Finding the Median and Quartiles

Find the median, quartiles and interquartile range from a list or from a cumulative frequency graph, and read the graph backwards.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 73 real past-paper questions below

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Quartiles Cut the Data Into Quarters

Put the data in order. The median cuts it in half. The quartiles cut each half in half again.

Word	What it means
Lower quartile	a quarter of the way along
Median	halfway along
Upper quartile	three quarters of the way along

Formula

IQR = UQ - LQ

The interquartile range, IQR, is the upper quartile take away the lower quartile. It is the spread of the middle half of the data.

Key idea

The IQR ignores the very smallest and biggest values. So one strange value cannot spoil it, unlike the range.

Key idea

Doctors check a baby's weight on a growth chart. The middle curve is the 50th percentile, the median. The 25th and 75th percentile curves are the lower and upper quartiles. A baby on the 75th curve weighs more than about 75% of babies of the same age.

Quartiles From a List

The 2 or 3-mark question: "Find the interquartile range" of a list. Order the list, then count to each position.

Dina cycled these distances, in km, on 11 days: 12, 7, 20, 4, 15, 9, 25, 11, 6, 18, 14.

Formula

LQ: \frac{n + 1}{4} th value

UQ: \frac{3 ( n + 1 )}{4} th value

Here $n$ is how many values there are.

Worked example

1
$4, 6, 7, 9, 11, 12,$ — Put the 11 distances in order, smallest first. Quartiles only work on an ordered list.
2
$14, 15, 18, 20, 25$ — The rest of the ordered list. Count the values: $n = 11$ .
3
$\frac{11 + 1}{4} = 3$ — Put $n = 11$ into the lower quartile formula. This gives a position, not the answer.
4
$3 rd value = 7$ — Count 3 along the ordered list: 4, 6, 7. So the lower quartile is 7 km.
5
$\frac{11 + 1}{2} = 6 th value = 12$ — The median is halfway along, the 6th value. You do not need it for the IQR, but questions often ask for it too.
6
$\frac{3 ( 11 + 1 )}{4} = 9$ — The upper quartile position is three times the lower one: 3 × 3 = 9.
7
$9 th value = 18$ — Count 9 along the list. So the upper quartile is 18 km.
8
$18 - 7 = 11 km$ — IQR = upper quartile − lower quartile. Keep the units of the data.

Common mistake

Forgetting to order the list, or giving the position (3rd) instead of the value (7).

Key idea

15 values: the quartiles are the $\frac{16}{4} = 4$ th and the $12$ th values.

When the Position Ends in .5

Exams often use 9 or 13 values. Then the position ends in .5, and the quartile is halfway between two values. Nine students spent these minutes on homework, already in order: 23, 25, 28, 31, 34, 36, 40, 42, 47.

Worked example

1
$n = 9$ — Count the values. They are already in order, so you can start counting positions.
2
$\frac{9 + 1}{4} = 2.5$ — The lower quartile position. 2.5th means halfway between the 2nd and the 3rd values.
3
$\frac{25 + 28}{2} = 26.5$ — The 2nd value is 25 and the 3rd is 28. Add them and halve to find the middle: LQ = 26.5.
4
$\frac{3 ( 9 + 1 )}{4} = 7.5$ — The upper quartile position: halfway between the 7th and the 8th values.
5
$\frac{40 + 42}{2} = 41$ — The 7th value is 40 and the 8th is 42. Halfway between them: UQ = 41.
6
$41 - 26.5 = 14.5 minutes$ — IQR = upper quartile − lower quartile.

Common mistake

Giving 25 or 28 as the lower quartile. Position 2.5 means halfway between them.

Key idea

A stem-and-leaf diagram lists the values in order for you. Count along the leaves. Write both quartiles down before you subtract: they earn a mark for working even if the subtraction goes wrong.

Median and IQR From a Graph

The 1 or 2-mark question: "Use your graph to find an estimate for the median" or "for the interquartile range". 40 people timed their wait for a bus.

Worked example

1
$\frac{1}{2} \times 40 = 20$ — The median is the middle person. Half of the 40 people is 20.
2
$across at 20, down: 15 min$ — Go across from 20 on the side axis to the curve, then straight down. The median is about 15 minutes.
3
$\frac{1}{4} \times 40 = 10 \to 10 min$ — Lower quartile: a quarter of 40 is 10. Across at 10, then down: 10 minutes.
4
$\frac{3}{4} \times 40 = 30 \to 20 min$ — Upper quartile: three quarters of 40 is 30. Across at 30, then down: 20 minutes.
5
$20 - 10 = 10 minutes$ — IQR = upper quartile − lower quartile. Use the two times from the bottom axis.

Rule

On a graph, use a half and quarters of the total: 20, 10 and 30. Go across from the side axis, then down.

Common mistake

Taking the coloured 10 and 30 away. Those are numbers of people. Subtract the times you read on the bottom axis.

Key idea

On a graph use $\frac{n}{4}$ , not $\frac{n + 1}{4}$ . With so many values the extra 1 makes no difference you could see, and the mark scheme expects 10, 20 and 30 here.

Percentiles From a Graph

The 2-mark question: “Find an estimate of the 80th percentile.” A percentile is the value that this percentage of the data is at or below. Use the same 40 waiting times.

Formula

p th percentile: across at \frac{p}{100} \times n

Here $p$ is the percentage and $n$ is the total number on the graph.

Worked example

1
$\frac{80}{100} \times 40 = 32$ — Turn the percentage into people: 80% of 40 is 32 people. This is a height on the side axis.
2
$across at 32, down: about 21 min$ — Go across at 32 to the curve, then straight down. 80% of the people waited about 21 minutes or less.

Common mistake

Going across at 80. That is the percentage, not a number of people. 80% of 40 is 32.

Key idea

The lower quartile is the 25th percentile, the median is the 50th and the upper quartile is the 75th.

Reading the Graph Backwards

The 3-mark question gives a number of people and asks for the value: 10 people waited longer than $w$ minutes. Find $w$ .

Worked example

1
$40 - 10 = 30$ — 10 people waited longer than $w$ . So the other 30 people waited $w$ minutes or less.
2
$across at 30, down: 20$ — The graph counts the people at or below each time. So go across at 30 and read the time below.
3
$w = 20$ — Check: 30 people waited 20 minutes or less, so 40 − 30 = 10 waited longer.

Common mistake

Going across at 10. The graph counts people at or below a time, so take the 10 from the total first.

Key idea

A percentage? Turn it into people first: 25% of 40 is 10 people.

Which Method? Read the Question

Check whether you have a list or a graph.

You see	You do
IQR of a list	order it, count to the quartile positions
Median from a graph	across at half the total, down
IQR from a graph	across at a quarter and three quarters
A percentile from a graph	across at that percentage of the total
How many more than a value	up from the value, take from the total (Drawing Cumulative Frequency Diagrams)
More than a value: find the value	take from the total, then across
Which is more consistent?	the smaller IQR
Draw the graph first	Drawing Cumulative Frequency Diagrams
Mean or mode of a list	Mean, Median, Mode and Range

Key idea

When you compare two sets, say it in words: "Team B's IQR is smaller, so their scores are more consistent."

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