{ const refresh = refresh_7; const experiment_1 = getExperiment({ ...pop1, numberOfSamples: 1 }); const experiment_2 = getExperiment({ ...pop2, numberOfSamples: 1 }); var trace_histogram = { x: ["First", "Second"], y: [experiment_1.means[0], experiment_2.means[0]], name: meanText, error_y: { type: "data", array: [experiment_1.stdErrors[0], experiment_2.stdErrors[0]], visible: true, color: "red" }, type: "bar", marker: { color: lighterRed } }; var trace_true_means = { x: ["First", "Second"], y: [pop1.mu, pop2.mu], name: muText, mode: "markers", marker: { color: "green", size: 7 }, type: "scatter" }; const data = [trace_histogram, trace_true_means]; const layout = { title: `Comparing means from samples of two different populations
Error bars show ${stdErrorSText}`, xaxis: { title: "population" }, yaxis: { title: scoreText }, width: width, annotations: [ { x: 0.25, y: 1.1, xref: "paper", yref: "paper", xanchor: "center", text: getInfo({ ...pop1 }), showarrow: false }, { x: 0.75, y: 1.1, xref: "paper", yref: "paper", xanchor: "center", text: getInfo({ ...pop2 }), showarrow: false } ], margin: { t: 150 } }; const div = DOM.element("div"); Plotly.newPlot(div, data, layout); return div; }37:T415,setPop = ({ name = "Inputs", muInitial = muDefault, sigmaInitial = sigmaDefault, nInitial = nDefault } = {}) => { const form = html` `; form.oninput = () => { const mu = form.mu.valueAsNumber; const sigma = form.sigma.valueAsNumber; const n = form.n.valueAsNumber; form.muResult.value = mu; form.sigmaResult.value = sigma; form.nResult.value = n; return (form.value = { mu, sigma, n }); }; form.oninput(); return form; }38:T453,getExperiment = function({ numberOfSamples = numberOfSamplesDefault, n = nDefault, mu = muDefault, sigma = sigmaDefault } = {}) { var samples = Array.from({ length: numberOfSamples }, (x, i) => i); var data = samples.map(() => { return d3.range(n).map(() => d3.randomNormal(mu, sigma)()); }); var means = data.map(sample => math.mean(sample)); var stds = data.map(sample => math.std(sample)); var stdErrors = stds.map(std => std / math.sqrt(n)); var stdErrorPopulation = math.std(means); var stdErrorsPopulation = Array(numberOfSamples).fill(stdErrorPopulation); // Now let's flatten the data for plotting const flatSamples = samples.reduce((res, current) => { return res.concat(Array(n).fill(current)); }, []); const flatData = [].concat.apply([], data); const mus = Array(numberOfSamples).fill(mu); const sigmas = Array(numberOfSamples).fill(sigma); return { numberOfSamples, n, samples, flatSamples, data, flatData, means, stds, stdErrors, stdErrorPopulation, stdErrorsPopulation, mus, sigmas }; }39:T68d,getTracesExperiment = ({ mu = muDefault, sigma = sigmaDefault, n = nDefault, numberOfSamples = numberOfSamplesDefault, axis = 1, meanMode = "markers", showMeanError = true }) => { const { xaxis, yaxis, showlegend } = getAxis(axis); const experiment = getExperiment({ mu, n, sigma, numberOfSamples }); const trace_x = { name: xText, x: experiment.flatSamples, y: experiment.flatData, type: "scatter", mode: "markers", marker: { color: "black" }, xaxis, yaxis, showlegend }; const trace_stdErrorsSample = getTraceStdErrorsEnvelopeExperiment({ experiment, axis, linecolor: lightRed, fillcolor: lighterRed }); var trace_mean = { name: meanText, x: experiment.samples, y: experiment.means, type: "scatter", mode: meanMode, marker: { color: "red" }, xaxis, yaxis, showlegend }; if (showMeanError) { trace_mean.error_y = { type: "data", array: experiment.stdErrors, visible: true, color: "red" }; } const trace_mu = { name: muText, x: experiment.samples, y: experiment.mus, type: "scatter", mode: "lines", marker: { color: "green" }, line: { width: 3 }, xaxis, yaxis, showlegend }; const trace_stdErrorPopulation = getTraceErrorsEnvelope({ x: experiment.samples, y: experiment.mus, yError: experiment.stdErrorsPopulation, name: stdErrorSText, fillcolor: lighterGreen, linecolor: lightGreen }); return { trace_x, trace_stdErrorsSample, trace_mean, trace_mu, trace_stdErrorPopulation }; }3a:[] 2f:{"id":"b3a2bcb93f33fc21","slug":"standard-error-of-the-mean","title":"Standard error of the mean","thumbnail":"315eda82d49dad8a84079034e4340acbdff155e744d78decd27d286a531bed13","access_level":"public","create_time":"$D2019-06-07T18:44:11.646Z","update_time":"$D2019-06-16T17:56:33.351Z","view_time":null,"first_public_time":"$D2019-06-07T20:52:51.265Z","fork":false,"stars":5,"starred":false,"owner":{"type":"team","id":"1dbca818aab3cef4","login":"bluprince13","name":"bluprince13","avatar_url":"https://avatars-next.observableusercontent.com/6293bad2ab82f32701b437bded214389444b0cc33ed2edcd76137e7acff2318f","tier":"starter_2024"},"version":3873,"publish_version":3873,"role":null,"authored":false,"authors":[{"type":"individual","id":"d8028b6d4a88d81e","login":"bluprince13","name":"bluprince13","avatar_url":"https://avatars-next.observableusercontent.com/6293bad2ab82f32701b437bded214389444b0cc33ed2edcd76137e7acff2318f","tier":null}],"members":[{"type":"individual","id":"d8028b6d4a88d81e","login":"bluprince13","name":"bluprince13","avatar_url":"https://avatars-next.observableusercontent.com/6293bad2ab82f32701b437bded214389444b0cc33ed2edcd76137e7acff2318f","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":9,"collection":{"id":"4aa6eb52a3549079","slug":"explorables","title":"Explorables","owner":{"id":"1dbca818aab3cef4","login":"bluprince13","name":"bluprince13"}},"type":"notebook","body":{"title":"Standard