Comparing Distributions
"Compare the two distributions" is one of the most common GCSE statistics questions, and one of the easiest to lose marks on. The recipe is short: compare one average and one measure of spread, quote the numbers, and say what each comparison means in the real situation. This page practises that recipe with dot plots, box plots, frequency polygons and stem-and-leaf diagrams.
GCSE subject content (Statistics): interpret, analyse and compare the distributions of data sets through appropriate graphical representation and appropriate measures of central tendency and spread (range, including consideration of outliers; Higher: quartiles, inter-quartile range and box plots)Before you start. You need the mean, median, mode and range (averages and range) and the ideas of shape, outliers and comparing from distributions and correlation. The grouped-data parts use estimated means and frequency polygons; the Higher parts use quartiles, IQR and box plots from cumulative frequency, box plots and histograms.
1 · What a comparison needs
A distribution is the whole pattern of a set of data: where it sits, how spread out it is, and its shape. Two numbers capture most of it:
- an average (a measure of central tendency: mean, median or mode) says where the data is centred: what's "typical";
- a measure of spread (range or inter-quartile range) says how spread out the data is. Small spread means consistent; large spread means variable.
Two bus companies were each late on 11 journeys. Here are the numbers of minutes late. Look at the dots first, then compare the measures.
Full-mark answer: "On average Bus B was less late, because its median was lower (3 minutes compared with 5 minutes). Bus A was more consistent, because its range was smaller (7 minutes compared with 14 minutes)." Two comparisons, each with the measure named, the numbers quoted, and a meaning in context. Saying only "B's median is lower" or "A's numbers are less spread out" without saying what that means about lateness loses the interpretation marks.
2 · Which measures? Outliers
An outlier is a value that is much bigger or smaller than the rest. Here is the weekly pocket money (£) of 9 friends. Drag the largest value and watch what happens to each measure.
| Measure | Good at | Weakness |
|---|---|---|
| Mean | Uses every value | Dragged towards outliers |
| Median | Ignores extreme values | Ignores most of the actual values |
| Mode | Works for non-numerical data | May not exist or not be typical |
| Range | Quick and simple | Uses only the two extremes, so one outlier ruins it |
| IQR Higher | Spread of the middle half, so it ignores outliers | Needs quartiles |
Rule of thumb: if there are outliers, compare the median and IQR (or median and range at Foundation, but mention the outlier). If the data has no outliers, the mean and range are fine. Match the pair: median goes with IQR, mean goes with range. And only remove an outlier if it is a mistake (a typing error, say); a genuine extreme value is part of the data.
3 · Shape and skew
A distribution is symmetrical if one side is a mirror image of the other. If it has a long tail on one side it is skewed: a tail to the right (towards big values) is positive skew; a tail to the left is negative skew. Incomes are positively skewed (a few people earn huge amounts); marks on an easy test are negatively skewed. Slide the skew and watch the three averages separate.
The mean is pulled furthest into the tail, because it adds up every value, including the extreme ones in the tail. The median stays nearer the bulk of the data. For a skewed distribution the median is usually the better "typical" value: it's why news reports quote median house prices and median incomes.
4 · Box plots side by side Higher
Two box plots on the same scale make comparing quick: compare the median lines for the average and the box lengths (IQR) for the spread. Here are the test marks of two classes, P and Q. Decide whether each statement is true, false, or can't be told from the box plots.
Box plot traps. A box plot shows the median, not the mean. It shows proportions (a quarter of the data in each section), not how many people: a longer section does not mean more people. And "75% scored more than the LQ" works for each class separately, but you can't compare actual numbers of pupils unless you know the class sizes.
5 · Frequency polygons and histograms
For large grouped data sets, draw both frequency polygons on the same axes. The polygon further to the right has the higher values; a flatter, wider polygon is more spread out. Back it up with numbers: estimated means and modal classes from the tables.
Different-sized groups? If one group has 50 people and the other 200, a frequency polygon of the bigger group sits higher everywhere just because it's bigger. Compare averages and spreads (which don't depend on group size), or compare relative frequencies (proportions) instead of frequencies.
6 · Back-to-back stem-and-leaf diagrams
A stem-and-leaf diagram splits each number into a stem (the tens digit) and a leaf (the units digit), so 73 is stem 7, leaf 3. A back-to-back diagram shares the stems down the middle: one group's leaves go to the right, the other's to the left. On the left the leaves are read backwards: the leaf nearest the stem is the smallest. Build the diagram for the resting pulse rates (beats per minute) of 13 boys and 13 girls.
Key: boys 8 | 5 means 58 bpm; girls 6 | 1 means 61 bpm
7 · Writing full-mark sentences
Build two sentences, one comparing an average and one comparing a spread. The feedback checks what an examiner checks: which group, which measure, the right direction, the numbers, and an interpretation in context.
Average
Spread
Worked examples
Compare using the median and IQR
Reaction times: boys median 0.42 s, IQR 0.10 s; girls median 0.38 s, IQR 0.15 s. Compare the distributions.
Choosing the right average when there's an outlier
Pocket money (£): 5, 6, 6, 7, 8, 8, 9, 10, 50. Which average best represents the data?
Comparing box plots Higher
Class P: min 22, LQ 41, median 52, UQ 60, max 78. Class Q: min 30, LQ 45, median 49, UQ 54, max 70. Compare the marks.
Back-to-back stem-and-leaf: pulse rates
Boys: 58, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 81, 85. Girls: 61, 66, 69, 72, 72, 75, 77, 78, 79, 80, 83, 86, 92 (bpm).
Comparing grouped data
Homework time t (minutes), 50 pupils in each year group. Classes 0–10, 10–20, …, 50–60. Year 9: 6, 14, 15, 9, 4, 2. Year 11: 2, 5, 10, 14, 12, 7.
Exam-style: spot what's missing
A student wrote: "The median for A is 12 and the median for B is 15. The range for A is bigger." The data is the time in seconds two groups took to solve a puzzle. Improve the answer.
Quick check
What you've learned
- To compare two distributions, compare one average and one measure of spread, quote both values, and say what each means in context.
- A smaller spread means more consistent data; a larger spread means more variable.
- Outliers distort the mean and the range. Use the median and IQR when there are outliers or skew; the mean and range otherwise.
- Positive skew has a tail to the right and usually mean > median > mode; negative skew is the reverse.
- Higher Box plots side by side: compare the medians and the IQRs (box lengths). They show proportions, not numbers of people, and not the mean.
- Frequency polygons on the same axes compare grouped data; back them up with estimated means and modal classes.
- In a back-to-back stem-and-leaf diagram, the left-hand leaves are read from the stem outwards; find medians by counting to the middle position.