Range and Spread: Why Two Data Sets Can Have the Same Mean

Two data sets can have the same mean and still behave very differently. One may be tightly clustered; the other may be widely spread. If we describe both only by their centre, we lose information about variation.

This guide introduces spread through range, comparison and distribution shape. The main idea is simple: a measure of centre tells us where data tends to sit, while a measure of spread tells us how far the values vary around that centre.

A first comparison

Consider:

Set A: 8, 9, 10, 11, 12
Set B: 2, 6, 10, 14, 18

Both sets have mean 10. But Set A is tightly clustered around 10, while Set B is much more dispersed.

The range of Set A is 12 − 8 = 4. The range of Set B is 18 − 2 = 16.

What range measures

Range is:

range = maximum value − minimum value

It records the total span from the smallest to the largest value. It is easy to calculate and useful for a first comparison, but it uses only two observations.

Why range can be sensitive to outliers

Suppose a data set is 9, 10, 10, 11, 12. Its range is 3. If the final value were 50 instead of 12, the range becomes 41 even though most values remain near 10.

This shows why a single extreme value can dominate the range.

Centre and spread answer different questions

The mean might tell us the average score of a class. The spread helps tell us whether most students performed similarly or whether the class contains very different levels of attainment.

In manufacturing, two machines may produce components with the same average size. The machine with lower spread is usually more consistent.

Worked example: same mean, different consistency

Machine A produces lengths 99, 100, 100, 100, 101 mm. Machine B produces 94, 97, 100, 103, 106 mm.

Both means are 100 mm. But Machine A has range 2 mm, while Machine B has range 12 mm. The same mean does not imply the same consistency.

Spread and decision-making

If two routes to school have the same average journey time, but one route varies between 20 and 25 minutes while the other varies between 10 and 35 minutes, the first may be more predictable even though the average is identical.

Spread therefore matters whenever reliability or consistency matters.

Why a graph can reveal what one number hides

A dot plot, histogram or box plot shows where observations lie. Two data sets with the same mean can have different clusters, gaps, symmetry or outliers. A numerical summary compresses data; a graph can preserve more of its shape.

A useful comparison language

• higher or lower centre
• greater or smaller spread
• more or less consistent
• presence of outliers
• symmetrical or skewed shape

A strong statistical comparison usually mentions more than one of these features rather than quoting a single average.

Common errors

Assuming the same mean means the same data. Many different distributions can share one mean.

Forgetting to order data when reading extremes from a messy list. Check the true minimum and maximum.

Calling range an average. Range is a measure of spread, not centre.

Ignoring outliers. A very large or small value can make the range look much larger than the spread of most observations.

Practice

1. Find the range of 7, 11, 9, 14, 8.
2. Compare the ranges of 4, 5, 6, 7, 8 and 0, 4, 6, 8, 12.
3. Two classes both have mean mark 70. Class A has range 8 and Class B range 30. Which class is more tightly clustered?
4. Explain why a single outlier can make range misleading.

Answers

1. 7. 2. First range 4; second range 12. 3. Class A. 4. Range depends only on the maximum and minimum, so one extreme value can greatly increase it even if the rest of the data stays clustered.

Connected routes

Use Mean, Median and Mode for centre, then continue to Box Plots and Distribution for a richer view of spread. Return to the Mathematics Learning Hub for the wider route.