// Numbas version: exam_results_page_options
{"name": "STAT8013 Sample standard deviation", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "STAT8013 Sample standard deviation", "tags": [], "metadata": {"description": "<p>Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>The following table gives the examination marks in {exam1} and in {exam2} and their total for a&nbsp;sample of $n=\\var{n}$ students.</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th align=\"left\">{exam1}</th>\n<td>$\\var{r0[0]}$</td>\n<td>$\\var{r0[1]}$</td>\n<td>$\\var{r0[2]}$</td>\n<td>$\\var{r0[3]}$</td>\n<td>$\\var{r0[4]}$</td>\n<td>$\\var{r0[5]}$</td>\n<td>$\\var{r0[6]}$</td>\n<td>$\\var{r0[7]}$</td>\n<td>$\\var{r0[8]}$</td>\n<td>$\\var{r0[9]}$</td>\n<th align=\"left\">Mean = $\\var{mean1}$</th>\n</tr>\n<tr>\n<th align=\"left\">{exam2}</th>\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<td>$\\var{r1[8]}$</td>\n<td>$\\var{r1[9]}$</td>\n<th align=\"left\">Mean = $\\var{mean2}$</th>\n</tr>\n<tr>\n<th align=\"left\">{total}</th>\n<td>$\\var{sscores[0]}$</td>\n<td>$\\var{sscores[1]}$</td>\n<td>$\\var{sscores[2]}$</td>\n<td>$\\var{sscores[3]}$</td>\n<td>$\\var{sscores[4]}$</td>\n<td>$\\var{sscores[5]}$</td>\n<td>$\\var{sscores[6]}$</td>\n<td>$\\var{sscores[7]}$</td>\n<td>$\\var{sscores[8]}$</td>\n<td>$\\var{sscores[9]}$</td>\n<th align=\"left\">Mean = $\\var{overallmean}$</th>\n</tr>\n</tbody>\n</table>\n<p>Find the sample standard deviation for each of {exam1}, {exam2} and Total Score.</p>", "advice": "<p>The solution to the first part is here &ndash; the other parts can be done in the same way. Note that in practice we use statistical software such as Minitab.</p>\n<p>For {exam1} we have the mean is:</p>\n<p>\\[\\simplify[]{({r0[0]} + {r0[1]} + {r0[2]} + {r0[3]} + {r0[4]} + {r0[5]} + {r0[6]} + {r0[7]} + {r0[8]} + {r0[9]}) / {n} = {mean1}}\\]</p>\n<p>The sample variance is given by the formula:</p>\n<p>\\[\\textrm{Sample Variance} = \\frac{1}{n-1}\\left(\\sum_{j=1}^{n}x_j^2 -n\\mu^2\\right)\\]</p>\n<p>where the $x_j$ are the exam scores for {exam1}, $n=\\var{n}$ the number of students and $\\mu=\\var{mean1}$ the sample mean.</p>\n<p>We find that<br/>\\[\\begin{eqnarray*}\\sum_{j=1}^{n}x_j^2 &amp;=&amp; \\simplify[]{({r0[0]}^2 + {r0[1]}^2 + {r0[2]}^2 + {r0[3]}^2 + {r0[4]}^2 + {r0[5]}^2 + {r0[6]}^2 + {r0[7]}^2 + {r0[8]}^2 + {r0[9]}^2)}\\\\ &amp;=&amp; \\var{ssq1}\\\\ \\\\ \\\\ n\\mu^2 &amp;=&amp;\\var{n} \\times\\var{mean1}^2\\\\ &amp;=&amp; \\var{n*mean1^2} \\end{eqnarray*} \\]<br/>Hence substituting these values into the formula we find that:</p>\n<p>\\[\\begin{eqnarray*} \\textrm{Sample Variance} &amp;=&amp; \\frac{1}{\\var{n-1}}\\left(\\var{ssq1}-\\var{n*mean1^2}\\right)\\\\ &amp;=&amp; \\var{var1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<p>The Sample Standard Deviation is then the square root of the Sample Variance i.e.</p>\n<p>Sample Standard Deviation = $\\sqrt{\\var{var1}} = \\var{stdev1}$ to one decimal place.</p>\n<p></p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"stdevoverall": {"name": "stdevoverall", "group": "Ungrouped variables", "definition": "precround(stdev(sscores,true),1)", "description": "", "templateType": "anything", "can_override": false}, "exam2": {"name": "exam2", "group": "Ungrouped variables", "definition": "'Cell Biology'", "description": "", "templateType": "anything", "can_override": false}, "mean2": {"name": "mean2", "group": "Ungrouped variables", 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