Averages, grouped data, quartiles & interquartile range – IGCSE Maths exam questions with mark scheme

1Grade 1–3FoundationCalculator4 marks

Ten people were asked how long, in minutes, they waited at a bus stop.

$$7 \quad 12 \quad 4 \quad 9 \quad 12 \quad 15 \quad 6 \quad 12 \quad 8 \quad 5$$

(a) Write down the mode. [1]

(b) Work out the median. [2]

(a) 12 (minutes)B1
(b) Lists the times in order (allow one error or omission), or identifies 8 and 9 as the middle pairM1
(b) 8.5 (minutes) caoA1
(c) 11 (minutes) cao (do not accept 4–15 or '4 to 15')B1
Answer: (a) 12 min (b) 8.5 min (c) 11 min
Examiner tip: With an even number of values the median is halfway between the two middle values, so order the list first and do not just pick the 5th value.

2Grade 4–5FoundationCalculator5 marks

The table gives information about the number of books read last month by each of 40 students.

Number of books	Frequency
0	5
1	9
2	12
3	8
4	4
5	2

(a) Write down the mode. [1]

(b) Find the median. [1]

(a) 2 (do not accept 12, the frequency)B1
(b) 2B1
(c) At least 4 correct products $f \times x$ seen, e.g. $1\times9,\ 2\times12,\ 3\times8,\ 4\times4$, or a total of 83M1
(c) (dep on first M1) Their $\sum fx \div 40$, e.g. $83 \div 40$M1
(c) 2.075 (accept 2.07, 2.08 or 2.1 if 2.075 or $83 \div 40$ seen)A1
Answer: (a) 2 (b) 2 (c) 2.075
Examiner tip: Divide by the total frequency (40), not by the number of rows (6); dividing by 6 gives 13.8, which is impossible for 0 to 5 books.

3Grade 4–5FoundationCalculator6 marks

The table gives information about the masses of 60 parcels.

Mass ($m$ kg)	Frequency
$0 < m \le 2$	14
$2 < m \le 4$	21
$4 < m \le 6$	13
$6 < m \le 10$	8
$10 < m \le 14$	4

(a) Write down the modal class. [1]

(b) Work out an estimate for the mean mass of the parcels. Give your answer correct to 3 significant figures. [4]

(a) $2 < m \le 4$ (accept 2–4)B1
(b) Products $f \times x$ for at least 4 rows, with $x$ consistently in each interval (including end points)M1
(b) (dep) Correct midpoints 1, 3, 5, 8, 12 used for at least 4 products, e.g. 14, 63, 65, 64, 48M1
(b) (dep on first M1) Their $\sum fx \div 60$, e.g. $254 \div 60$M1
(b) 4.23 (kg) (accept 4.233...)A1
(c) $2 < m \le 4$B1
Answer: (a) $2 < m \le 4$ (b) 4.23 kg (c) $2 < m \le 4$
Examiner tip: The last two classes are wider, so their midpoints are 8 and 12, not 7 and 11; check each midpoint against its own interval.

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4Grade 4–5Calculator4 marks

Leila plays five rounds of a quiz. Her mean score for the five rounds is 9.

Her scores for four of the rounds are 6, 11, 4 and 13.

(a) Work out her score in the fifth round. [2]

Leila then plays a sixth round. Her mean score for all six rounds is 10.

(b) Work out her score in the sixth round. [2]

(a) $5 \times 9 = 45$ seen, or $45 - 34$M1
(a) 11 caoA1
(b) $6 \times 10 = 60$ seen, or $60 - 45$ (ft their total from (a))M1
(b) 15 caoA1
Answer: (a) 11 (b) 15
Examiner tip: Work with totals: mean × number of values gives the total, then subtract the known scores; averaging the two means does not work.

5Grade 6–7HigherCalculator4 marks

Here are the heights, in cm, of 15 tomato plants.

$$34 \quad 26 \quad 41 \quad 23 \quad 37 \quad 30 \quad 48 \quad 28 \quad 36 \quad 42 \quad 25 \quad 33 \quad 39 \quad 31 \quad 45$$

(a) Find the median height. [1]

(b) Work out the interquartile range of the heights. [3]

(a) 34 (cm)B1
(b) Heights written in order (allow one error), or the lower quartile 28 or upper quartile 41 identifiedM1
(b) Both 28 and 41 identified, or $41 - 28$M1
(b) 13 (cm) caoA1
Answer: (a) 34 cm (b) 13 cm
Examiner tip: Quartiles are the $\tfrac{1}{4}(n+1)$th and $\tfrac{3}{4}(n+1)$th values of the ordered list (4th and 12th here); reading from the unordered list loses every mark.

