S4, S5StatisticsFoundation & HigherGrades 1–5

Averages & Range

Q: Can a data set have more than one mode?

Yes — two modes is called bimodal. If every value appears once, there is no mode.

Q: Does the mean have to be one of the data values?

No. The mean of 1, 2 and 4 is 2. 3, which is not in the data.

Q: What is an outlier?

A value much larger or smaller than the rest. Outliers affect the mean and range, but not the median much.

An average summarises a set of data with one typical value; the range measures how spread out it is. There are three averages — mean, median and mode — and each suits different situations. AQA questions test the calculations, working backwards from a mean, and comparing two data sets in words.

40 min Video by Maths Genie AQA GCSE Maths

The big picture

The mean uses every value, so it is sensitive to extreme values (outliers). One house worth £5 million can drag the mean house price far above what most people pay. The median, the middle value, ignores extremes — which is why house prices and salaries are usually reported as medians.

Comparing distributions always needs two statements: one about an average ("on average, class A scored higher") and one about spread ("class A's scores were more consistent, as the range was smaller"), both in context.

Finding the Mean, Median, Mode and RangeCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Calculate the mean, median, mode and range
Choose the most appropriate average for a data set
Find a missing value given the mean, and combine means
Compare two distributions using an average and the range

The three averages and the range

$\textbf{Mean}$ : add all the values and divide by how many there are. $\textbf{Median}$ : the middle value when the data is in order; with an even number of values, it is halfway between the two middle ones. $\textbf{Mode}$ : the most common value (there can be more than one, or none).

$\textbf{Range}$ : largest value minus smallest value.

Example 1Find the mean, median, mode and range of

\ 7, 3, 9, 4, 3, 10, 6, 8

1In order:

3, 3, 4, 6, 7, 8, 9, 10

2Mean:

50 \div 8 = 6.25

. Median: halfway between 6 and 7, so 6.5.

3Mode: 3. Range:

10 - 3 = 7

Answer

Mean 6.25, median 6.5, mode 3, range 7

Your turnSame method, new numbers — try it before moving on.

Find the median of $\ 12, 5, 8, 15, 9, 11, 6$ .

Your answer

Show worked solution

Solution

1In order: $5, 6, 8, 9, 11, 12, 15$ .
2Seven values, so the middle is the 4th: $9$ .

Answer: $9$

Example 2The temperatures (°C) one week were

-3, 2, -1, 4, 0, -2, 3

. Find the mean and range.

1Sum:

-3 + 2 - 1 + 4 + 0 - 2 + 3 = 3

. Mean

= 3 \div 7 = 0.43

(2 d.p.).

2Range:

4 - (-3) = 7

Answer

Mean

0.43^\circ

C, range

7^\circ

Your turnSame method, new numbers — try it before moving on.

Find the mean of $\ 14, 18, 11, 20, 17$ .

Your answer

Show worked solution

Solution

1Sum: $80$ .
2 $80 \div 5 = 16$ .

Answer: $16$

Tip — Always put the data in order before finding the median — and the range.

Choosing an average

Use the $\textbf{mean}$ when there are no extreme values and every value matters. Use the $\textbf{median}$ when there are outliers. Use the $\textbf{mode}$ for non-numerical data (favourite colour) or to find the most popular item (shoe size in a shop).

Example 1Salaries at a small firm are £22k, £24k, £25k, £26k and £140k. Which average best represents a typical salary?

1Mean:

237 \div 5 = £47.4

k — higher than four of the five salaries.

2Median: £25k — typical of most employees.

Answer

The median, because the £140k salary is an outlier that distorts the mean.

What the mean really measures

The mean is the "fair share" value: if everyone pooled their data and shared it equally, each would get the mean. That is exactly why one huge value pulls it up — the big earner is "sharing" a lot.

Working backwards from a mean

Total $=$ mean $\times$ number of values. Use totals to find missing values or to combine groups: the mean of two groups combined is (total of both) ÷ (number in both) — not the average of the two means.

Example 1The mean of five numbers is 12. Four of them are 8, 15, 10 and 14. Find the fifth.

1Total

= 5 \times 12 = 60

2Known total:

8 + 15 + 10 + 14 = 47

3Fifth:

60 - 47 = 13

Answer

13

Your turnSame method, new numbers — try it before moving on.

The mean of six test scores is 70. After a seventh test, the mean is 72. What was the seventh score?

Your answer

Show hint

Compare the total before and after.

Show worked solution

Solution

1Total of six: $6 \times 70 = 420$ . Total of seven: $7 \times 72 = 504$ .
2Seventh score: $504 - 420 = 84$ .

Answer: $84$

Example 2Class A (20 students) has a mean score of 62. Class B (30 students) has a mean of 72. Find the mean of all 50 students.

1Totals:

20 \times 62 = 1240

and

30 \times 72 = 2160

2Combined:

3400 \div 50 = 68

Answer

68

Tip — The combined mean is 68, not 67 (the average of 62 and 72), because class B is larger.

Comparing distributions

Make one comparison of an average and one comparison of spread. Use the numbers and the context, and say what they mean: a smaller range means more consistent.

Example 1Team X: mean 18 goals, range 12. Team Y: mean 15 goals, range 5. Compare the teams.

1Average: Team X scored more goals on average (18 > 15).

