The interquartile range (IQR) measures the spread of the middle 50% of a data set. First order the values, find the lower quartile (Q₁) and the upper quartile (Q₃), then subtract: IQR = Q₃ − Q₁. A small IQR means the data are tightly clustered; a large IQR means they are spread out.
What is the difference between range and interquartile range?
Both measure spread, but they behave very differently when data contains extreme values:
| Measure | What it uses | Effect of outliers |
|---|---|---|
| Range | Highest value − lowest value | Very sensitive — one extreme value makes it large |
| Interquartile range | Q₃ − Q₁ (middle 50%) | Not affected by the top 25% or bottom 25% |
| Standard deviation | All values | Sensitive to all values (studied post-GCSE) |
The IQR is preferred when the data contain outliers or are skewed, because it focuses only on the central half of the data set.
How do you find the median and quartiles from a list?
The three quartiles divide ordered data into four equal parts. Here is the method:
- Order all values from smallest to largest.
- Find the median (Q₂): the middle value. For n values, the median is at position (n+1)/2.
- Find Q₁ (lower quartile): the median of the lower half of the data (values below the median).
- Find Q₃ (upper quartile): the median of the upper half of the data (values above the median).
- IQR = Q₃ − Q₁
How do you find the IQR for an odd number of values?
Worked example 1: Find the IQR of: 3, 7, 8, 12, 14, 17, 21, 25, 30
Step 1: Already in order. n = 9. Step 2: Median position = (9+1)/2 = 5th value. Median = 14. Step 3: Lower half (values below 14): 3, 7, 8, 12. Median of these 4 values = (7+8)/2 = 7.5 → Q₁ = 7.5 Step 4: Upper half (values above 14): 17, 21, 25, 30. Median of these 4 values = (21+25)/2 = 23 → Q₃ = 23 Step 5: IQR = 23 − 7.5 = 15.5
How do you find the IQR for an even number of values?
Worked example 2: Find the IQR of: 5, 9, 11, 15, 18, 22, 26, 30
Step 1: Already in order. n = 8. Step 2: Median = average of 4th and 5th values = (15+18)/2 = 16.5. (Not used in the quartile calculation — the halves are found differently.) Step 3: Lower half (lower 4 values): 5, 9, 11, 15. Q₁ = (9+11)/2 = 10 Step 4: Upper half (upper 4 values): 18, 22, 26, 30. Q₃ = (22+26)/2 = 24 Step 5: IQR = 24 − 10 = 14
How do you compare the spread of two data sets using the IQR?
Worked example 3: Two classes sit the same test. Compare their results.
| Statistic | Class A | Class B |
|---|---|---|
| Median | 62 | 64 |
| Q₁ | 51 | 55 |
| Q₃ | 73 | 67 |
| IQR | 22 | 12 |
| Range | 60 | 38 |
Comparison: Class A and Class B have very similar medians, so their typical performance is almost identical. However, Class A has a much larger IQR (22 vs 12), indicating that its results are far more spread out — some students did very well and others did poorly. Class B is more consistent.
When writing a comparison at KS3, always compare both a measure of average (e.g. median) and a measure of spread (e.g. IQR). One sentence about each.
What does the IQR tell you about outliers?
An outlier is a value that is unusually far from the rest of the data. A common rule (used at GCSE and beyond) is:
- Any value below Q₁ − 1.5 × IQR is a possible lower outlier.
- Any value above Q₃ + 1.5 × IQR is a possible upper outlier.
Example: Q₁ = 10, Q₃ = 24, IQR = 14.
Lower fence: 10 − 1.5 × 14 = 10 − 21 = −11
Upper fence: 24 + 1.5 × 14 = 24 + 21 = 45
Any value below −11 or above 45 would be flagged as a possible outlier. This calculation is most commonly examined in the context of box plots at GCSE.
Frequently asked questions
Why do we not include the median when finding Q₁ and Q₃ for odd n?
When there is an odd number of values, the middle value is the median. It belongs to neither the lower half nor the upper half — it sits exactly in the centre. Excluding it from both halves keeps the split fair and symmetric. Including it would inflate either Q₁ or Q₃.
Is the IQR the same as the middle 50% of the data?
Yes, by definition. Q₁ marks the point below which the lowest 25% of data falls, and Q₃ marks the point below which the lowest 75% falls. The interval between Q₁ and Q₃ therefore contains the middle 50% of values.
When is the range more useful than the IQR?
The range is simple and captures the full spread from the very smallest to the very largest value. If you are specifically interested in the most extreme values (for example, in quality control where even a single outlier matters), the range is more informative. For typical performance with possible extreme values, the IQR is more reliable.
How does the IQR relate to box plots?
A box plot (box-and-whisker diagram) is a visual display of five statistics: minimum, Q₁, median, Q₃ and maximum. The box in the box plot stretches from Q₁ to Q₃, so its width is equal to the IQR. A wide box means large IQR (spread out data); a narrow box means small IQR (tightly clustered data).
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