IGCSE · Practice questions · Averages, range and interquartile range

Centre and spread.

Six original Cambridge-style questions on averages and measures of spread for lists of data, including Extended quartiles and interquartile range.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Order first, then count positions.

Order the data before finding the median or quartiles.
Mean: total \(\div\) number of values; to find a missing value, work with totals (mean \(\times\) n).
Median at position \(\dfrac{n+1}{2}\); (Extended) quartiles at \(\dfrac{n+1}{4}\) and \(\dfrac{3(n+1)}{4}\).
Spread: range = max \(-\) min; (Extended) IQR = UQ \(-\) LQ.
Choosing: median when there is an outlier; mode for categories.

Recall

[4 marks]

Find the mean, median, mode and range of 6, 2, 9, 4, 9, 3, 8.

In order: 2, 3, 4, 6, 8, 9, 9 (\(n = 7\)). ✓
Mean \(= \dfrac{41}{7} = 5.86\) (3 s.f.) ✓
Median = 4th value = 6. Mode = 9. Range \(= 9 - 2 = 7\). ✓

Calculation

[2 marks]

The mean of 8 numbers is 12.5. One of the numbers, 20, is removed. Find the mean of the remaining 7 numbers.

Total \(= 8 \times 12.5 = 100\). New total \(= 100 - 20 = 80\). ✓
New mean \(= \dfrac{80}{7} = 11.4\) (3 s.f.) ✓

Calculation

[3 marks]

(Extended) The marks of 15 students are

41 35 52 47 38 60 44 49 55 33 46 58 40 51 43

Find the median and the interquartile range.

In order: 33, 35, 38, 40, 41, 43, 44, 46, 47, 49, 51, 52, 55, 58, 60. ✓
Median = 8th value = 46. ✓
LQ = 4th value = 40, UQ = 12th value = 52, so IQR \(= 52 - 40 = 12\). ✓

Analysis

[2 marks]

The numbers of minutes eight buses were late are 2, 0, 3, 1, 2, 4, 1, 45.

Explain why the median is a better average to use than the mean, and find it.

45 is an extreme value that drags the mean up (mean \(= \dfrac{58}{8} = 7.25\) minutes, more than seven of the eight values). ✓
In order: 0, 1, 1, 2, 2, 3, 4, 45. Median \(= \dfrac{2+2}{2} = 2\) minutes. ✓

Proof

[3 marks]

Four whole numbers have a mode of 5, a median of 6 and a mean of 7. Find the four numbers.

Total \(= 4 \times 7 = 28\). ✓
The mode is 5, so 5 appears at least twice. The median 6 is more than 5, so the two smallest numbers are 5 and 5. ✓
Median \(= \dfrac{5 + c}{2} = 6\), so the third number \(c = 7\). ✓
Fourth number \(= 28 - 5 - 5 - 7 = 11\). The numbers are 5, 5, 7, 11. ✓

Analysis

[5 marks]

(Extended) The heights, in cm, of plants grown with fertiliser X and fertiliser Y are

X: 31 24 28 35 22 30 26

Y: 27 33 18 29 40 21 36

(a) Find the median and IQR for each fertiliser. (b) A gardener wants tall plants of a similar height. Which fertiliser should she choose? Give two reasons.

X in order: 22, 24, 26, 28, 30, 31, 35. Median 28, LQ 24, UQ 31, IQR 7. ✓
Y in order: 18, 21, 27, 29, 33, 36, 40. Median 29, LQ 21, UQ 36, IQR 15. ✓
(b) The medians are almost equal (28 cm and 29 cm), so Y is only slightly taller on average. ✓
X has a much smaller IQR (7 cm against 15 cm), so its plants are far more consistent in height. ✓
Fertiliser X is the better choice for plants of a similar height. ✓

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