Compare two distributions in context

Centre, spread and shape

In 3 of 20 recent exams
Paper 2, calculator
1 to 2 marks

How it is set in the exam

Paper 2 (calculator), 1–2 marks a part, after the statistics have been found. Typical (3 of 20 exam versions, all on Paper 2 (calculator); 8 logged parts).

Spot it in the exam

Exam phrases to spot (4)

“compare the … of the two groups”
“does the diagram support the claim that …”
“explain why the distribution is not symmetric”
“which graph could represent …”

In disguise (2)

"interpret" or "comment on" two medians or two IQRs: a comparison is still expected, with the numbers
"which group … represents": the second half of a matching part, and it scores only if the matching is right

Variants

asked: compare the medians, compare the IQRs, judge a claim, explain the skew, match a graph to a mean and a variance (then name the group it shows)

Looks like it, but isn't

a claim that one variable affects another (as $x$ goes up, $y$ goes up): that is correlation (Regression on the GDC), and "correlation is not causation" is not a comparison of two distributions.

Worked example

Medians and IQRs of two groups, in context

Paper 2 (calculator)

Question

A puzzle was given to two classes. The times, in seconds, that the $11$ students of class A took to solve it were

34, \ 41, \ 28, \ 52, \ 45, \ 39, \ 31, \ 47, \ 36, \ 58, \ 43

For class B, the median time was $36$ seconds and the interquartile range was $7$ seconds.
(a) Find the median and the interquartile range of the times for class A. [3]
(b) Compare the times of the two classes, using (i) the medians (ii) the interquartile ranges. [4]

Try it yourself first, then compare.

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Steps

Choose the measures:
a centre (the median for box plots; the mean if means are given) and a spread (the IQR for box plots; the standard deviation with means).
Compare explicitly:
" $41 > 36$ " is the safest form ("A's median is higher" also earns it). This is the reason mark, and it must match your own values.
Conclude in context:
the centre is about a typical value (who is usually quicker, heavier, higher); the spread is about how varied or how consistent the values are. Keep the two conclusions apart.
Skew:
where the median sits in the box, and which whisker is longer. A median near $Q_1$ with a long right whisker is positive skew.
Match a graph:
for a bell-shaped (symmetric) curve, the peak is at the mean and the width comes from $\sigma = \sqrt{\text{variance}}$ : it covers about $3\sigma$ either side of the mean (the 68–95–99.7 figures).
One statement is enough.
An extra statement that is wrong loses the mark, even beside a right one.

Common mistakes

Comparing means from box plots
box plots show medians and quartiles only.
A conclusion with no comparison
"class B were quicker" on its own scores . The reason mark needs the two measures compared: $13 > 7$ , or "class A's IQR is larger".
A spread conclusion about the average
("class A's times are more spread out, so class A was slower"): spread says nothing about which group is quicker.

More mistakes (4)

No context
"A is higher" earns less than "class B were typically quicker at solving the puzzle".
An extra, wrong statement
next to a right one: the mark is lost.
The range as the measure of spread
when one extreme value drives it: use the IQR.
Variance as the width
take the square root first.

Easy to confuse with

Next typeUnknown mean or standard deviation

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Keep going

Easy to confuse with

More in Descriptive statistics: sampling, averages, spread and data displays