Introduction
You already know that the median splits a dataset exactly in half — 50% of values fall below it and 50% above it. But what if you want to split the data into four equal parts? That is exactly what quartiles do.
Quartiles are one of the most important tools in statistics. They tell you not just what the middle looks like, but how the data is distributed across its entire range. They are used to calculate the interquartile range, draw box plots, identify outliers, and compare datasets — skills that appear repeatedly at GCSE, A-Level, and beyond.
What Are Quartiles?
Quartiles are three values that divide an ordered dataset into four equal parts. Each part contains 25% of the data.
- Q1 (Lower Quartile): The median of the lower half of the data. 25% of values fall below Q1.
- Q2 (Median): The middle value of the entire dataset. 50% of values fall below Q2.
- Q3 (Upper Quartile): The median of the upper half of the data. 75% of values fall below Q3.
Key Terms
Interquartile Range (IQR) — the difference between the upper and lower quartiles. It measures the spread of the middle 50% of the data.
IQR = Q3 − Q1
Semi-interquartile Range (SIQR) — half of the interquartile range.
SIQR = (Q3 − Q1) ÷ 2
How to Find Quartiles of Ungrouped Data
There are two methods depending on whether the dataset has an odd or even number of values.
Step 1 — Arrange the data in ascending order.
Step 2 — Find Q2 (the median) of the entire dataset.
Step 3 — Find Q1 (the median of the lower half).
Step 4 — Find Q3 (the median of the upper half).
Method 1 — Odd Number of Values
When there is an odd number of values, the median (Q2) is the exact middle value. When finding Q1 and Q3, exclude the median from both halves.
Worked Example 1
Find Q1, Q2, Q3 and the IQR of:
7, 3, 11, 5, 9, 1, 15, 4, 8
Step 1 — Arrange in order:
1, 3, 4, 5, 7, 8, 9, 11, 15
Step 2 — Find Q2:
There are 9 values. Median position = (9 + 1) ÷ 2 = 5th value
Q2 = 7
Step 3 — Find Q1:
Lower half (excluding Q2): 1, 3, 4, 5
Wait — 4 values in the lower half. Median of lower half = mean of 2nd and 3rd values = (3 + 4) ÷ 2 = 3.5
Q1 = 3.5
Step 4 — Find Q3:
Upper half (excluding Q2): 8, 9, 11, 15
Median of upper half = mean of 2nd and 3rd values = (9 + 11) ÷ 2 = 10
Q3 = 10
IQR = Q3 − Q1 = 10 − 3.5 = 6.5
Worked Example 2
Find Q1, Q2, Q3 and the IQR of:
12, 5, 18, 9, 25, 3, 14, 21, 7, 16, 10
Step 1 — Arrange in order:
3, 5, 7, 9, 10, 12, 14, 16, 18, 21, 25
Step 2 — Find Q2:
11 values. Median position = (11 + 1) ÷ 2 = 6th value
Q2 = 12
Step 3 — Find Q1:
Lower half (excluding Q2): 3, 5, 7, 9, 10
5 values → median is 3rd value
Q1 = 7
Step 4 — Find Q3:
Upper half (excluding Q2): 14, 16, 18, 21, 25
5 values → median is 3rd value
Q3 = 18
IQR = 18 − 7 = 11
SIQR = 11 ÷ 2 = 5.5
Method 2 — Even Number of Values
When there is an even number of values, Q2 is the mean of the two middle values. When finding Q1 and Q3, split the data exactly in half — each half gets an equal number of values.
Worked Example 3
Find Q1, Q2, Q3 and the IQR of:
4, 10, 6, 14, 8, 18, 2, 12
Step 1 — Arrange in order:
2, 4, 6, 8, 10, 12, 14, 18
Step 2 — Find Q2:
8 values. Two middle values are 4th and 5th: 8 and 10
Q2 = (8 + 10) ÷ 2 = 9
Step 3 — Find Q1:
Lower half: 2, 4, 6, 8
4 values → median = mean of 2nd and 3rd: (4 + 6) ÷ 2 = 5
Q1 = 5
Step 4 — Find Q3:
Upper half: 10, 12, 14, 18
4 values → median = mean of 2nd and 3rd: (12 + 14) ÷ 2 = 13
Q3 = 13
IQR = 13 − 5 = 8
Worked Example 4
Find Q1, Q2, Q3, IQR and SIQR of:
45, 72, 38, 91, 55, 63, 28, 84, 47, 69, 52, 76
Step 1 — Arrange in order:
28, 38, 45, 47, 52, 55, 63, 69, 72, 76, 84, 91
Step 2 — Find Q2:
12 values. Two middle values are 6th and 7th: 55 and 63
Q2 = (55 + 63) ÷ 2 = 59
Step 3 — Find Q1:
Lower half: 28, 38, 45, 47, 52, 55
6 values → median = mean of 3rd and 4th: (45 + 47) ÷ 2 = 46
Q1 = 46
Step 4 — Find Q3:
Upper half: 63, 69, 72, 76, 84, 91
6 values → median = mean of 3rd and 4th: (72 + 76) ÷ 2 = 74
Q3 = 74
IQR = 74 − 46 = 28
SIQR = 28 ÷ 2 = 14
Using Quartile Positions — The Formula Method
Some exam boards and textbooks use a position formula to find quartiles directly. This is especially useful for larger datasets.
For a dataset of n values arranged in order:
- Q1 position = (n + 1) ÷ 4
- Q2 position = (n + 1) ÷ 2
- Q3 position = 3(n + 1) ÷ 4
If the position is a whole number, that value is the quartile.
If the position is a decimal (e.g. 3.5), find the mean of the values at either side (3rd and 4th).