error of the mean","cells":[{"id":0,"value":"md`# Standard error of the mean`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2376,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2211,"value":"toc()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2382,"value":"md`## Normal distributions can be characterised using the mean and the standard deviation.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3522,"value":"md`\nGood things may come in twos, but most things in life come in normal distributions. A normal distribution, sometimes called the bell curve, is a distribution that occurs naturally in many situations, e.g., heights and IQ scores. We can describe such a population using parameters such as the mean and the standard deviation. \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2478,"value":"md`The population mean, ${getTex(\n muTex\n)}, is an average of a group characteristic.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2579,"value":"md`\n${tex.block`\n${muTex}=\\frac{\\sum_{i=1}^{${NTex}} x_{i}}{${NTex}}\n`}\n\nwhere:\n- ${getTex(NTex)} is the total number of individuals or cases in the population,\n- ${tex`x_i`} is a single observation, where i can vary from 1 to ${getTex(\n NTex\n)}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2576,"value":"md`The population standard deviation, ${getTex(\n sigmaTex\n)}, is a measure of the spread (variability) of the scores.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2482,"value":"md`\n${tex.block`\n${sigmaTex}=\\sqrt{\\frac{\\sum_{i=1}^{${NTex}}\\left(x_{i}-${muTex}\\right)^{2}}{${NTex}}}\n`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3532,"value":"md`\nThe mean and standard deviation are pretty easy to understand visually. In the plot below, we can see a 'standard' normal distribution, i.e., a normal distribution with a mean of 0 and a standard deviation of 1. Use the slider below to see how the standard deviation helps us describe the spread of the data. It'll be useful while going through the rest of this article if you can remember that approximately 68% of the population lies within 1 standard deviation of the mean.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2878,"value":"viewof numberOfStdDev_0 = setNumberOfStdDev()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2769,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2385,"value":"md`## Sampling allows us to estimate characteristics of a population.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3564,"value":"md`Unfortunately, population parameters such as ${getTex(muTex)} and ${getTex(\n sigmaTex\n)} are usually unknown because it’s generally impossible to measure an entire population. However, you can use random samples to calculate estimates of these parameters.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3554,"value":"new Promise((resolve, reject) => {\n let image = new Image();\n image.onload = () => resolve(image);\n image.onerror = reject;\n image.crossOrigin = \"anonymous\";\n image.src = \"https://i.imgur.com/F1Y3KqB.jpg\";\n image.width = width / 4;\n image.height = width / 4;\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2682,"value":"md`\nThe sample mean, ${getTex(\n meanTex\n)}, is the average score of a sample on a given variable,\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2525,"value":"md`\n${tex.block`\n${meanTex}=\\frac{\\sum_{i=1}^{${nTex}} x_{i}}{${nTex}}\n`}\n\nwhere ${getTex(nTex)} is the samples size.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2694,"value":"md`\nThe sample standard deviation, ${getTex(\n sTex\n)}, is a measure of the spread (variability) of the scores in the sample on a given variable.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2691,"value":"md`\n${tex.block`\n${sTex}=\\sqrt{\\frac{\\sum_{i=1}^{${nTex}}\\left(x_{i}-${meanTex}\\right)^{2}}{${nTex} - 1}}\n`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2703,"value":"md`\nDon't worry too much about the 'n - 1' term in the denominator. If you are puzzled by why it's not just 'n', read up on the [Bessel's correction](https://en.wikipedia.org/wiki/Bessel%27s_correction).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3662,"value":"md`The visualisation below shows us what a random sample from a population with a set mean and standard deviation looks like. Did you notice that the sample mean isn't always very accurate?