What is S4: Averages and spread?

A comparison scores when it names a measure of centre and a measure of spread, quotes the figures, and says what they mean in context. From a stem-and-leaf diagram, count to the middle value for the median and subtract the smallest from the largest for the range.

How many marks are S4 questions worth?

In verified past-paper appearances, S4 questions on Edexcel GCSE Maths 1MA1 have been worth 1–7 marks.

Which past exam papers has S4 appeared in?

S4 has appeared in 52 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Interpret, analyse and compare data-set distributions via graphs (incl. box plots), central tendency (median, mean, mode, modal class) and spread (range, outliers, quartiles, inter-quartile range)

Averages and spread

Worked answers, methods and verified real exam appearances for S4 on Edexcel GCSE Maths 1MA1.

Definition

Specification S4: interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate graphical representation involving discrete, continuous and grouped data, including box plots, and through appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers, quartiles and inter-quartile range).

Explanation

A comparison scores when it names a measure of centre and a measure of spread, quotes the figures, and says what they mean in context. From a stem-and-leaf diagram, count to the middle value for the median and subtract the smallest from the largest for the range.
Higher tier: on a cumulative frequency graph read the median at half the total and the quartiles at a quarter and three quarters; the interquartile range $Q_3-Q_1$ measures the middle half and ignores outliers.
A box plot shows both directly.
Estimate a mean from a grouped table with midpoints, $\sum fx\div\sum f$ ; with two known group means, multiply each by its size to recover totals.
Compare median with median and range with range.

Worked example

For a group of Year 9 girls, the least height is

152

cm, the median height is

166

cm and the greatest height is

176

cm. The heights, in cm, of

15

Year 9 boys are

156,\ 158,\ 161,\ 163,\ 165,\ 166,\ 169,\ 171,\ 172,\ 174,\ 175,\ 178,\ 179,\ 181,\ 183

. Compare the distribution of the girls' heights with the distribution of the boys' heights.

(3)

(Total for Question 1 is 3 marks)

Working

1.The boys' median is the $8$ th of the $15$ ordered values: $171$ cm.
2.The boys' range is $183-156=27$ cm; the girls' range is $176-152=24$ cm.
3.Compare centres: the boys' median height of $171$ cm is greater than the girls' $166$ cm, so the boys are typically taller.
4.Compare spreads: the boys' range of $27$ cm is greater than the girls' $24$ cm, so the boys' heights are more spread out.

Answer: Boys' median $171$ cm and range $27$ cm; the boys are typically taller than the girls and their heights are more varied.

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Additional guidance
1	for identification of the median (171) or the range (27) of the boys, or the range (24) of the girls	B1
	for a correct comparison of medians and a correct comparison of ranges supported by correct figures, eg the median height for boys (171) is greater than the median for girls (166) and the range for boys (27) is greater than the range for girls (24)	C2	At least one comparison must be in context. (C1 for a correct comparison of medians or of ranges that could ft their figures)

Common mistakes

Don't fall into the trap of giving comparison figures without stating what they mean in context.
Don't fall into the trap of finding the median before putting raw data in order.

Mark-scheme wording to use

“At least one comparison must be in context referring to height or quoting cm. Simply quoting values for median, range is insufficient; they must be compared” — Pearson 1MA1, June 2020 Paper 1H, Q7
“B1 for correctly identifying the median of (School) A as 57 C1 (ft, dep on value for the median being stated) for making a correct comparison of median eg ‘the median of (School) A is higher than the median of (School) B’” — Pearson 1MA1, November 2024 Paper 3H, Q6
“for a first step in the process … for a complete process eg (50 × 167.6 - 20 × 182) ÷ 30” — Pearson 1MA1, November 2022 Paper 3H, Q8

What the examiner wants

Stated values and a direction. The November 2024 Paper 3H report says the median of $57$ had to be clearly stated, not just circled on the stem-and-leaf diagram, and that a comparison must say which group is greater; simply giving the difference between the values was not enough.

Exam tip

A full comparison usually needs both a typical-value statement and a spread statement.

Worked practice

Tier 1 · Easy

For the data

4,6,6,9,10

, work out the mean and the range.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	Mean $=7$ ; range $=6$ .	2	The total is $4+6+6+9+10=35$ , so the mean is $35/5=7$ . The range is $10-4=6$ .