Worked Example 5 — Formula method
Find Q1, Q2 and Q3 of:
5, 8, 12, 15, 19, 23, 27, 31, 35, 40, 44, 48
(already in order, n = 12)
Q1 position = (12 + 1) ÷ 4 = 13 ÷ 4 = 3.25
→ between 3rd and 4th values: 12 and 15
→ Q1 = 12 + 0.25(15 − 12) = 12 + 0.75 = 12.75
Q2 position = (12 + 1) ÷ 2 = 6.5
→ between 6th and 7th values: 23 and 27
→ Q2 = (23 + 27) ÷ 2 = 25
Q3 position = 3(12 + 1) ÷ 4 = 39 ÷ 4 = 9.75
→ between 9th and 10th values: 35 and 40
→ Q3 = 35 + 0.75(40 − 35) = 35 + 3.75 = 38.75
IQR = 38.75 − 12.75 = 26
Note: different exam boards use slightly different methods for quartile positions. Always check which method your syllabus requires. The splitting method (Examples 1–4) is the most widely taught at GCSE.
Identifying Outliers Using Quartiles
Quartiles are used to identify outliers — values that are unusually high or low compared to the rest of the data.
A value is considered an outlier if:
- It is less than Q1 − 1.5 × IQR, or
- It is greater than Q3 + 1.5 × IQR
Worked Example
Q1 = 20, Q3 = 50, IQR = 30
Lower boundary = 20 − 1.5 × 30 = 20 − 45 = −25
Upper boundary = 50 + 1.5 × 30 = 50 + 45 = 95
Any value below −25 or above 95 is an outlier.
If the dataset includes the value 110 → 110 is an outlier (above 95).
If the dataset includes the value −30 → −30 is an outlier (below −25).
Box Plots and Quartiles
Quartiles are the foundation of a box plot (also called a box-and-whisker diagram). A box plot gives a visual summary of a dataset using five values: Minimum value, Q1 (Lower Quartile), Q2 (Median), Q3 (Upper Quartile), and Maximum value. These five values are called the five-number summary.
Example:
Dataset: 28, 38, 45, 47, 52, 55, 63, 69, 72, 76, 84, 91
Five-number summary:
- Minimum = 28
- Q1 = 46
- Q2 = 59
- Q3 = 74
- Maximum = 91
This would be plotted as a box from Q1 to Q3, with a line at Q2, and whiskers extending to the minimum and maximum.
Comparing Datasets Using Quartiles
Quartiles allow you to compare two datasets meaningfully — not just by their averages but by their spread and consistency.
Example:
Interpretation:
- Class A and Class B have similar medians (62 vs 64) — their typical performance is comparable.
- Class A has a much smaller IQR (26 vs 50) — Class A's scores are more consistent and less spread out.
- Class B has a wider spread — there is more variation in performance across students.
Common Mistakes to Avoid
① Forgetting to arrange data in ascending order before finding any quartile.
② Including the median (Q2) in both halves when the dataset has an odd number of values — the median must be excluded from both halves.
③ Confusing Q1 and Q3, Q1 is always the lower quartile (the smaller value), Q3 is always the upper quartile (the larger value).
④ Calculating IQR as Q1 − Q3 instead of Q3 − Q1, IQR is always positive.
✗ IQR = Q1 − Q3 = 5 − 13 = −8
✓ IQR = Q3 − Q1 = 13 − 5 = 8
⑤ Confusing IQR with range, the range = maximum − minimum; IQR = Q3 − Q1. They measure different things.
⑥ Using the wrong position formula — always check which method your exam board requires before choosing an approach.
Frequently Asked Questions
Q1. What is the difference between range and interquartile range?
The range measures the spread of the entire dataset — it is the difference between the maximum and minimum values. The interquartile range measures the spread of the middle 50% of the data — it is the difference between Q3 and Q1. The range is simple but heavily affected by outliers. The IQR is more reliable because it ignores the extreme values at both ends and focuses only on the central portion of the data.
Q2. Why do we exclude the median when finding Q1 and Q3 for odd datasets?
When a dataset has an odd number of values, the median is a single data point that sits exactly in the middle. Including it in both halves would skew the quartile calculations because one half would have more values than the other. By excluding the median, we ensure both halves are equal in size and the quartiles are calculated fairly and symmetrically.
Q3. Can Q1, Q2 and Q3 all be the same value?
Yes, in theory. If all values in a dataset are identical, then Q1 = Q2 = Q3 = that value, and the IQR would be zero. An IQR of zero means there is no spread at all in the middle 50% of the data — every value is the same.
Q4. What is the five-number summary and when is it used?
The five-number summary consists of the minimum value, Q1, Q2 (median), Q3, and the maximum value. Together these five values give a complete picture of the distribution of a dataset. The five-number summary is used to draw box plots (box-and-whisker diagrams), which provide a visual comparison of datasets and make it easy to spot outliers, skewness, and spread at a glance.
Q5. How is the IQR used to identify outliers?
A value is classified as an outlier if it falls more than 1.5 × IQR below Q1 or more than 1.5 × IQR above Q3. The boundaries are: lower boundary = Q1 − 1.5 × IQR and upper boundary = Q3 + 1.5 × IQR. Any data point outside these boundaries is considered an outlier. This method is more robust than simply using the range because it is based on the spread of the central data rather than the extreme values.
Q6. Why is the IQR considered a better measure of spread than the range?
The range uses only two values — the maximum and minimum — and is therefore extremely sensitive to outliers. A single unusually large or small value can make the range very large even if the rest of the data is tightly grouped. The IQR avoids this problem entirely by ignoring the top 25% and bottom 25% of values and measuring only the spread of the central 50%. This makes it a far more reliable and representative measure of spread, especially when the data contains outliers or is skewed.