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":527,"value":"viewof mu_1 = setMu()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":557,"value":"viewof sigma_1 = setSigma()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":113,"value":"viewof n_1 = setn()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3767,"value":"viewof refresh_1 = button({ value: \"Resample\" })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":37,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2390,"value":"md`## Increasing ${nText} increases the accuracy of the ${meanText}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3649,"value":"md`The sample mean won't be exactly equal to the population mean that we're trying to estimate. If our sample size is small, our estimate of the mean won't be as good as an estimate based on a larger sample size.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3652,"value":"md`Below we have a sample size of 3 compared with a sample size of 10. Both samples are from the same population.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":582,"value":"viewof numberOfSamples_2 = setNumberOfSamples()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":253,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2396,"value":"md`## The ${stdErrorSigmaText} tells us how accurate our estimate of the ${muText} is likely to be.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2755,"value":"md`The standard error of the mean (also called just 'standard error') is the standard deviation of the different sample means if you took mutliple samples from the same population. Thus, we know that approximately 68% of the sample means would be within one standard error of the population mean.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":621,"value":"viewof sigma_3 = setSigma()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":637,"value":"viewof n_3 = setn()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":350,"value":"viewof numberOfSamples_3 = setNumberOfSamples()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":346,"value":"{\n const n = n_3;\n const numberOfSamples = numberOfSamples_3;\n const sigma = sigma_3;\n const mu = muDefault;\n\n const {\n trace_x,\n trace_mean,\n trace_stdErrorPopulation,\n trace_mu\n } = getTracesExperiment({\n mu,\n sigma,\n n,\n numberOfSamples,\n showMeanError: false\n });\n\n const data = [trace_x, trace_mean, trace_stdErrorPopulation, trace_mu];\n\n var layout = {\n title: `${stdErrorSText} is the std deviation of the different sample means
${getInfo(\n {\n mu,\n sigma,\n n,\n numberOfSamples\n }\n )}`,\n xaxis: {\n title: sampleText\n },\n yaxis: {\n title: scoreText,\n range: range.slice()\n },\n width: width,\n showlegend: true\n };\n\n var div = DOM.element(\"div\");\n Plotly.newPlot(div, data, layout);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3346,"value":"md`However, in reality, we won't have multiple samples to use to estimate the standard error. Luckily, we can calculate the standard error of the mean from just one sample using the following formula.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3468,"value":"tex.block`\n${stdErrorSigmaTex}\n=\\frac{${sigmaTex}}{\\sqrt{${nTex}}}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3479,"value":"md`Since we seldom know the ${sigmaText}, we use an approximation using the ${sText}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3473,"value":"tex.block`\n${stdErrorSigmaTex}\n\\approx \\frac{${sTex}}{\\sqrt{${nTex}}}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3488,"value":"md`In practice, the standard error of the mean tends to be simply defined as such.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2739,"value":"tex.block`\n\\textcolor{red}{${stdErrorSTex}}=\\frac{${sTex}}{\\sqrt{${nTex}}}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3691,"value":"md`In the following visualisation, we can see the standard error calculated for each sample. Approximately 68% of the samples will include the population mean within ±1 standard error of the sample mean.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":649,"value":"viewof sigma_4 = setSigma()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":642,"value":"viewof n_4 = setn()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":644,"value":"viewof numberOfSamples_4 = setNumberOfSamples()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":382,"value":"{\n const mu = muDefault;\n const sigma = sigma_4;\n const n = n_4;\n const numberOfSamples = numberOfSamples_4;\n\n const { trace_stdErrorsSample, trace_mean, trace_mu } = getTracesExperiment({\n mu,\n sigma,\n n,\n numberOfSamples,\n linecolor: lightRed,\n fillcolor: lighterRed\n });\n\n const data = [trace_stdErrorsSample, trace_mean, trace_mu];\n\n const layout = {\n title: `${stdErrorSText} calculated for each sample
