// Numbas version: exam_results_page_options
{"name": "Measures of central tendency from a list", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Measures of central tendency from a list", "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>The average number of tackles per game was measured during the last Rugby Union World Cup.</p>\n<p>Here is a random sample from 20 forwards:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$\\var{x[0]}$</td>\n<td>$\\var{x[1]}$</td>\n<td>$\\var{x[2]}$</td>\n<td>$\\var{x[3]}$</td>\n<td>$\\var{x[4]}$</td>\n<td>$\\var{x[5]}$</td>\n<td>$\\var{x[6]}$</td>\n<td>$\\var{x[7]}$</td>\n<td>$\\var{x[8]}$</td>\n<td>$\\var{x[9]}$</td>\n</tr>\n<tr>\n<td>$\\var{x[10]}$</td>\n<td>$\\var{x[11]}$</td>\n<td>$\\var{x[12]}$</td>\n<td>$\\var{x[13]}$</td>\n<td>$\\var{x[14]}$</td>\n<td>$\\var{x[15]}$</td>\n<td>$\\var{x[16]}$</td>\n<td>$\\var{x[17]}$</td>\n<td>$\\var{x[18]}$</td>\n<td>$\\var{x[19]}$</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{x[0]} + \\var{x[1]} +\\var{x[2]} +\\var{x[3]} +\\var{x[4]} +\\var{x[5]} +\\var{x[6]} +\\var{x[7]} +\\var{x[8]} +\\var{x[9]} + \\var{x[10]} + \\var{x[11]} +\\var{x[12]} +\\var{x[13]} +\\var{x[14]} +\\var{x[15]} +\\var{x[16]} +\\var{x[17]} +\\var{x[18]} +\\var{x[19]}&nbsp;\\\\<br/>&amp;= \\var{sum(x)} \\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(x)}}{20}&nbsp;\\\\[0.5em]<br/>&amp;= \\var{meanx} \\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 ascending order:</p>\n<p style=\"text-align: center;\">\\[ \\var{s[0]}, \\quad \\var{s[1]}, \\quad&nbsp;\\var{s[2]}, \\quad \\var{s[3]}, \\quad \\var{s[4]}, \\quad \\var{s[5]}, \\quad \\var{s[6]}, \\quad \\var{s[7]}, \\quad \\var{s[8]}, \\quad \\var{s[9]}, \\quad \\var{s[10]}, \\quad \\var{s[11]}, \\quad&nbsp;\\var{s[12]}, \\quad \\var{s[13]}, \\quad \\var{s[14]}, \\quad \\var{s[15]}, \\quad \\var{s[16]}, \\quad \\var{s[17]}, \\quad \\var{s[18]}, \\quad \\var{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 $\\var{s[9]}$ and $\\var{s[10]}$:</p>\n<p>\\begin{align} <br/>\\frac{\\var{s[9]} + \\var{s[10]}}{2} &amp;= &nbsp;\\frac{\\var{s[9] + s[10]}}{2} \\\\<br/>&amp;= \\var{medianx} \\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;\">$\\var{s[0]}, \\var{s[1]}, \\var{s[2]}, \\var{s[3]}, \\var{s[4]}, \\var{s[5]}, \\var{s[6]}, \\var{s[7]}, \\var{s[8]}, \\var{s[9]}, \\var{s[10]}, \\var{s[11]}, \\var{s[12]}, \\var{s[13]}, \\var{s[14]}, \\var{s[15]}, \\var{s[16]}, \\var{s[17]}, \\var{s[18]}, \\var{s[19]}$.</p>\n<p>We notice that the values $\\var{modex}$ occur the most. The lowest of these values is $\\var{modex[0]}$.</p>\n<p>&nbsp;</p>\n<h4>d)</h4>\n<p>Range is the difference between the highest and the lowest value in the data.</p>\n<p>To find this, we subtract the lowest value from the highest value:</p>\n<p>\\[ \\var{max(x)} - \\var{min(x)} = \\var{rangex} \\text{.}\\]</p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"modea1": {"name": "modea1", "group": "Ungrouped variables", "definition": "mode(a1)", "description": "", "templateType": "anything", "can_override": false}, "mean": {"name": "mean", "group": "final list", "definition": "mean(a)", "description": "", "templateType": "anything", "can_override": false}, "modea2": {"name": "modea2", "group": "Ungrouped variables", "definition": "mode(a2)", "description": "", "templateType": "anything", "can_override": false}, "sigm": {"name": "sigm", "group": "Ungrouped variables", "definition": "5", "description": "", "templateType": "anything", "can_override": false}, "medianx": {"name": "medianx", "group": "Ungrouped variables", "definition": "median(x)", "description": "", "templateType": "anything", "can_override": false}, "meanx": {"name": "meanx", "group": "Ungrouped variables", "definition": "mean(x)", "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. Ensures there is only one mode (2) while still randomising the data.</p>", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "repeat(precround(normalsample(mu,sigm),0),20)", "description": "", "templateType": "anything", "can_override": false}, "mu": {"name": "mu", "group": "Ungrouped variables", "definition": "23", "description": "", "templateType": "anything", "can_override": false}, "rangex": {"name": "rangex", "group": "Ungrouped variables", "definition": "range(x)", "description": "", "templateType": "anything", "can_override": false}, "range": {"name": "range", "group": "final list", "definition": "max(a) - min(a)", "description": "", "templateType": "anything", "can_override": false}, "modex": {"name": "modex", "group": "Ungrouped variables", "definition": "mode(x)", "description": "", "templateType": "anything", "can_override": false}, "mode1": {"name": "mode1", "group": "final list", "definition": "mode[0]", "description": "<p>Mode as a value.</p>", "templateType": "anything", "can_override": false}, "modetimes": {"name": "modetimes", "group": "final list", "definition": "map(\nlen(filter(x=j,x,a)),\nj, 0..8)", "description": "<p>The vector of&nbsp;number of times of each value in the data.</p>", "templateType": "anything", "can_override": false}, "median": {"name": "median", "group": "final list", "definition": "median(a)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "repeat(random(0..8), 20)", "description": "<p>Option 1 for the list. Only used if there is only one mode.</p>", "templateType": "anything", "can_override": false}, "mode": {"name": "mode", "group": "final list", "definition": "mode(a)", "description": "<p>Mode as a vector.</p>", "templateType": "anything", "can_override": false}, "as": {"name": "as", "group": "final list", "definition": "sort(a)", "description": "<p>Sorted list.</p>", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "sort(x)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "repeat(random(0..8), 20)", "description": "<p>Option&nbsp;2 for the list. Only used if there is only one mode and option 1 was not used.</p>", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "final list", "definition": "if(len(modea1) = 1, a1, if(len(modea2) = 1, a2, a3))", "description": "<p>The final list.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["modea1", "modea2", "a1", "a2", "a3", "mu", "sigm", "x", "modex", "meanx", "medianx", "rangex", "s"], "variable_groups": [{"name": "final list", "variables": ["a", "as", "mean", "median", "mode", "mode1", "range", "modetimes"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the mean number of tackles.</p>", "minValue": "meanx", "maxValue": "meanx", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the median number of tackles.</p>", "minValue": "medianx", "maxValue": "medianx", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the mode number of tackles. If there is more than one modal value, enter the smallest modal value.</p>", "minValue": "modex[0]", "maxValue": "modex[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the range of the number of tackles.</p>", "minValue": "rangex", "maxValue": "rangex", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}, {"name": "Pete Philipson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/13881/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}, {"name": "Pete Philipson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/13881/"}]}