// Numbas version: exam_results_page_options
{"name": "Measures of central tendency from frequency tables", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>The total number of goals scored during a random sample of $30$ Premier League football matches are shown below:</p>\n<table border=\"1\" cellspacing=\"10\" cellpadding=\"10\">\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<tr>\n<td>$\\var{a[20]}$</td>\n<td>$\\var{a[21]}$</td>\n<td>$\\var{a[22]}$</td>\n<td>$\\var{a[23]}$</td>\n<td>$\\var{a[24]}$</td>\n<td>$\\var{a[25]}$</td>\n<td>$\\var{a[26]}$</td>\n<td>$\\var{a[27]}$</td>\n<td>$\\var{a[28]}$</td>\n<td>$\\var{a[29]}$</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "variables": {"modea3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "mode(a3)", "name": "modea3"}, "modea2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "mode(a2)", "name": "modea2"}, "a3": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "shuffle(repeat(0, 7) + repeat(1, 10) + repeat(2, 5) + repeat(3, 4) + repeat(4, 2) + repeat(5,1) + repeat(6,1))", "name": "a3"}, "mode": {"templateType": "anything", "description": "", "group": "Final data", "definition": "m[0]", "name": "mode"}, "a1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "shuffle(repeat(random(0..1), 13) + 0 + 2 + 2 + repeat(random(2..3), 10) + repeat(random(4..5), 3) + random(0..6))", "name": "a1"}, "freq": {"templateType": "anything", "description": "", "group": "Final data", "definition": "map(\nlen(filter(x=j,x,a)),\nj, 0..6)", "name": "freq"}, "median": {"templateType": "anything", "description": "", "group": "Final data", "definition": "median(a)", "name": "median"}, "a2": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "shuffle(repeat(random(0..1), 13) + 0 + 2 + 2 + repeat(random(2..3), 10) + repeat(random(4..5), 3) + random(0..6))", "name": "a2"}, "mean": {"templateType": "anything", "description": "", "group": "Final data", "definition": "mean(a)", "name": "mean"}, "as": {"templateType": "anything", "description": "", "group": "Final data", "definition": "sort(a)", "name": "as"}, "a": {"templateType": "anything", "description": "", "group": "Final data", "definition": "if(len(modea1) = 1, a1, if(len(modea2) = 1, a2, a3))", "name": "a"}, "m": {"templateType": "anything", "description": "", "group": "Final data", "definition": "mode(a)", "name": "m"}, "modea1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "mode(a1)", "name": "modea1"}}, "tags": [], "extensions": ["stats"], "name": "Measures of central tendency from frequency tables", "metadata": {"description": "<p>Given a table of data, calculate the mean, mode and median, and complete a frequency table.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "advice": "<h4>a)</h4>\n<p>Organising the data in a frequency table helps to make mistakes less likely when calculating statistics from our data, summarising the responses all in one place with fewer numbers.</p>\n<p>Each row of the frequency column gives&nbsp;the number of Premier League football matches with the corresponding number of goals.</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th>Total number of goals scored</th>\n<th>Frequency</th>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$0$</td>\n<td style=\"text-align: left;\">$\\var{freq[0]}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$1$</td>\n<td style=\"text-align: left;\">$\\var{freq[1]}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$2$</td>\n<td style=\"text-align: left;\">$\\var{freq[2]}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$3$</td>\n<td style=\"text-align: left;\">$\\var{freq[3]}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$4$</td>\n<td style=\"text-align: left;\">$\\var{freq[4]}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$5$</td>\n<td style=\"text-align: left;\">$\\var{freq[5]}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$6$</td>\n<td style=\"text-align: left;\">$\\var{freq[6]}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">Total</td>\n<td style=\"text-align: left;\">$30$</td>\n</tr>\n</tbody>\n</table>\n<p>Always remember to check whether your frequency column adds up to the total (here, it is $30$) to make sure you have not left out any responses.