To compare two sets of data, compare an average, such as the median or mean, and a measure of spread, such as the range or interquartile range. Say which set is higher on average and which is more consistent, and always write the comparison in the context of the question. A smaller range or interquartile range means the data is more consistent.
Making a comparison
- Compare an average: median or mean
- Compare a spread: range or interquartile range (IQR)
- Higher average: higher on average in context
- Smaller spread: more consistent
- Write each comparison as a sentence about the context
Two parts to every comparison
A full comparison has two parts: one about the average and one about the spread. Saying only that one set has a higher median is half an answer. For example: “Team P scored higher on average, because its median is higher, and Team P's scores were more consistent, because its interquartile range is smaller.”
Which average and which spread?
Compare like with like: medians with medians, means with means. The median and the interquartile range are not affected by extreme values, so they are better when there are outliers. The range is quick to calculate but depends only on the two most extreme values.
Comparing from raw data
If you are given the raw data, work out the average and the spread for each set first, then compare. Put each set in order before finding the median, and use the same measures for both sets.
Writing in context
Exam marks depend on comparisons written in context. Instead of “A has a bigger median”, write “the times for route A were longer on average”. Instead of “B has a smaller range”, write “the times for route B were more consistent”. Refer to the actual quantities in the question.
Comparing box plots and cumulative frequency graphs
The same method works with box plots: compare the medians, shown by the middle line in each box, and the interquartile ranges, shown by the widths of the boxes. With cumulative frequency graphs, read off the median and quartiles for each group first.
Worked examples
Example 1. Team P: median 23 points, IQR 5. Team Q: median 19 points, IQR 11. Compare the two teams.
- Average: P's median is higher.
- Spread: P's IQR is smaller.
Answer: Team P scored more points on average, and its scores were more consistent.
Example 2. Group 1: 12, 15, 15, 18, 20. Group 2: 10, 13, 17, 19, 26. Compare the groups using the median and range.
- Group 1: median 15, range 8.
- Group 2: median 17, range 16.
Answer: Group 2 scored higher on average, but Group 1's scores were more consistent.
Example 3. Two buses have the same median journey time, 32 minutes. Bus A has a range of 4 minutes and bus B a range of 18 minutes. Which bus is more reliable?
- The averages are equal, so compare the spread.
- Bus A has the smaller range.
Answer: Bus A, because its journey times are more consistent.
Example 4. The salaries in a small company are £18 000, £20 000, £21 000, £22 000 and £90 000. Explain why the median is a better average to use in a comparison than the mean.
- The mean is pulled up by the outlier of £90 000.
- The median, £21 000, is typical of most salaries.
Answer: The median is not distorted by the extreme value.
Example 5. Box plots show that Shop X has a median wait of 6 minutes and an IQR of 3 minutes, and Shop Y has a median of 9 minutes and an IQR of 2 minutes. Compare the waiting times.
- Average: X's median is lower.
- Spread: Y's IQR is smaller.
Answer: Customers at Shop X waited less time on average, but Shop Y's waiting times were more consistent.
Common mistakes
- Comparing only the averages. Always compare a measure of spread as well.
- Writing without context. Say what the numbers mean, such as “longer journey times”.
- Saying a larger range is more consistent. A smaller spread means more consistent.
- Comparing a median with a mean. Use the same measure for both sets.
- Forgetting to order raw data. Put the data in order before finding the median.
How comparing distributions is examined
Comparing two data sets is usually the final, 2-mark part of a statistics question, on Foundation and Higher papers alike. Questions ask you to:
- compare two sets of data given their averages and spreads
- work out the measures from raw data, then compare
- compare two box plots or cumulative frequency graphs
- choose and justify the more suitable average
Practise with the comparing data worksheet, which has model answers, and the GCSE data revision engine.
You need averages and range first. At Higher tier, the interquartile range comes from cumulative frequency.
Frequently asked questions
How do you compare two distributions?
Compare an average and a measure of spread, and write each comparison in context.
What does a smaller range mean?
The data is more consistent, or less spread out.
Should I use the mean or the median to compare?
Either, as long as you use the same one for both sets. The median is better when there are outliers.