6Grade 6–7HigherCalculator5 marks

80 students each completed a puzzle. The cumulative frequency graph shows information about the times, $t$ minutes, they took.

(a) Use the graph to find an estimate for the median time. [1]

(b) Use the graph to find an estimate for the interquartile range of the times. [2]

(a) Answer in the range 27.5 to 29 (minutes)B1
(b) Readings at cumulative frequencies 20 and 60 (or 20.25 and 60.75) shown, e.g. 20 and 38, or one of the quartiles correctM1
(b) Answer in the range 17 to 19 (minutes)A1
(c) Reading from graph at $t = 45$ (68 to 70) seen, or $80 - $ their readingM1
(c) Answer in the range 10 to 12A1
Answer: (a) about 28 min (b) about 18 min (c) about 11 students
Examiner tip: In (c) the graph gives the number who took 45 minutes or less; you must subtract that reading from 80.

7Grade 6–7HigherCalculator4 marks

Two cafés, North and South, recorded the number of cups of coffee they sold each day for 50 days. The table gives information about the numbers sold.

	North	South
Median	142	151
Lower quartile	128	146
Upper quartile	160	157
Range	96	58

(a) Work out the interquartile range for each café. [1]

(b) Compare the distributions of the numbers of cups sold at the two cafés. [2]

On one day North sold far more cups than usual because of a local event.

(a) North 32 and South 11 (both required)B1
(b) Correct comparison of medians in context, e.g. 'on average South sold more cups per day (median 151 > 142)'C1
(b) Correct comparison of spread using IQR or range in context, e.g. 'the numbers sold at North were more varied / less consistent (IQR 32 > 11)'C1
(c) e.g. 'the IQR uses only the middle 50% of the data so it is not affected by the extreme value'; accept 'the range is affected by outliers'B1
Answer: (a) North 32, South 11 (b) South higher on average; North more spread out (c) IQR ignores the extreme value
Examiner tip: A comparison needs one statement about an average and one about spread, each said in context (cups sold), not just two numbers quoted.

8Grade 6–7Calculator4 marks

There are 30 students in a class: 18 boys and 12 girls. All the students sat a test.

The mean score of the boys was 62.
The mean score of all 30 students was 65.6.

Work out the mean score of the girls.

[4]

$30 \times 65.6 = 1968$M1
$18 \times 62 = 1116$M1
(dep on both M1) $(1968 - 1116) \div 12$, e.g. $852 \div 12$M1
71 caoA1
Answer: 71
Examiner tip: You cannot find the girls' mean by doubling or averaging the two means; work with the totals for all students and for the boys.

9Grade 4–5Calculator3 marks

Five positive whole numbers have

a mode of 4
a median of 5
a mean of 6.2
a range of 8

Find the five numbers.

[3]

Total of the numbers $5 \times 6.2 = 31$ seenM1
Deduces the two smallest numbers are 4 and 4 and the largest is 12 (from range $4 + 8$)M1
4, 4, 5, 6, 12 (any order)A1
Answer: 4, 4, 5, 6, 12
Examiner tip: Start from the total ($5 \times 6.2 = 31$): with 4, 4, 5 and 12 fixed, the fourth number must be $31 - 25 = 6$; then check the list against every condition.

10Grade 8–9HigherCalculator6 marks

The table shows information about the scores, out of 5, that some customers gave a hotel.

Score	Frequency
1	4
2	$x$
3	7
4	$2x$
5	3

The mean score is 3.2

(a) Work out the value of $x$. [4]

(b) Find the median score. [2]

(a) Total frequency $14 + 3x$ seen or $\sum fx = 40 + 10x$ seen (allow unsimplified, e.g. $4 + 2x + 21 + 8x + 15$)M1
(a) Forms an equation, e.g. $\dfrac{40 + 10x}{14 + 3x} = 3.2$M1
(a) Correct rearrangement to $ax = b$, e.g. $0.4x = 4.8$M1
(a) $x = 12$ caoA1
(b) Uses frequencies 4, 12, 7, 24, 3 (total 50) and identifies the 25th/26th (or 25.5th) value, e.g. cumulative frequencies 4, 16, 23, 47M1
(b) 4 (ft their integer $x$ if M1 awarded)A1
Answer: (a) $x = 12$ (b) 4
Examiner tip: The mean is $\sum fx$ divided by the total frequency, which also contains $x$; dividing by 5 (the number of scores) is the most common error.

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