</p>\n<h4>b)</h4>\n<h4>Mean</h4>\n<p>The mean number of goals is the total number of goals, $\\sum x$, divided by the number of football matches in the sample, $n$.</p>\n<p>\\begin{align} &nbsp;<br/>\\sum x &amp;= 0 \\times \\var{freq[0]} +&nbsp;1 \\times \\var{freq[1]} +&nbsp;2 \\times \\var{freq[2]} +&nbsp;3 \\times \\var{freq[3]} +&nbsp;4 \\times \\var{freq[4]} +&nbsp;5 \\times \\var{freq[5]} +&nbsp;6 \\times \\var{freq[6]}<br/> \\\\<br/>&amp;=&nbsp;0 + \\var{1*freq[1]} + \\var{2*freq[2]}&nbsp;+ \\var{3*freq[3]}&nbsp;+ \\var{4*freq[4]}&nbsp;+ \\var{5*freq[5]}&nbsp;+ \\var{6*freq[6]}&nbsp;\\\\&amp;= \\var{sum(a)} \\text{.}<br/>\\end{align}</p>\n<p>The total number of football matches $n$ is $30$.</p>\n<p>Therefore the mean is</p>\n<p><span face=\"Courier New\">\\begin{align} <br/>\\bar{x} &amp;= \\frac{\\sum x}{n}&nbsp;\\\\<br/>&amp;= \\frac{\\var{sum(a)}}{30}&nbsp;\\\\<br/>&amp;= \\var{precround(mean, 2)} \\text{.} <br/>\\end{align}</span></p>\n<p></p>\n<h4>Mode</h4>\n<p>The mode is the value with the highest frequency. Here, the mode is $\\var{mode}$ goal<span data-jme-visible=\"mean &lt;&gt; 1\">s</span>, with frequency $\\var{freq[mode]}$.</p>\n<h4>Median</h4>\n<p>The median is the \"middle\" value in the sample, when&nbsp;arranged in ascending order.</p>\n<p>To find the middle position within a data set, we take the sample size, add $1$, then divide by $2$. For our data set, the middle position is</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{n+1}{2}=\\frac{30+1}{2}=15.5.$</p>\n<p>As there is not actually a $15.5$<sup>th</sup>&nbsp;position, we need to find the mean of the&nbsp;$15$<sup>th</sup>&nbsp;and $16$<sup>th</sup>&nbsp;values. We can count from the top of the table until we locate rows where these values lie, as the numbers in the table are already sorted by order.</p>\n<p><span data-jme-visible=\"as[14] = as[15]\">Here, both $15$<sup>th</sup> and $16$<sup>th</sup> value lie in the row $\\var{as[14]}$.</span><span data-jme-visible=\"as[14] &lt;&gt; as[15]\">Here, the $15$<sup>th</sup> value lies in the row $\\var{as[14]}$ while the $16$<sup>th</sup> value lies in the row $\\var{as[15]}$.</span></p>\n<p><span data-jme-visible=\"as[14] = as[15]\">As $15$<sup>th</sup> value $= 16$<sup>th</sup> value $= \\var{as[14]}$, the median is $\\var{as[14]}$.</span><span data-jme-visible=\"as[14] &lt;&gt; as[15]\">&nbsp;The $15$<sup>th</sup> value $= \\var{as[14]}$ and $16$<sup>th</sup> value $= \\var{as[15]}$ and their mean is given by</span></p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"as[14] &lt;&gt; as[15]\">$ \\displaystyle \\frac{\\var{as[14]} + \\var{as[15]}}{2} = &nbsp;\\frac{\\var{as[14] + as[15]}}{2} = \\var{median}{.}$</span></p>\n<p><span data-jme-visible=\"as[14] &lt;&gt; as[15]\">This is the median for this data.</span></p>\n<p></p>", "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": "1000"}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a1", "modea1", "a2", "modea2", "a3", "modea3"], "parts": [{"extendBaseMarkingAlgorithm": true, "gaps": [{"extendBaseMarkingAlgorithm": true, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "unitTests": [], "variableReplacements": [], "maxValue": "freq[0]", "correctAnswerStyle": "plain", "minValue": "freq[0]"}, {"extendBaseMarkingAlgorithm": true, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "unitTests": [], "variableReplacements": [], "maxValue": "freq[1]", "correctAnswerStyle": "plain", "minValue": "freq[1]"}, {"extendBaseMarkingAlgorithm": true, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "unitTests": [], "variableReplacements": [], "maxValue": "freq[2]", "correctAnswerStyle": "plain", "minValue": "freq[2]"}, {"extendBaseMarkingAlgorithm": true, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "unitTests": [], "variableReplacements": [], "maxValue": "freq[3]", "correctAnswerStyle": "plain", "minValue": "freq[3]"}, {"extendBaseMarkingAlgorithm": true, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "unitTests": [], "variableReplacements": [], "maxValue": "freq[4]", "correctAnswerStyle": "plain", "minValue": "freq[4]"}, {"extendBaseMarkingAlgorithm": true, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "unitTests": [], "variableReplacements": [], "maxValue": "freq[5]", "correctAnswerStyle": "plain", "minValue": "freq[5]"}, {"extendBaseMarkingAlgorithm": true, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "unitTests": [], "variableReplacements": [], "maxValue": "freq[6]", "correctAnswerStyle": "plain", "minValue": "freq[6]"}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "marks": 0, "unitTests": [], "variableReplacements": [], "prompt": "<p>Complete the following frequency table:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th>Number goals scored</th>\n<th>Frequency</th>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$0$</td>\n<td style=\"text-align: left;\">[[0]]</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$1$</td>\n<td style=\"text-align: left;\">[[1]]</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$2$</td>\n<td style=\"text-align: left;\">[[2]]</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$3$</td>\n<td style=\"text-align: left;\">[[3]]</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$4$</td>\n<td style=\"text-align: left;\">[[4]]</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$5$</td>\n<td style=\"text-align: left;\">[[5]]</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">$6$</td>\n<td style=\"text-align: left;\">[[6]]</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">Total</td>\n<td style=\"text-align: left;\">$30$</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "sortAnswers": false}, {"extendBaseMarkingAlgorithm": true, "gaps": [{"allowFractions": false, "mustBeReducedPC": 0, "minValue": "mean", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "marks": 1, "precision": "2", "strictPrecision": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "maxValue": "mean", "type": "numberentry", "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacements": [], "precisionType": "dp", "showPrecisionHint": true, "precisionMessage": "You have not given your answer to the correct precision."}, {"allowFractions": false, "mustBeReducedPC": 0, "minValue": "mode", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "marks": 1, "precision": "2", "strictPrecision": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "maxValue": "mode", "type": "numberentry", "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacements": [], "precisionType": "dp", "showPrecisionHint": true, "precisionMessage": "You have not given your answer to the correct precision."}, {"allowFractions": false, "mustBeReducedPC": 0, "minValue": "median", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "marks": 1, "precision": "2", "strictPrecision": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "maxValue": "median", "type": "numberentry", "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacements": [], "precisionType": "dp", "showPrecisionHint": true, "precisionMessage": "You have not given your answer to the correct precision."}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "marks": 0, "unitTests": [], "variableReplacements": [], "prompt": "<p>Find the mean, mode and median for this data.</p>\n<p>Mean = [[0]]</p>\n<p>Mode =&nbsp; [[1]]</p>\n<p>Median&nbsp;=&nbsp; [[2]]</p>", "sortAnswers": false}], "variable_groups": [{"variables": ["a", "mean", "median", "m", "mode", "freq", "as"], "name": "Final data"}], "functions": {}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"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/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"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/"}]}