2Spread: Team Y was more consistent (range 5 < 12).

Answer

Team X scored more on average; Team Y was more consistent.

Think like an examiner

"Compare" answers need both an average and a spread, each in context. "Team Y's range is smaller" without "so more consistent" may lose the mark.
Show the total when finding a mean — it earns the method mark.
State the median position (e.g. "the 8th value") for larger data sets.

Averages

\bar{x} = \dfrac{\sum x}{n}

Mean: total ÷ number of values.

\text{median position} = \tfrac{n + 1}{2}

In ordered data.

\text{range} = \text{max} - \text{min}

A simple measure of spread.

Watch out for these

Finding the median without ordering the data.

Order the values first.

Median of 8 values = the 4th value.

It is halfway between the 4th and 5th.

Combined mean = average of the two means.

Use totals:

(n_1\bar{x}_1 + n_2\bar{x}_2) \div (n_1 + n_2)

Range given as "3 to 10".

The range is a single number:

10 - 3 = 7

Stretch yourself

Find five positive whole numbers with mode 4, median 5 and mean 6.

Hint — Median 5 means the middle (3rd) value is 5. Mode 4 means 4 appears most often. The total must be 30.

Reveal solution

SolutionIn order the numbers are

a, b, 5, d, e

. For mode 4, let

a = b = 4

. The total must be

5 \times 6 = 30

, so

d + e = 30 - 13 = 17

. To keep 4 as the only mode,

d

and

e

must be different and not 5 — for example

d = 6

e = 11

. Data:

4, 4, 5, 6, 11

. Check: mode 4, median 5, mean

30 \div 5 = 6

✓. (Other answers, such as

4, 4, 5, 7, 10

, also work.)

\blacksquare

Questions students ask

Can a data set have more than one mode?

Yes — two modes is called bimodal. If every value appears once, there is no mode.

Does the mean have to be one of the data values?

No. The mean of 1, 2 and 4 is

2.\dot{3}

, which is not in the data.

What is an outlier?

A value much larger or smaller than the rest. Outliers affect the mean and range, but not the median much.

Key takeaways

Mean = total ÷ number; median = middle of ordered data; mode = most common.
Range = largest − smallest.
Use the median when there are outliers; the mode for categories.
Total = mean × number: use it for missing values and combined means.
Compare with one average and one measure of spread, in context.

Exam-style questions

Practise it the way it is examined

3 original questions · 9 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 25 marks

Here are the numbers of goals scored in 10 matches: $\ 2, 0, 3, 1, 2, 4, 2, 1, 0, 5$

(a)Write down the mode.[1]
Show mark scheme (a)
- B1 $2$
Answer: $2$
(b)Work out the median.[2]
Show mark scheme (b)
- M1Orders the data: $0, 0, 1, 1, 2, 2, 2, 3, 4, 5$
- A1 $2$
Answer: $2$
(c)Work out the mean.[2]
Show mark scheme (c)
- M1Total $20$
- A1 $2$
Answer: $2$

Question 2Grade 42 marks

Four friends have a mean height of 158 cm. A fifth friend joins, and the mean height becomes 160 cm.

Work out the height of the fifth friend.[2]
Show mark scheme
- M1 $4 \times 158 = 632$ or $5 \times 160 = 800$
- A1 $168$ cm
Answer: $168$ cm

Question 3Grade 52 marks

Two machines fill bags of rice. Machine A: median 502 g, range 14 g. Machine B: median 499 g, range 6 g. Bags should contain 500 g.

Which machine is better? Give two reasons, referring to the data.[2]
Show mark scheme
- B1A comparison of averages in context, e.g. B's median (499 g) is closer to 500 g than A's (502 g)
- B1B is more consistent because its range is smaller (6 g < 14 g), so B is better

Examiner note

Either machine can be argued on the average, but the consistency point clearly favours B. A decision supported by both measures scores full marks.

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Mode of $3, 5, 5, 7, 8, 5, 3$ ?
- $5$
- $3$
- $36$
- $7$
Show answer and explanation
Answer: $5$
5 appears three times.
Range of $12, 4, 19, 7, 10$ ?
- $15$
- $19$
- $10$
- $4$
Show answer and explanation
Answer: $15$
$19 - 4$ .
Mean of $6, 9, 12, 13$ ?
- $10$
- $10.5$
- $40$
- $9$
Show answer and explanation
Answer: $10$
$40 \div 4$ .
Median of $4, 9, 2, 7, 5, 8$ ?
- $6$
- $5$
- $7$
- $5.5$
Show answer and explanation
Answer: $6$
Ordered: 2, 4, 5, 7, 8, 9; halfway between 5 and 7.
Best average for favourite pizza topping?
- mode
- mean
- median
- range
Show answer and explanation
Answer: mode
The data is not numerical.
Mean of 4 numbers is 9. Three are 5, 10, 12. The fourth?
- $9$
- $36$
- $27$
- $8$
Show answer and explanation
Answer: $9$
$36 - 27$ .
Which average is most affected by an outlier?
- mean
- median
- mode
- none of them
Show answer and explanation
Answer: mean
It uses every value.
10 boys have mean 5; 15 girls have mean 7. Overall mean?
- $6.2$
- $6$
- $6.5$
- $12$
Show answer and explanation
Answer: $6.2$
$(50 + 105) \div 25$ .

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