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{"name": "Harry's copy of Find sample mean, standard deviation, median and interquartile range, , ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"IQR": {"templateType": "anything", "description": "", "definition": "0.5*(r1[11]+r1[12])-0.5*(r1[3]+r1[4])", "group": "Ungrouped variables", "name": "IQR"}, "median": {"templateType": "anything", "description": "", "definition": "0.5*(r1[7]+r1[8])", "group": "Ungrouped variables", "name": "median"}, "mean": {"templateType": "anything", "description": "", "definition": "precround(mean(r0),2)", "group": "Ungrouped variables", "name": "mean"}, "r0": {"templateType": "anything", "description": "", "definition": "repeat(random(23..36),16)", "group": "Ungrouped variables", "name": "r0"}, "r1": {"templateType": "anything", "description": "", "definition": "sort(r0)", "group": "Ungrouped variables", "name": "r1"}, "upper_quartile": {"templateType": "anything", "description": "", "definition": "0.5*(r1[11]+r1[12])", "group": "Ungrouped variables", "name": "upper_quartile"}, "lower_quartile": {"templateType": "anything", "description": "", "definition": "0.5*(r1[3]+r1[4])", "group": "Ungrouped variables", "name": "lower_quartile"}, "standard_deviation": {"templateType": "anything", "description": "", "definition": "precround(stdev(r0,true),2)", "group": "Ungrouped variables", "name": "standard_deviation"}}, "parts": [{"variableReplacements": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{mean}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "mustBeReduced": false, "variableReplacements": [], "minValue": "{mean}", "showCorrectAnswer": true, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "type": "numberentry", "allowFractions": false, "marks": "1"}, {"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{standard_deviation}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "mustBeReduced": false, "variableReplacements": [], "minValue": "{standard_deviation}", "showCorrectAnswer": true, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "type": "numberentry", "allowFractions": false, "marks": "1"}, {"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{median}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "mustBeReduced": false, "variableReplacements": [], "minValue": "{median}", "showCorrectAnswer": true, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "type": "numberentry", "allowFractions": false, "marks": "1"}, {"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "maxValue": "{IQR}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "mustBeReduced": false, "variableReplacements": [], "minValue": "{IQR}", "showCorrectAnswer": true, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "type": "numberentry", "allowFractions": false, "marks": "1"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "prompt": "<table border=\"1\" align=\"center\">\n<tbody>\n<tr>\n<th>Sample Mean (2 dp)</th>\n<th>Sample Standard Deviation (2 dp)</th>\n<th>Median (exact value)</th>\n<th>Interquartile Range (exact value)</th>\n</tr>\n<tr align=\"center\">\n<th>[[0]]</th>\n<th>[[1]]</th>\n<th>[[2]]</th>\n<th>[[3]]</th>\n</tr>\n</tbody>\n</table>\n<p></p>", "marks": 0, "scripts": {}}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Sample of size $24$ is given in a table. Find sample mean, sample standard deviation, sample median and the interquartile range.