${getInfo({\n mu,\n sigma,\n n,\n numberOfSamples\n })}`,\n xaxis: {\n title: sampleText\n },\n yaxis: {\n title: scoreText,\n range: range.slice()\n },\n width: width,\n showlegend: true\n };\n\n const div = DOM.element(\"div\");\n Plotly.newPlot(div, data, layout);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2425,"value":"md`## Increasing ${nText} reduces ${stdErrorSText}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3694,"value":"md`With increasing sample size, the sample mean becomes a more accurate estimate of the population mean. Therefore, the standard error of the mean becomes smaller. Below we have a sample size of 3 compared with a sample size of 20. Both samples are from the same population.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":903,"value":"viewof sigma_5 = setSigma()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":897,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3703,"value":"md`In the visualisation below, we can see how the standard error reduces with sample size. Since it is inversely proportional to the square root of ${getTex(\n nTex\n)}, it reduces drastically at first but, beyond a sample size of 20, the reduction is less pronounced. By contrast, the ${sText} will not tend to change as we increase the size of our sample; it tends to ${sigmaText}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1114,"value":"viewof s_6 = setS()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1101,"value":"{\n const s = s_6 * math.sqrt(3);\n const ns = _.range(3, 100);\n const stdErrorsMean = ns.map(n => s / math.sqrt(n));\n\n const data = [\n {\n x: ns,\n y: stdErrorsMean,\n mode: \"lines\",\n marker: {\n color: \"red\"\n }\n }\n ];\n\n var layout = {\n title: `Increasing ${nText} reduces ${stdErrorSText}`,\n xaxis: {\n title: nText\n },\n yaxis: {\n title: stdErrorSText\n },\n width: width\n };\n\n var div = DOM.element(\"div\");\n Plotly.newPlot(div, data, layout);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2988,"value":"md`## The standard error allows us to compare the means between two populations.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3324,"value":"md`Let's take a sample from two different populations. You can define the two populations using the sliders below. Use the button to resample.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3065,"value":"viewof pop1 = setPop({\n name: \"Population 1\",\n muInitial: 5,\n sigmaInitial: 1,\n nInitial: 10\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3070,"value":"viewof pop2 = setPop({\n name: \"Population 2\",\n muInitial: 6,\n sigmaInitial: 1,\n nInitial: 10,\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3164,"value":"viewof refresh_7 = button({ value: \"Resample\" })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2991,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3758,"value":"md`Knowing the standard errors allows us to make a meaningful comparison of the populations means, not just the sample means.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3711,"value":"md`Visually comparing standard error bars still isn't a very good way of testing the significance of the difference of means between two samples. There is a statistical tool called the [t-test](https://www.youtube.com/watch?v=0Pd3dc1GcHc) which we can use for this. Perhaps a topic for another explorable :)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3364,"value":"md`## Conclusion`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3367,"value":"md`\nI hope you enjoyed that :) Here's what we learnt so far.\n\n- Normal distributions can be characterised using the ${sigmaText} and the ${muText}.\n- Sampling allows us to estimate characteristics of a population.\n- Increasing ${nText} increases the accuracy of the ${meanText}.\n- The ${stdErrorSText} tells us how accurate our estimate of the ${muText} is likely to be.\n- Increasing ${nText} reduces ${stdErrorSText}.\n- The standard error allows us to compare the means between two populations.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2434,"value":"md`## References`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2437,"value":"md`\n1. McDonald, J.H. [Standard error of the mean](http://www.biostathandbook.com/standarderror.html), 2014. Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland.\n2. Altman DG, Bland JM. [Standard deviations and standard errors](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/), 2005. BMJ. [doi:10.1136/bmj.331.7521.903](https://dx.doi.org/10.1136%2Fbmj.331.7521.903)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1121,"value":"md`## Appendix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1123,"value":"md`I've captured all of the setup and calculations in this section. To see the code, head over to my Observable notebook [here](https://observablehq.com/@bluprince13/standard-error-of-the-mean). You can also make a copy and play with it!