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S2.3StatisticsCore

Measures of Spread

An average alone doesn’t describe data — you also need to know how spread out it is. The range, interquartile range and interpercentile range each measure spread, with different sensitivity to extreme values.

20 min Measures of Location and Spread
Edexcel AS Level Maths: 2.3 Measures of SpreadCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Calculate the range
  • Calculate the interquartile range (IQR)
  • Calculate an interpercentile range
  • Choose a suitable measure of spread
1

Range and interquartile range

The range\textbf{range} is simply the largest minus the smallest value — easy, but distorted by outliers. The interquartile range\textbf{interquartile range} (IQR) is Q3−Q1Q_3 - Q_1, the spread of the middle 50%, and is resistant to outliers.

range=max−min,IQR=Q3−Q1\text{range} = \text{max} - \text{min}, \qquad \text{IQR} = Q_3 - Q_1
IQR ignores the extremes, so it is more robust.
Example 1If Q1=12Q_1 = 12 and Q3=20Q_3 = 20, find the IQR.
120−1220 - 12.
Answer88
2

Interpercentile range

An interpercentile range\textbf{interpercentile range} is the difference between two percentiles, e.g. the 10th to 90th percentile range, P90−P10P_{90} - P_{10}. It captures the spread of the central part of the data while trimming the extremes.

3

Choosing a measure

Use the range for a quick, simple spread; use the IQR or an interpercentile range when outliers would distort the picture — they focus on the bulk of the data.

Tip — Outliers around? Prefer the IQR or interpercentile range over the range.

Formula recap

range=max−min\text{range} = \text{max} - \text{min}
Simplest spread (outlier-sensitive).
IQR=Q3−Q1\text{IQR} = Q_3 - Q_1
Middle 50% — robust.
P90−P10P_{90} - P_{10}
An interpercentile range.

Common mistakes to avoid

Computing IQR as Q₁ − Q₃.
It is Q₃ − Q₁ (upper minus lower).
Using the range when there are clear outliers.
The range is dominated by extremes; use the IQR instead.

Key takeaways

  • Range = max − min (sensitive to outliers).
  • IQR = Q₃ − Q₁ — the middle 50%, robust to outliers.
  • Interpercentile ranges (e.g. P₉₀ − P₁₀) trim the extremes.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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