// Numbas version: exam_results_page_options
{"name": "Mean\u00b1error 1", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Mean\u00b1error 1", "tags": [], "metadata": {"description": "<p>A measurement is performed multiple times for the same object, the student will</p>\n<ul>\n<li>calculate the mean result</li>\n<li>calculate the standard error on the mean&nbsp;</li>\n<li>write the mean&plusmn;error to the correct precision as defined by the error written to 1 significant figure&nbsp;</li>\n</ul>\n<p>Advice is provided including on performing the calculations in Python or spreedsheets together with further reading.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Using a metre rule, you performed serveral repeat measurements of your friend's height :</p>\n<div data-jme-visible=\"n=3\">\n<table style=\"width: 200px; height: 108px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95.3594px; height: 18px;\">Measurement</td>\n<td style=\"width: 100.828px; height: 18px;\">Height [m]</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95.3594px; height: 18px;\">1</td>\n<td style=\"width: 100.828px; height: 18px;\">$\\var{data[0]}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95.3594px; height: 18px;\">2</td>\n<td style=\"width: 100.828px; height: 18px;\">$\\var{data[1]}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95.3594px; height: 18px;\">3</td>\n<td style=\"width: 100.828px; height: 18px;\">$\\var{data[2]}$</td>\n</tr>\n</tbody>\n</table>\n</div>\n<div data-jme-visible=\"n=5\">\n<table style=\"width: 195px; height: 108px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95px; height: 18px;\">Measurement</td>\n<td style=\"width: 100px; height: 18px;\">Height [m]</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95px; height: 18px;\">1</td>\n<td style=\"width: 100px; height: 18px;\">$\\var{data[0]}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95px; height: 18px;\">2</td>\n<td style=\"width: 100px; height: 18px;\">$\\var{data[1]}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95px; height: 18px;\">3</td>\n<td style=\"width: 100px; height: 18px;\">$\\var{data[2]}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95px; height: 18px;\">4</td>\n<td style=\"width: 100px; height: 18px;\">$\\var{data[3]}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 95px; height: 18px;\">5</td>\n<td style=\"width: 100px; height: 18px;\">$\\var{data[4]}$</td>\n</tr>\n</tbody>\n</table>\n</div>", "advice": "<p>When studying a parameter of interest, scientists record many repeat measurements. This is to help account for the imperfect precision and accuracy of measurements. <br/>The best estimate of a parameter, $x$, is the mean of \ud835\udc41 repeat measurements: <br/>\\[ \\bar{x} = \\frac{1}{N}(x_1 + x_2 +\\cdots+ x_N) = \\frac{1}{N}\\sum_{i=1}^N x_i \\]</p>\n<p>In this example of measuring the height of your friend, the values of $x_i$ are:</p>\n<table style=\"width: 238px; height: 72px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"width: 68px; height: 18px;\">$x_1$ =</td>\n<td style=\"width: 137px; height: 18px;\">$\\var{data[0]}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 68px; height: 18px;\">$x_2$ =</td>\n<td style=\"width: 137px; height: 18px;\">$\\var{data[1]}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 68px; height: 18px;\">$x_3$ =</td>\n<td style=\"width: 137px; height: 18px;\">$\\var{data[2]}$</td>\n</tr>\n</tbody>\n</table>\n<p>Using Python, this can be calculated via:</p>\n<blockquote>\n<p>import numpy as np</p>\n<p>x = np.array([$\\var{data[0]}$, $\\var{data[1]}$, $\\var{data[2]}$])</p>\n<p>mean = np.mean(x)</p>\n</blockquote>\n<p>Or if using a spreadsheet:</p>\n<blockquote>\n<p>=average($\\var{data[0]}$, $\\var{data[1]}$, $\\var{data[2]}$)</p>\n</blockquote>\n<p>To determine how precise the mean is we need to calculate the average deviation of our measurements $x_i$ with the mean value $\\bar{x}$.&nbsp;</p>\n<p>In this case, we have repeated measurements of the <strong>same object</strong> (height of your friend) so we can use the <strong>population</strong> standard deviation:</p>\n<p>\\[ \\sigma = \\sqrt{\\frac{1}{N} \\sum_{i=1}^N (x_i - \\overline{x})^2} \\]</p>\n<p>which can be calculated using Python via:</p>\n<blockquote>\n<p>sd = np.std(x)</p>\n</blockquote>\n<p>or via a spreadsheet approach:</p>\n<blockquote>\n<p>=stdevp($\\var{data[0]}$, $\\var{data[1]}$, $\\var{data[2]}$)</p>\n</blockquote>\n<p>We can use this standard deviation to determine the error on the mean value:</p>\n<p>\\[ \\alpha = \\frac{\\sigma}{\\sqrt{N}} \\]</p>\n<p>which indicates that the more repeat measurements we perform, the smaller the error on the mean becomes.&nbsp;</p>\n<p>Typically, the error on the mean is only precise to 1 significant figure. It requires 1000's of repeat measurements for an additional significant figure of precision. Hence we round $\\alpha$ to 1 significant figure and then round the mean to the same number of decimal places to produce the final result:</p>\n<p>\\[ \\bar{x} \\pm \\alpha \\]&nbsp;</p>\n<p>See this useful book for more details:</p>\n<p>Hughes, I., Hase, T. , \"Measurements and Their Uncertainties\", Oxford University Press (2010)</p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"data": {"name": "data", "group": "Ungrouped variables", "definition": "repeat(precround(normalsample(mu,sigma), precision), n)", "description": "<p>Produce values of measurements by repeatively sampling a normal distribution with mean (mu) and standard deviation (sigma).</p>\n<p></p>", "templateType": "anything", "can_override": false}, "result": {"name": "result", "group": "Ungrouped variables", "definition": "mean(data)", "description": "<p>Mean of repeat measurements&nbsp;</p>", "templateType": "anything", "can_override": false}, "error": {"name": "error", "group": "Ungrouped variables", "definition": "population_stdev(data)/sqrt(n)", "description": "<p>Error on the mean of repeat measurements.</p>", "templateType": "anything", "can_override": false}, "finalresult": {"name": "finalresult", "group": "Ungrouped variables", "definition": "precround(result, finalprecision)", "description": "<p>Round mean result to the final precision.&nbsp;</p>", "templateType": "anything", "can_override": false}, "finalerror": {"name": "finalerror", "group": "Ungrouped variables", "definition": "siground(error, 1)", "description": "<p>Round the error to 1 significant figure.&nbsp;</p>", "templateType": "anything", "can_override": false}, "mu": {"name": "mu", "group": "Ungrouped variables", "definition": "random(180..190#0.5)*0.01", "description": "<p>Mean of normal distribution used to generate measurement data.&nbsp;</p>", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "5", "description": "<p>Number of repeat measurements. 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