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2205,"value":"md`#### Imports`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":73,"value":"import {slider, button} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":95,"value":"d3 = require(\"d3@5\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":98,"value":"Plotly = require(\"https://cdn.plot.ly/plotly-latest.min.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":363,"value":"math = require(\"https://cdnjs.cloudflare.com/ajax/libs/mathjs/5.4.0/math.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1103,"value":"_ = require(\"underscore\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2208,"value":"import {toc} from \"@bluprince13/toc/2\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2924,"value":"stdlib = require(\"https://unpkg.com/@stdlib/stdlib@0.0.32/dist/stdlib-flat.min.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2927,"value":"// https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/base/dists/normal/pdf\n\npdf = stdlib.base.dists.normal.pdf","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2214,"value":"md`#### Defaults`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":473,"value":"muMin = -10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":474,"value":"muMax = 10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":600,"value":"muDefault = 0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":480,"value":"sigmaMin = 0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":484,"value":"sigmaMax = 10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":603,"value":"sigmaDefault = 5","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":471,"value":"rangeVarMean = [muMin - sigmaMax, muMax + sigmaMax]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":655,"value":"range = [-sigmaMax, sigmaMax]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3055,"value":"nMin = 3","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3057,"value":"nMax = 100","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":611,"value":"nDefault = 10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":615,"value":"numberOfSamplesDefault = 10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":751,"value":"xText = \"observations (x)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":728,"value":"muText = \"pop mean (μ)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":737,"value":"sigmaText = \"pop std dev (σ)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":784,"value":"sampleText = \"sample (i)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":741,"value":"meanText = \"sample mean (x̅)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":744,"value":"sText = \"sample std dev (s)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":765,"value":"nText = \"sample size (n)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2544,"value":"NText = \"population size (N)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":761,"value":"numberOfSamplesText = \"no. of samples\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":861,"value":"stdErrorSigmaText = \"std error (σ_x̅)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3435,"value":"stdErrorSText = \"std error (s_x̅)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2960,"value":"probabilityDensityText = \"Probability Density Function (PDF)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2968,"value":"frequencyText = \"frequency (f)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2971,"value":"scoreText = \"score\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2459,"value":"meanTex = `\\\\textcolor{red}{\\\\overline{x}}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2670,"value":"sigmaTex = \"\\\\textcolor{green}{\\\\sigma}\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2672,"value":"muTex = \"\\\\textcolor{green}{\\\\mu}\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2676,"value":"sTex = \"\\\\textcolor{red}{s}\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2712,"value":"nTex = \"\\\\textcolor{red}{n}\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2743,"value":"stdErrorSigmaTex = \"\\\\textcolor{green}{{\\\\sigma_{\\\\overline{x}}}}\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3429,"value":"stdErrorSTex = \"\\\\textcolor{red}{{s_{\\\\overline{x}}}}\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2713,"value":"NTex = \"\\\\textcolor{green}{N}\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2220,"value":"md`#### Slider generators`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":70,"value":"setMu = () =>\n slider({\n min: muMin,\n max: muMax,\n value: muDefault,\n step: 0.1,\n description: muText\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":547,"value":"setSigma = () =>\n slider({\n min: sigmaMin,\n max: sigmaMax,\n value: sigmaDefault,\n step: 0.1,\n description: