// Numbas version: exam_results_page_options
{"name": "1 Calculate the measures of central tendency for a sample", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1 Calculate the measures of central tendency for a sample", "tags": [], "metadata": {"description": "<p>This question provides a list of data to the student. They are asked to find the&nbsp;mean, median, mode and range.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>A random sample of 20 residents from Newcastle were asked about the number of times they&nbsp;went&nbsp;to see a play at&nbsp;the&nbsp;theatre&nbsp;last year.</p>\n<p>Here is the list of their answers:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$\\var{a[0]}$</td>\n<td>$\\var{a[1]}$</td>\n<td>$\\var{a[2]}$</td>\n<td>$\\var{a[3]}$</td>\n<td>$\\var{a[4]}$</td>\n<td>$\\var{a[5]}$</td>\n<td>$\\var{a[6]}$</td>\n<td>$\\var{a[7]}$</td>\n<td>$\\var{a[8]}$</td>\n<td>$\\var{a[9]}$</td>\n</tr>\n<tr>\n<td>$\\var{a[10]}$</td>\n<td>$\\var{a[11]}$</td>\n<td>$\\var{a[12]}$</td>\n<td>$\\var{a[13]}$</td>\n<td>$\\var{a[14]}$</td>\n<td>$\\var{a[15]}$</td>\n<td>$\\var{a[16]}$</td>\n<td>$\\var{a[17]}$</td>\n<td>$\\var{a[18]}$</td>\n<td>$\\var{a[19]}$</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: center;\"></p>", "advice": "<h4>a)</h4>\n<p>The mean is the sum of all the&nbsp;responses ($\\sum x$) divided by the number of responses ($n$).</p>\n<p>Here, $n = 20$.</p>\n<p>\\begin{align} <br/>\\sum x &amp;= \\var{a[0]} + \\var{a[1]} +\\var{a[2]} +\\var{a[3]} +\\var{a[4]} +\\var{a[5]} +\\var{a[6]} +\\var{a[7]} +\\var{a[8]} +\\var{a[9]} + \\var{a[10]} + \\var{a[11]} +\\var{a[12]} +\\var{a[13]} +\\var{a[14]} +\\var{a[15]} +\\var{a[16]} +\\var{a[17]} +\\var{a[18]} +\\var{a[19]}&nbsp;\\\\<br/>&amp;= \\var{sum(a)} \\text{.}<br/>\\end{align}</p>\n<p><span face=\"Courier New\">Therefore we calculate the mean</span></p>\n<p><span face=\"Courier New\">\\begin{align} <br/>\\overline{x} &amp;= \\frac{\\sum x}{n}&nbsp;\\\\[0.5em]<br/>&amp;= \\frac{\\var{sum(a)}}{20}&nbsp;\\\\[0.5em]<br/>&amp;= \\var{mean} \\text{.}<br/>\\end{align}</span></p>\n<p><span face=\"Courier New\">&nbsp;</span></p>\n<h4>b)</h4>\n<p>The median is&nbsp;the middle value. We need to sort the list in order:</p>\n<p style=\"text-align: center;\">\\[ \\var{a_s[0]}, \\quad \\var{a_s[1]}, \\quad&nbsp;\\var{a_s[2]}, \\quad \\var{a_s[3]}, \\quad \\var{a_s[4]}, \\quad \\var{a_s[5]}, \\quad \\var{a_s[6]}, \\quad \\var{a_s[7]}, \\quad \\var{a_s[8]}, \\quad \\var{a_s[9]}, \\quad \\var{a_s[10]}, \\quad \\var{a_s[11]}, \\quad&nbsp;\\var{a_s[12]}, \\quad \\var{a_s[13]}, \\quad \\var{a_s[14]}, \\quad \\var{a_s[15]}, \\quad \\var{a_s[16]}, \\quad \\var{a_s[17]}, \\quad \\var{a_s[18]}, \\quad \\var{a_s[19]} \\]</p>\n<p>There is an even number of responses, so there are&nbsp;two numbers in the middle (10<sup>th</sup> and 11<sup>th</sup> place). To&nbsp;find the median, we need to&nbsp;find the mean of these two numbers $\\var{a_s[9]}$ and $\\var{a_s[10]}$:</p>\n<p>\\begin{align} <br/>\\frac{\\var{a_s[9]} + \\var{a_s[10]}}{2} &amp;= &nbsp;\\frac{\\var{a_s[9] + a_s[10]}}{2} \\\\<br/>&amp;= \\var{median} \\text{.}&nbsp;<br/>\\end{align}</p>\n<p>&nbsp;</p>\n<h4>c)</h4>\n<p>The mode is the value that occurs the most often in the data.</p>\n<p>To find a mode, we can look at our sorted list:</p>\n<p style=\"text-align: center;\">$\\var{a_s[0]}, \\var{a_s[1]}, \\var{a_s[2]}, \\var{a_s[3]}, \\var{a_s[4]}, \\var{a_s[5]}, \\var{a_s[6]}, \\var{a_s[7]}, \\var{a_s[8]}, \\var{a_s[9]}, \\var{a_s[10]}, \\var{a_s[11]}, \\var{a_s[12]}, \\var{a_s[13]}, \\var{a_s[14]}, \\var{a_s[15]}, \\var{a_s[16]}, \\var{a_s[17]}, \\var{a_s[18]}, \\var{a_s[19]}$.</p>\n<p>We&nbsp;notice that $\\var{mode1}$ occurs the most ($\\var{modetimes[mode1]}$ times) so $\\var{mode1}$ is the mode.</p>\n<p></p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a2": {"name": "a2", "group": "Ungrouped variables", "definition": "repeat(random(0..8), 20)", "description": "<p>Option&nbsp;2 for the list. 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Only used if there is only one mode.</p>", "templateType": "anything", "can_override": false}, "a_s": {"name": "a_s", "group": "final list", "definition": "sort(a)", "description": "<p>Sorted list.</p>", "templateType": "anything", "can_override": false}, "modea2": {"name": "modea2", "group": "Ungrouped variables", "definition": "mode(a2)", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "shuffle([ random(0..1),\n  2, \n  random(4..6),\n  random(0..3 except 2), \n  random(0..3 except 2),\n  random(4..6),\n  2,\n  2,\n  random(4..6),\n  random(7..8),\n  random(0..3 except 2 except 1), \n  random(4..6),\n  2,\n  random(1..3 except 2), \n  random(7..8),\n  2,\n  random(7..8),\n  random(4..6), \n  random(0..3 except 2), \n  2\n])", "description": "<p>Option 3 for the list. 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