Range and Interquartile Range
The range and the interquartile range are the two measures of spread you need for IGCSE Maths. They tell you how spread out a set of data is, rather than where its centre lies. This page explains how to find the range, how to locate the lower and upper quartiles in a discrete data set, and how to calculate the interquartile range, with worked examples and a practice room of randomised, auto-marked questions.
An average tells you where the centre of a data set is, but it says nothing about how spread out the values are. Two classes could have the same mean test score while one is tightly bunched and the other is all over the place. A measure of spread captures that difference. On this page we look at the two measures of spread you need for IGCSE: the range and the interquartile range.
Range
The range is the simplest measure of spread. It is the gap between the largest and the smallest value in the data set. A larger range means the data is more spread out.
Because it only uses the two most extreme values, the range is quick to find but easily distorted: a single unusually high or low value (an outlier) can make the range look far bigger than the spread of the bulk of the data really is. That weakness is exactly why the interquartile range exists.
💡 Example 1: Range from a list
Find the range of: 14, 9, 22, 17, 11, 25, 16.
Identify the extremes:
Largest = 25, smallest = 9
Subtract:
\[ \text{Range} = 25 - 9 = 16 \]💡 Example 2: How an outlier distorts it
Find the range of: 31, 33, 34, 35, 36, 38, 96.
Subtract the extremes:
\[ \text{Range} = 96 - 31 = 65 \]Six of the seven values sit between 31 and 38, yet the single value 96 makes the range look like 65. The range is misleading here.
Quartiles Higher tier
Just as the median splits ordered data into two halves, the quartiles split it into four equal parts. There are three quartiles:
- The lower quartile (\(Q_1\)): one quarter of the way through the ordered data.
- The median (\(Q_2\)): the middle, halfway through.
- The upper quartile (\(Q_3\)): three quarters of the way through.
📌 Finding the quartile positions
For \(n\) values written in ascending order:
\[ Q_1 \text{ at position } \frac{n+1}{4}, \qquad Q_3 \text{ at position } \frac{3(n+1)}{4} \]
- If the position is a whole number, the quartile is that value in the list.
- If the position is not a whole number, the quartile lies between two values. Take the appropriate fraction of the way from one to the next (for a position like 3.5, take the value halfway between the 3rd and 4th).
The Interquartile Range Higher tier
The interquartile range (IQR) is the spread of the middle half of the data: the gap between the upper and lower quartiles. Because it ignores the bottom quarter and the top quarter, outliers at either end have no effect on it. That makes the IQR a far more reliable measure of spread than the range when extreme values are present.
💡 Example 3: IQR with a whole-number position
Find the lower quartile, upper quartile and interquartile range of: 6, 2, 9, 4, 12, 7, 15, 5, 11, 8, 14.
Order the data (\(n = 11\)):
Q₁ = 3rd value median = 6th Q₃ = 9th
Lower quartile position:
\[ \frac{n+1}{4} = \frac{12}{4} = 3 \Rightarrow Q_1 = 5 \]Upper quartile position:
\[ \frac{3(n+1)}{4} = \frac{36}{4} = 9 \Rightarrow Q_3 = 12 \]Interquartile range:
\[ \text{IQR} = 12 - 5 = 7 \]What's happening?
With 11 values, both quartile positions work out to whole numbers, so you simply read the 3rd and 9th values straight off the ordered list. The IQR of 7 describes the spread of the central six values, ignoring the extremes 2 and 15.
💡 Example 4: IQR with a position between two values
Find the interquartile range of: 20, 13, 28, 16, 24, 31, 18, 26.
Order the data (\(n = 8\)):
Lower quartile position:
\[ \frac{n+1}{4} = \frac{9}{4} = 2.25 \]Quarter of the way from the 2nd value (16) to the 3rd (18):
\[ Q_1 = 16 + 0.25(18 - 16) = 16.5 \]Upper quartile position:
\[ \frac{3(n+1)}{4} = \frac{27}{4} = 6.75 \]Three quarters from the 6th value (26) to the 7th (28):
\[ Q_3 = 26 + 0.75(28 - 26) = 27.5 \]Interquartile range:
\[ \text{IQR} = 27.5 - 16.5 = 11 \]What's happening?
With 8 values the positions are not whole numbers, so each quartile sits between two list values. You move the stated fraction of the way from the lower value to the next one up. A position of 2.25 means a quarter of the step; 6.75 means three quarters of the step.
Comparing Two Data Sets
Exam questions often ask you to compare two sets of data using an average and a measure of spread. The reliable approach is to make one statement about the centre (using the median) and one about the spread (using the IQR), each written in the context of the question.
💡 Example 5: Writing a comparison
Two routes to school were timed over several mornings. Route A has median 24 minutes and IQR 6 minutes. Route B has median 21 minutes and IQR 14 minutes. Compare the two routes.
Compare the centre (median):
Route B is typically quicker, since its median (21) is lower than Route A's (24).
Compare the spread (IQR):
Route A is more consistent, since its IQR (6) is much smaller than Route B's (14).
What's happening?
A lower median means a faster typical journey; a smaller IQR means the times vary less from day to day. Route B is usually quicker but less predictable. Always phrase both points in context, not just "the IQR is smaller".
Range or Interquartile Range?
Both measure spread, but they behave differently. This comparison comes up directly in exam questions.
| Measure | Advantage | Disadvantage | Best used when... |
|---|---|---|---|
| Range | Very quick to calculate | Uses only two values; badly distorted by outliers | You want a rough, fast measure and there are no extreme values |
| Interquartile range | Ignores the extremes, so not affected by outliers | More steps; does not use all the data | You want a reliable measure of the central spread, especially with outliers present |
🔑 Key Points
- Range = largest − smallest. It uses only two values.
- Always put the data in ascending order before finding any quartile.
- \(Q_1\) is at position \(\frac{n+1}{4}\); \(Q_3\) is at position \(\frac{3(n+1)}{4}\).
- If a position is not whole, interpolate the stated fraction of the way to the next value.
- IQR = \(Q_3 - Q_1\): the spread of the middle 50% of the data.
- The IQR is not affected by outliers; the range is.
- To compare data sets, give one statement on the median and one on the IQR, both in context.
⚠ Common Mistakes
- Forgetting to order the data before reading off quartiles.
- Confusing position with value: position 3 means the 3rd value, not the number 3.
- Mixing up the quartiles: \(Q_1\) is the lower quartile, \(Q_3\) the upper.
- Adding instead of subtracting for the IQR; it is \(Q_3 - Q_1\).
- Ignoring the fraction when a position is not a whole number.
- Comparing only the spread and forgetting to also compare the centre, or stating values with no context.