// Numbas version: exam_results_page_options {"name": "Calculate the measures of central tendency for a sample", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "

This question provides a list of data to the student. They are asked to find the mean, median, mode and range.

", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Calculate the measures of central tendency for a sample", "ungrouped_variables": ["modea1", "modea2", "a1", "a2", "a3"], "rulesets": {}, "variable_groups": [{"name": "final list", "variables": ["a", "a_s", "mean", "median", "mode", "mode1", "range", "modetimes"]}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

A random sample of 20 residents from Newcastle were asked about the number of times they went to see a play at the theatre last year.

\n

Here is the list of their answers:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\var{a[0]}$ $\\var{a[1]}$ $\\var{a[2]}$ $\\var{a[3]}$ $\\var{a[4]}$ $\\var{a[5]}$ $\\var{a[6]}$ $\\var{a[7]}$ $\\var{a[8]}$ $\\var{a[9]}$ $\\var{a[10]}$ $\\var{a[11]}$ $\\var{a[12]}$ $\\var{a[13]}$ $\\var{a[14]}$ $\\var{a[15]}$ $\\var{a[16]}$ $\\var{a[17]}$ $\\var{a[18]}$ $\\var{a[19]}$
\n

", "variables": {"mode": {"templateType": "anything", "name": "mode", "definition": "mode(a)", "group": "final list", "description": "

Mode as a vector.

"}, "modetimes": {"templateType": "anything", "name": "modetimes", "definition": "map(\nlen(filter(x=j,x,a)),\nj, 0..8)", "group": "final list", "description": "

The vector of number of times of each value in the data.

"}, "modea1": {"templateType": "anything", "name": "modea1", "definition": "mode(a1)", "group": "Ungrouped variables", "description": ""}, "a": {"templateType": "anything", "name": "a", "definition": "if(len(modea1) = 1, a1, if(len(modea2) = 1, a2, a3))", "group": "final list", "description": "

The final list.

"}, "median": {"templateType": "anything", "name": "median", "definition": "median(a)", "group": "final list", "description": ""}, "a1": {"templateType": "anything", "name": "a1", "definition": "repeat(random(0..8), 20)", "group": "Ungrouped variables", "description": "

Option 1 for the list. Only used if there is only one mode.

"}, "range": {"templateType": "anything", "name": "range", "definition": "max(a) - min(a)", "group": "final list", "description": ""}, "mode1": {"templateType": "anything", "name": "mode1", "definition": "mode[0]", "group": "final list", "description": "

Mode as a value.

"}, "a2": {"templateType": "anything", "name": "a2", "definition": "repeat(random(0..8), 20)", "group": "Ungrouped variables", "description": "

Option 2 for the list. Only used if there is only one mode and option 1 was not used.

"}, "mean": {"templateType": "anything", "name": "mean", "definition": "mean(a)", "group": "final list", "description": ""}, "a3": {"templateType": "anything", "name": "a3", "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])", "group": "Ungrouped variables", "description": "

Option 3 for the list. Ensures there is only one mode (2) while still randomising the data.

"}, "modea2": {"templateType": "anything", "name": "modea2", "definition": "mode(a2)", "group": "Ungrouped variables", "description": ""}, "a_s": {"templateType": "anything", "name": "a_s", "definition": "sort(a)", "group": "final list", "description": "

Sorted list.

"}}, "preamble": {"js": "", "css": ""}, "extensions": ["stats"], "tags": ["mean", "measures of average and spread", "median", "mode", "range", "taxonomy"], "parts": [{"allowFractions": false, "useCustomName": false, "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "prompt": "

Find the mean.

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Find the median.

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Find the mode.

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Find the range.

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#### a)

\n

The mean is the sum of all the responses ($\\sum x$) divided by the number of responses ($n$).

\n

Here, $n = 20$.

\n

\\begin{align}
\\sum x &= \\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]} \\\\
&= \\var{sum(a)} \\text{.}
\\end{align}

\n

Therefore we calculate the mean

\n

\\begin{align}
\\overline{x} &= \\frac{\\sum x}{n} \\\0.5em] &= \\frac{\\var{sum(a)}}{20} \\\\[0.5em] &= \\var{mean} \\text{.} \\end{align} \n \n #### b) \n The median is the middle value. We need to sort the list in order: \n \\[ \\var{a_s[0]}, \\quad \\var{a_s[1]}, \\quad \\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 \\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]} \

\n

There is an even number of responses, so there are two numbers in the middle (10th and 11th place). To find the median, we need to find the mean of these two numbers $\\var{a_s[9]}$ and $\\var{a_s[10]}$:

\n

\\begin{align}
\\frac{\\var{a_s[9]} + \\var{a_s[10]}}{2} &=  \\frac{\\var{a_s[9] + a_s[10]}}{2} \\\\
&= \\var{median} \\text{.}
\\end{align}

\n

\n

#### c)

\n

The mode is the value that occurs the most often in the data.

\n

To find a mode, we can look at our sorted list:

\n

$\\var{a_s[0]}, \\var{a_s[1]}, \\var{a_s[2]}, \\var{a_s[3]}, \\var{a_s[4]}, \\var{a_s[5]}, \\var{a_s[6]}, \\var{a_s[7]}, \\var{a_s[8]}, \\var{a_s[9]}, \\var{a_s[10]}, \\var{a_s[11]}, \\var{a_s[12]}, \\var{a_s[13]}, \\var{a_s[14]}, \\var{a_s[15]}, \\var{a_s[16]}, \\var{a_s[17]}, \\var{a_s[18]}, \\var{a_s[19]}$.

\n

We notice that $\\var{mode1}$ occurs the most ($\\var{modetimes[mode1]}$ times) so $\\var{mode1}$ is the mode.

\n

\n

#### d)

\n

Range is the difference between the highest and the lowest value in the data.

\n

To find this, we subtract the lowest value from the highest value:

\n

\$\\var{max(a)} - \\var{min(a)} = \\var{range} \\text{.}\$

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