</p>"}, "ungrouped_variables": ["r0", "median", "mean", "standard_deviation", "r1", "IQR", "lower_quartile", "upper_quartile"], "tags": [], "name": "Harry's copy of Find sample mean, standard deviation, median and interquartile range, , ", "variable_groups": [], "preamble": {"css": "", "js": ""}, "statement": "<h2>The following data are taken from a survey of primary school class sizes</h2>\n<table width=\"979\" height=\"55\" border=\"1\">\n<tbody>\n<tr>\n<td style=\"width: 100px;\">\n<h2>$\\var{r0[0]}$</h2>\n</td>\n<td style=\"width: 100px;\">\n<h2>$\\var{r0[1]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[2]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[3]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[4]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[5]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[6]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[7]}$</h2>\n</td>\n</tr>\n<tr>\n<td style=\"width: 100px;\">\n<h2>$\\var{r0[8]}$</h2>\n</td>\n<td style=\"width: 100px;\">\n<h2>$\\var{r0[9]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[10]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[11]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[12]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[13]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[14]}$</h2>\n</td>\n<td style=\"width: 109px;\">\n<h2>$\\var{r0[15]}$</h2>\n</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "advice": "<p><strong>Sample mean</strong></p>\n<p>The sample mean is $\\frac{\\var{sum(r0)}}{\\var{len(r0)}} = \\var{mean} $, to 2 decimal places.</p>\n<p><strong>&nbsp;</strong></p>\n<p><strong>Sample standard deviation</strong></p>\n<p>The formula required may be written as</p>\n<p>\\[s = \\sqrt{\\frac{ \\Sigma(x-\\bar{x})^2}{n-1}}\\]</p>\n<p>&nbsp; \\text{where }\\bar{x}\\text{ is the sample mean and n is the&nbsp; number of items in the data set (16 in this example). }</p>\n<p>This is very time consuming for all but the simplest of questions so you should use your calculator, in statistics mode, to find this.</p>\n<p><strong>&nbsp;</strong></p>\n<p></p>\n<p>If you rearrange the data in ascending order you get the following table:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n<td>$\\var{r1[5]}$</td>\n<td>$\\var{r1[6]}$</td>\n<td>$\\var{r1[7]}$</td>\n</tr>\n<tr>\n<td>$\\var{r1[8]}$</td>\n<td>$\\var{r1[9]}$</td>\n<td>$\\var{r1[10]}$</td>\n<td>$\\var{r1[11]}$</td>\n<td>$\\var{r1[12]}$</td>\n<td>$\\var{r1[13]}$</td>\n<td>$\\var{r1[14]}$</td>\n<td>$\\var{r1[15]}$</td>\n</tr>\n</tbody>\n</table>\n<p>Denote the ordered data by $x_j$, thus $x_{10}=\\var{r1[9]}$ for example.</p>\n<p><strong>Median:</strong> The median is the middle of the data once it has been ordered, it therefore lies between the 8th and 9th entries in the ordered table and is given by:</p>\n<p>\\[ \\frac{x_{8}+ x_{9}}{2}&nbsp; =&nbsp; \\frac{\\var{r1[7]}+ \\var{r1[8]}}{2}=\\var{median}\\]</p>\n<p><strong>Interquartile range:&nbsp;&nbsp; </strong>There<strong> </strong>is <strong>not</strong> one definitive method for finding the the interquartile range (IQR). We will use the following: as there is an even number of values, the Lower Quartile will be the 'median' of the first 8 values. Its position is calculated by finding</p>\n<p>\\[\\frac{n+1}{2}=\\frac{\\var{8+1}}{2}=4\\frac{1}{2}\\]</p>\n<p>Hence the Lower Quartile lies between the 4th and 5th entries in the ordered table.</p>\n<p>It is \\[x_4 + \\frac{1}{2}(x_5 - x_4) = \\var{r1[3]}+ \\frac{1}{2}(\\var{r1[4]} - \\var{r1[3]})=\\var{lower_quartile}\\]</p>\n<p>Once again as there is an even number of values, the Upper Quartile will lie between two values and its position is calculated by finding</p>\n<p>\\[x_{12} + \\frac{1}{2}(x_{13} - x_{12}) = \\var{r1[11]}+ \\frac{1}{2}(\\var{r1[12]} - \\var{r1[11]})=\\var{upper_quartile}\\]</p>\n<p></p>\n<p>The interquartile range (IQR) is defined to be</p>\n<p>\\[ \\text{Upper Quartile} &ndash; \\text{Lower Quartile} \\]</p>\n<p>and so in this case we have:</p>\n<p>\\[ \\text{Interquartile range} = \\var{upper_quartile}-\\var{lower_quartile}=\\var{iqr} \\]</p>", "extensions": ["stats"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}