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{"name": "nuExam06 - Mean & Standard deviation", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["data", "sigma", "x", "mean", "var"], "name": "nuExam06 - Mean & Standard deviation", "preamble": {"js": "", "css": ""}, "rulesets": {}, "extensions": ["stats"], "statement": "", "parts": [{"prompt": "<p>Find the <strong>population </strong>standard deviation $\\sigma$ of the following list&nbsp;of numbers {data}.</p>\n<p>Give your answer correct to 2 d.p.</p>", "variableReplacements": [], "showPrecisionHint": false, "showCorrectAnswer": true, "stepsPenalty": "0", "customMarkingAlgorithm": "", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "marks": "5", "maxValue": "{sigma}", "steps": [{"prompt": "<p>To calculate the <strong>population</strong> standard deviation, we can use the formula:</p>\n<p>Standard deviation = $\\sqrt{\\frac{\\Sigma (x-\\text{mean})^2}{n}}$</p>\n<p></p>\n<p>So we must:</p>\n<ul>\n<li>find the mean</li>\n<li>subtract the mean individually from each of&nbsp;our values</li>\n<li>square each result</li>\n<li>add up these results. This is the '$\\Sigma (x-\\text{mean})^2$' part in the formula</li>\n<li>divide by the number of values</li>\n<li>finally, find the square root to&nbsp;obtain the standard deviation</li>\n</ul>", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "unitTests": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "customName": "", "useCustomName": false, "customMarkingAlgorithm": ""}], "minValue": "{sigma}", "customName": "", "useCustomName": false, "strictPrecision": false, "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "precisionMessage": "<p>You have not given your answer to the correct number of decimal places.</p>", "variableReplacementStrategy": "originalfirst", "precision": "2", "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "scripts": {}, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "precisionType": "dp", "mustBeReducedPC": 0, "unitTests": []}], "metadata": {"description": "<p>Just showing how to use the stdev function from the stats extension&nbsp;to calculate the standard deviation of a list of numbers.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "variables": {"x": {"description": "", "name": "x", "templateType": "anything", "definition": "random(4,5,8)", "group": "Ungrouped variables"}, "mean": {"description": "", "name": "mean", "templateType": "anything", "definition": "mean(data)", "group": "Ungrouped variables"}, "sigma": {"description": "", "name": "sigma", "templateType": "anything", "definition": "stdev(data)", "group": "Ungrouped variables"}, "data": {"description": "", "name": "data", "templateType": "anything", "definition": "repeat(random(1..30),x)", "group": "Ungrouped variables"}, "var": {"description": "", "name": "var", "templateType": "anything", "definition": "sigma^2", "group": "Ungrouped variables"}}, "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<p>To calculate the <strong>population </strong>standard deviation, $\\sigma$, we can use the formula:</p>\n<p>Population standard deviation, $\\sigma = \\sqrt{\\frac{\\Sigma (x-\\text{mean})^2}{n}}$</p>\n<p></p>\n<p>(note this is slightly different to the formula used to calculate the <strong>sample</strong> standard deviation, $s$)</p>\n<p></p>\n<p>So we must:</p>\n<ul>\n<li>Find the mean:&nbsp;\n<ul>\n<li>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;mean = $\\frac{(\\var{data[0]}+\\var{data[1]}+...)}{\\var{x}}=\\var{mean}$</li>\n</ul>\n</li>\n<li>subtract the mean individually from each of&nbsp;our values:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\n<ul>\n<li>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;we obtain $\\var{data[0]}-\\var{mean}=\\var{data[0]-mean}$,&nbsp; $\\var{data[1]}-\\var{mean}=\\var{data[1]-mean}$, etc.</li>\n</ul>\n</li>\n<li>square each result\n<ul>\n<li>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(\\var{data[0]-mean})^2=\\var{(data[0]-mean)^2}$,&nbsp;&nbsp;$(\\var{data[1]-mean})^2=\\var{(data[1]-mean)^2}$,&nbsp; etc.</li>\n</ul>\n</li>\n<li>add up these results. This is the '$\\Sigma (x-\\text{mean})^2$' part in the formula\n<ul>\n<li>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\\var{(data[0]-mean)^2}+\\var{(data[1]-mean)^2}+... = \\var{x*var}$</li>\n</ul>\n</li>\n<li>divide by the number of values\n<ul>\n<li>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\\var{x*var} \\div \\var{x} = \\var{var}$</li>\n</ul>\n</li>\n<li>finally, find the square root&nbsp;to obtain the standard deviation:\n<ul>\n<li>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;$\\sigma$= $\\sqrt{\\var{var}}=\\var{precround(sigma,2)}$</li>\n</ul>\n</li>\n</ul>", "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}