Comparing Distributions

Real statistics questions rarely stop at calculating one average — they ask you to compare two sets of data and say what the comparison means. The golden rule examined throughout the OCR GCSE Mathematics (J560) Statistics strand is simple but strict: compare one measure of average (mean, median or mode) and one measure of spread (range or IQR), and always phrase the comparison in context. This lesson pulls together everything from earlier in the strand and turns it into confident, mark-winning comparison statements for both Foundation and Higher tiers.

The topic is dominated by AO2 (interpreting and comparing) and AO3 (drawing a justified conclusion), with AO1 supplying the underlying calculations. OCR command words here are above all "Compare", together with "Give a reason for your answer", "Work out" and "Describe". The marks are earned by what you say about the numbers, not just the numbers themselves.

Key Vocabulary

Term	Meaning
Average	A measure of the centre: mean, median or mode
Spread	A measure of variability: range or interquartile range
In context	Phrased using the real situation (times, marks, goals), not bare numbers
Consistent	Having a smaller spread; values closer together
Higher on average	Having a larger mean or median
Like-for-like	Comparing the same measure for both data sets

The Rule: One Average and One Spread

A complete comparison has two statements:

Average: which data set is higher on average, using the same average for both (median with median, or mean with mean).
Spread: which data set is more consistent, using the same spread for both (range with range, or IQR with IQR).

Both statements must be in context — refer to what the data actually measures.

A common mistake is to compare a mean for one set with a median for the other, or to give two averages and no spread. Match like for like, and cover both centre and spread.

Worked Example 1

Two basketball players' points per game are summarised. Player A: mean 18, range 12. Player B: mean 15, range 4. Compare the two players.

Solution:

Average: Player A has the higher mean (18 > 15), so on average Player A scores more points per game.
Spread: Player B has the smaller range (4 < 12), so Player B is more consistent from game to game.

Worked Example 2

Rewrite this weak comparison so it would earn full marks: "Class A's mean is bigger and Class B's median is smaller."

Solution: The weak version mixes mean and median and says nothing about spread or context. A full-mark version: "Class A has a higher mean test mark than Class B, so Class A scored higher on average. Class A also has a larger range, so its marks were less consistent than Class B's." (Same average for both, plus a spread comparison, both in context.)

Common error: comparing different measures for the two groups — always compare like with like.

Choosing Which Average and Spread

Use the median and IQR when the data may contain outliers or be skewed; use the mean and range when the data is fairly symmetric with no extreme values. The median–IQR pairing is more robust, which is why Higher-tier box-plot comparisons rely on it.

Worked Example 3

Two shops' daily takings (£) over a month are summarised. Shop X: median £820, IQR £90. Shop Y: median £760, IQR £300, with one day of £4,000 (a one-off event). Which pairing of measures should you use, and compare the shops.

Solution: Because Shop Y has an extreme value (£4,000), use the median and IQR, which resist outliers.

Average: Shop X has the higher median (£820 > £760), so on a typical day Shop X takes more.
Spread: Shop X has the smaller IQR (£90 < £300), so Shop X's daily takings are much more consistent.

Comparing from Lists

When given raw lists, calculate the matching measures yourself before comparing.

Worked Example 4

Two students' homework scores (out of 20) over five weeks: Amir 12, 14, 15, 15, 19; Beth 8, 14, 16, 16, 16. Compare their performance using the median and the range.

Solution: Order each (already ordered). Amir: median (3rd of 5) $= 15$ , range $= 19 - 12 = 7$ . Beth: median $= 16$ , range $= 16 - 8 = 8$ .

Average: Beth has the higher median (16 > 15), so Beth scored higher on average.
Spread: Amir has the slightly smaller range (7 < 8), so Amir was a little more consistent.

Worked Example 5

Two delivery firms' times (minutes) for six parcels: FastVan 22, 25, 26, 28, 30, 51; QuickShip 30, 31, 33, 34, 35, 37. Decide which average and spread are most appropriate, and compare.

Solution: FastVan has a possible outlier (51), so the median and IQR are safer than the mean and range.

FastVan: median $= \dfrac{26 + 28}{2} = 27$ min. QuickShip: median $= \dfrac{33 + 34}{2} = 33.5$ min.
Average: FastVan has the lower median (27 < 33.5), so FastVan is faster on average.
Spread: ignoring the outlier, FastVan's central times (22–30) are also tightly grouped, so it is at least as consistent; QuickShip's times are close together too. A fair conclusion: FastVan is faster on average, with both firms fairly consistent apart from FastVan's single slow delivery.

Worked Example 5b

Two classes' scores out of 10 are: Class M 5, 6, 6, 7, 7, 7, 8, 8; Class N 2, 4, 6, 7, 8, 8, 9, 10. Compare the two classes using the mean and the range.

Solution: Class M mean $= \dfrac{5+6+6+7+7+7+8+8}{8} = \dfrac{54}{8} = 6.75$ ; range $= 8 - 5 = 3$ . Class N mean $= \dfrac{2+4+6+7+8+8+9+10}{8} = \dfrac{54}{8} = 6.75$ ; range $= 10 - 2 = 8$ .

Average: the means are equal (both 6.75), so on average the two classes performed the same.
Spread: Class M has the smaller range (3 < 8), so Class M's scores were much more consistent; Class N had both the lowest and the highest scores.

This example shows why a spread comparison is essential: identical averages can hide very different distributions.

Worked Example 5c

Explain why, when comparing two data sets that each contain an outlier, the median and IQR are usually preferred to the mean and range.

Solution: The mean and the range both use the extreme values directly, so a single outlier can distort each of them — pulling the mean towards itself and inflating the range. The median and the IQR depend on the middle of the data and ignore the extremes, so they give a fairer, more stable comparison when outliers are present in either set.

Comparing with Box Plots (Higher)

Box plots are built for comparison: line them up on the same scale and compare medians and IQRs (box widths). The diagram below compares the times (minutes) for two routes to school.

Worked Example 6

From the box plots above, Route A has median 24 and IQR $30 - 18 = 12$ ; Route B has median 30 and IQR $36 - 22 = 14$ . Compare the two routes.

Solution:

Average: Route A has the lower median (24 < 30), so Route A is quicker on average.
Spread: Route A has the smaller IQR (12 < 14), so Route A's times are slightly more consistent. Route A is the better choice on both counts.

Worked Example 7

A box plot for Route A reaches a maximum of 44 minutes, while Route B's maximum is 40 minutes. Does this change your conclusion? Explain.

Comparing Distributions

Comparing Distributions

Key Vocabulary

The Rule: One Average and One Spread

Worked Example 1

Worked Example 2

Choosing Which Average and Spread

Worked Example 3

Comparing from Lists

Worked Example 4

Worked Example 5

Worked Example 5b

Worked Example 5c

Comparing with Box Plots (Higher)

Worked Example 6

Worked Example 7

More in Mathematics