Tier 2 · Standard

Data set A has median

24

and range

33

. Data set B has median

25

and range

29

. Compare the two distributions.

(2)

(Total for Question 2 is 2 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	B has the slightly higher median: $25$ compared with $24$ . B has the smaller range: $29$ compared with $33$ , so its values are less spread out overall.	2	Compare the typical values using the medians, then compare spread using the ranges. B has both the larger centre and the smaller spread.

Tier 3 · Hard

The data are

12,13,13,14,15,15,16,52

. Work out the mean, median and range. Decide which of the mean or median better describes a typical value for these data. Give a reason.

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	Mean $=18.75$ , median $=14.5$ , range $=40$ . The median is more representative because $52$ is an outlier that pulls up the mean.	4	The total is $150$ , giving mean $150/8=18.75$ . The median is $(14+15)/2=14.5$ and the range is $52-12=40$ . The isolated value $52$ has a strong effect on the mean but not on the median, so the median better represents the main cluster.

Tier 1 · Easy

For the data

2,4,4,6,9,10,12

, write down the mode and the median.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	Mode $=4$ ; median $=6$ .	2	$4$ occurs most often, so it is the mode. The data are ordered and the fourth of the seven values is $6$ , so the median is $6$ .

Tier 2 · Standard

Five readings have mean

18

. A sixth reading of

24

is added. Work out the new mean and explain why it is greater than

18

(3)

(Total for Question 5 is 3 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	New mean $=19$ . The added reading is greater than the original mean, so it raises the mean.	3	The original total is $5\times18=90$ . The new total is $90+24=114$ , so the new mean is $114/6=19$ . Since $24$ is above the original mean, the mean increases.

Tier 3 · Hard

The ordered data are

6,9,11,x,18,20

. Their mean is

13

. Work out

x

, the median and the range.

(4)

(Total for Question 6 is 4 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	$x=14$ ; median $=12.5$ ; range $=14$ .	4	The total of all six values is $6\times13=78$ . The known values total $6+9+11+18+20=64$ , so $x=78-64=14$ . The median is the mean of the third and fourth values: $(11+14)/2=12.5$ . The range is $20-6=14$ .

Tier 2 · Standard

A data set has mean

12

, median

11

and range

9

. Every value in the data set is doubled and then

3

is added. Work out the mean, median and range of the new data set.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	Mean $=27$ ; median $=25$ ; range $=18$ .	3	The mean and median both undergo the same transformation as every value: $2\times12+3=27$ and $2\times11+3=25$ . Doubling every value doubles the range, while adding the same amount to every value does not change the range, so the new range is $2\times9=18$ .

Tier 3 · Hard

Seven integer values are written in order of size:

3,7,a,11,b,16,20

. Their mean is

11

and their only mode is

7

. Work out

a

b

and the range.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	$a=7$ , $b=13$ and the range is $17$ .	4	For $7$ to be the only mode, it must occur again, so $a=7$ . A mean of $11$ for seven values gives a total of $77$ . The known values, including $a=7$ , total $64$ , so $b=77-64=13$ . This is consistent with the stated order. The range is $20-3=17$ .

Tier 3 · Hard

The values

4

7

10

and

13

have frequencies

2

5

k

and

1

respectively. The mean is

8

. Work out

k

, the median, the mode and the range.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	$k=4$ , median $=7$ , mode $=7$ and range $=9$ .	4	The total frequency is $8+k$ and the total of the values is $2(4)+5(7)+10k+13=56+10k$ . Therefore $56+10k=8(8+k)$ , so $2k=8$ and $k=4$ . There are $12$ values; the sixth and seventh are both $7$ , so the median is $7$ . The greatest frequency is $5$ , so the mode is $7$ , and the range is $13-4=9$ .

Q10

Tier 3 · Hard

Higher only: Nine values are written in order:

4,7,9,12,15,18,x,26,31

. The upper quartile is

24

. The median is excluded when the lower and upper halves are formed. Work out

x

, the lower quartile and the interquartile range.