sigmaText\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":567,"value":"setn = () =>\n slider({\n min: nMin,\n max: nMax,\n value: nDefault,\n step: 1,\n description: nText\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":593,"value":"setNumberOfSamples = () =>\n slider({\n min: 3,\n max: 100,\n value: numberOfSamplesDefault,\n step: 1,\n description: \"Number of samples\"\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1110,"value":"setS = () =>\n slider({\n min: sigmaMin,\n max: sigmaMax,\n value: sigmaDefault,\n step: 0.1,\n description: sText\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2872,"value":"setNumberOfStdDev = () =>\n slider({\n min: 1,\n max: 3,\n value: 1,\n step: 1,\n description: \"Number of standard deviations\"\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3177,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2247,"value":"md`#### Helper functions`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3597,"value":"getTex = symbolTex => html`${tex`${symbolTex}`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":757,"value":"getInfo = ({ mu, sigma, n, numberOfSamples }) => {\n var info = [];\n mu != undefined && info.push(`μ = ${mu}`);\n sigma != undefined && info.push(`σ = ${sigma}`);\n n != undefined && info.push(`n = ${n}`);\n numberOfSamples != undefined &&\n info.push(`${numberOfSamplesText} = ${numberOfSamples}`);\n return info.join(\", \");\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":837,"value":"getTraceErrorsEnvelope = ({\n x,\n y,\n yError,\n name,\n axis,\n fillcolor = lighterBlue,\n linecolor = lightBlue\n}) => {\n const { xaxis, yaxis, showlegend } = getAxis(axis);\n\n const polygonX = [...x, ...[...x].reverse()];\n\n const polygonY = polygonX.map((item, idx) => {\n if (idx < x.length) {\n return y[item] + yError[item];\n } else {\n return y[item] - yError[item];\n }\n });\n\n var traceStds = {\n x: polygonX,\n y: polygonY,\n fill: \"tozerox\",\n fillcolor,\n line: { color: linecolor },\n name: name,\n type: \"scatter\",\n xaxis,\n yaxis,\n showlegend\n };\n\n return traceStds;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2889,"value":"getPDF = ({ mu, sigma, numberOfStdDev }) => {\n const getX = (mu, sigma, numberOfStdDev) => {\n const min = mu - numberOfStdDev * sigma;\n const max = mu + numberOfStdDev * sigma;\n const unit = (max - min) / 100;\n return _.range(min, max, unit);\n };\n const x = getX(mu, sigma, numberOfStdDev);\n\n const getPDFs = (x, mu, sigma) => pdf(x, mu, sigma);\n const y = x.map(score => getPDFs(score, mu, sigma));\n\n const fractionsOfPopulation = {\n 1: 68,\n 2: 95,\n 3: 99.7\n };\n\n const fractionOfPopulation = fractionsOfPopulation[numberOfStdDev];\n\n return { x, y, fractionOfPopulation };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":931,"value":"getAxis = axis => {\n let xaxis, yaxis, showlegend;\n\n if (axis == 1) {\n xaxis = \"x1\";\n yaxis = \"y1\";\n showlegend = true;\n } else if (axis == 2) {\n xaxis = \"x2\";\n yaxis = \"y2\";\n showlegend = false;\n }\n\n // if axis is not defined, then it returns undefined which is required when there aren't subplots\n return { xaxis, yaxis, showlegend };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2242,"value":"md`#### Generate samples and traces`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":106,"value":"getRandomNormal = () => {\n const mu = mu_1;\n const sigma = sigma_1;\n const n = n_1;\n\n return {\n x: d3.range(n).map(() => d3.randomNormal(mu, sigma)()),\n y: d3.range(n).map(() => d3.randomNormal(muDefault, sigmaDefault)())\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":214,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":907,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":826,"value":"getTraceStdErrorsEnvelopeExperiment = ({\n experiment,\n axis,\n linecolor,\n fillcolor\n}) =>\n getTraceErrorsEnvelope({\n x: experiment.samples,\n y: experiment.means,\n yError: experiment.stdErrors,\n name: stdErrorSText,\n axis,\n linecolor,\n fillcolor\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2370,"value":"md`#### Colors`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1016,"value":"lighterGreen = \"rgba(0, 255, 0, 0.2)\";\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1018,"value":"lightGreen = \"rgba(0, 255, 0, 0.4)\";","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1024,"value":"lighterBlue = \"rgba(0, 199, 255, 0.2)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1025,"value":"lightBlue = \"rgba(0, 199, 255, 0.4)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1037,"value":"lighterRed = \"rgba(255, 0, 0, 0.2)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1038,"value":"lightRed = \"rgba(255, 0, 0, 0.2)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2833,"value":"transparent = \"rgba(0, 0, 0, 0)\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W3a","stdlib":"1","titling":"auto"}} 30:"$2f:body"

${name}