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{"name": "Mean, SD, median and IQR", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>Find the mean, SD, median and IQR for the following sample of $\\var{n}$ data values:</p>\n<table>\n<tbody>\n<tr>\n<td>{r0[0]}</td>\n<td>{r0[1]}</td>\n<td>{r0[2]}</td>\n<td>{r0[3]}</td>\n<td>{r0[4]}</td>\n<td>{r0[5]}</td>\n<td>{r0[6]}</td>\n<td>{r0[7]}</td>\n<td>{r0[8]}</td>\n<td>{r0[9]}</td>\n<td>{r0[10]}</td>\n<td>{r0[11]}</td>\n</tr>\n<tr>\n<td>{r0[12]}</td>\n<td>{r0[13]}</td>\n<td>{r0[14]}</td>\n<td>{r0[15]}</td>\n<td>{r0[16]}</td>\n<td>{r0[17]}</td>\n<td>{r0[18]}</td>\n<td>{r0[19]}</td>\n<td>{r0[20]}</td>\n<td>{r0[21]}</td>\n<td>{r0[22]}</td>\n<td>{r0[23]}</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "name": "Mean, SD, median and IQR", "advice": "<p><strong>Sample mean:</strong> The sample mean is $\\frac{\\var{sum(r0)}}{\\var{len(r0)}} = \\var{precround(mean(r0),1)}$ to 1 decimal place.</p>\n<p><strong>Sample standard deviation:</strong> The sample standard deviation is $\\var{stdev(r0,true)}=\\var{siground(stdev(r0,true),3)}$ to 3 significant figures.</p>\n<p>If you order the data in increasing 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<tr>\n<td>$\\var{r1[16]}$</td>\n<td>$\\var{r1[17]}$</td>\n<td>$\\var{r1[18]}$</td>\n<td>$\\var{r1[19]}$</td>\n<td>$\\var{r1[20]}$</td>\n<td>$\\var{r1[21]}$</td>\n<td>$\\var{r1[22]}$</td>\n<td>$\\var{r1[23]}$</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 lies between the 12th and 13th entries in the ordered table and is given by:</p>\n<p>\\[0.5\\times x_{12}+0.5\\times x_{13} = 0.5\\times\\var{r1[11]}+0.5\\times \\var{r1[12]}=\\var{median}\\]</p>\n<p><strong>Interquartile range:&nbsp;</strong>As there is an even number of values, the Lower Quartile will lie between two values. Its position is calculated by finding</p>\n<p>\\[\\frac{n+1}{4}=\\frac{\\var{n+1}}{4}=6\\frac{1}{4}\\]</p>\n<p>Hence the Lower Quartile lies between the 6th and 7th entries in the ordered table.</p>\n<p>It is \\[0.75\\times x_6+0.25\\times x_7 = 0.75\\times\\var{r1[5]}+0.25\\times \\var{r1[6]}=\\var{lquartile}\\]</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>\\[\\frac{3(n+1)}{4}=\\frac{\\var{3*(n+1)}}{4}=18\\frac{3}{4}\\]</p>\n<p>Hence the Upper Quartile lies between the 18th and 19th entries in the ordered table.</p>\n<p>We find it is \\[0.25\\times x_{18}+0.75\\times x_{19} = 0.25\\times\\var{r1[17]}+0.75\\times \\var{r1[18]}=\\var{uquartile}\\]</p>\n<p>The interquartile range 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} = 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