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	$x=22$ , lower quartile $=8$ and interquartile range $=16$ .	4	After excluding the median $15$ , the upper half is $18,x,26,31$ . Its median is $(x+26)/2=24$ , so $x=22$ , which is consistent with the stated order. The lower half is $4,7,9,12$ , so the lower quartile is $(7+9)/2=8$ . The interquartile range is $24-8=16$ .

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2019-11	2F	Q9	6	Allowed	Foundation	QP MS
2023-11	3F	Q22	4	Allowed	Foundation	QP MS
2019-11	3H	Q4	3	Allowed	Higher	QP MS
2019-06	1F	Q7	3	Non-calculator	Foundation	QP MS
2019-06	1H	Q11	4	Non-calculator	Higher	QP MS
2024-06	2H	Q11	7	Allowed	Higher	QP MS
2024-11	2F	Q7	5	Allowed	Foundation	QP MS
2024-06	3H	Q11	5	Allowed	Higher	QP MS
2023-11	1F	Q21	4	Non-calculator	Foundation	QP MS
2022-06	1H	Q8	3	Non-calculator	Higher	QP MS
2019-11	2H	Q11	3	Allowed	Higher	QP MS
2024-11	3F	Q15	4	Allowed	Foundation	QP MS
2022-11	1F	Q7	3	Non-calculator	Foundation	QP MS
2023-11	1F	Q8	6	Non-calculator	Foundation	QP MS
2023-11	3H	Q3	4	Allowed	Higher	QP MS
2022-06	2H	Q10	5	Allowed	Higher	QP MS
2022-11	3H	Q8	3	Allowed	Higher	QP MS
2024-11	1H	Q9	6	Non-calculator	Higher	QP MS
2023-06	3F	Q9	6	Allowed	Foundation	QP MS
2019-06	3H	Q3	3	Allowed	Higher	QP MS
2022-11	2F	Q15	5	Allowed	Foundation	QP MS
2024-06	3F	Q7	2	Allowed	Foundation	QP MS
2022-06	2F	Q15	4	Allowed	Foundation	QP MS
2022-06	1H	Q10	6	Non-calculator	Higher	QP MS
2023-11	1F	Q1	1	Non-calculator	Foundation	QP MS
2021-11	2H	Q9	4	Allowed	Higher	QP MS
2023-11	1H	Q4	4	Non-calculator	Higher	QP MS
2019-06	3F	Q26	3	Allowed	Foundation	QP MS
2019-11	3H	Q12	3	Allowed	Higher	QP MS
2023-06	1F	Q12	2	Non-calculator	Foundation	QP MS
2022-06	2F	Q3	1	Allowed	Foundation	QP MS
2023-06	2F	Q17	4	Allowed	Foundation	QP MS
2022-11	2F	Q17	3	Allowed	Foundation	QP MS
2023-11	2H	Q15	3	Allowed	Higher	QP MS
2019-11	3F	Q13	2	Allowed	Foundation	QP MS
2024-11	3F	Q22	4	Allowed	Foundation	QP MS
2019-11	1H	Q10	6	Non-calculator	Higher	QP MS
2019-11	1H	Q6	3	Non-calculator	Higher	QP MS
2019-11	3F	Q25	3	Allowed	Foundation	QP MS
2024-06	1H	Q8	3	Non-calculator	Higher	QP MS
2024-06	1F	Q15	4	Non-calculator	Foundation	QP MS
2022-06	2F	Q18	3	Allowed	Foundation	QP MS
2024-11	3H	Q6	4	Allowed	Higher	QP MS
2021-11	1H	Q12	3	Non-calculator	Higher	QP MS
2022-11	3F	Q8	3	Allowed	Foundation	QP MS
2023-06	3F	Q12	4	Allowed	Foundation	QP MS
2022-06	3F	Q14	4	Allowed	Foundation	QP MS
2022-11	2H	Q11	3	Allowed	Higher	QP MS
2019-11	1F	Q25	3	Non-calculator	Foundation	QP MS
2022-11	3F	Q28	3	Allowed	Foundation	QP MS
2019-06	3F	Q18	3	Allowed	Foundation	QP MS
2024-06	2F	Q28	4	Allowed	Foundation	QP MS

Other points in S Statistics

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