When do the mean and the median differ a lot?

An extreme value drags the mean away while the median only looks at the middle position, so comparing them shows whether the data are skewed.

Why is the interquartile range more reliable than the range?

The range uses only the largest and smallest values, so one outlier inflates it; the interquartile range uses the middle half and reflects the whole spread better.

What must be done first before calculating?

Sort the data from smallest to largest first, otherwise the positions of the median and quartiles are all wrong.

S4 Maths statistics paper 1: mean, median, mode and range

2026-11-10

Tutor Prep Guide · S4 Maths Statistics and Dispersion

Ten items start from the basic statistics: the three averages, quartiles, the effect of adding or multiplying, and a slip about spread.

Direct answers　Items 01 to 03 cover mean, median and mode, 04 to 06 the range, a missing value and quartiles, 07 to 09 how shifting and scaling change the mean and range, and 10 a forgotten division.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S4 Maths Statistics and Dispersion
Level	S4

02　Questions (10 items)

FoundationFind the mean of 4, 7, 9, 10, 15.
FoundationFind the median of 3, 5, 8, 12, 14, 20.
FoundationFind the mode of 2, 3, 3, 5, 7, 7, 7, 9.
IntermediateFind the range of 12, 18, 25, 31, 40.
IntermediateFive numbers have mean 12. Four of them are 8, 10, 13 and 17. Find the fifth.
IntermediateFind the lower quartile, upper quartile and interquartile range of 2, 4, 6, 8, 10, 12, 14, 16.
ChallengeAdd 3 to every value of 4, 7, 9, 10, 15. How do the mean and the range change?
ChallengeMultiply every value of 4, 7, 9, 10, 15 by 2. How do the mean and the range change?
ChallengeWhich set has the larger dispersion: 5, 5, 5, 5 or 2, 5, 5, 8?
Error redoThe total of five numbers is 45 and a student answers that the mean is 45. What is the correct mean?

03　Answers and one-line notes

#	Answer	One-line note
1	9	Mean = sum divided by count = 45 / 5 = 9, so add all five values first.
2	10	With six ordered values the median is the average of the two middle ones, 8 and 12, giving 10.
3	7	The value 7 appears three times, more than any other, and the mode is simply the most frequent value.
4	28	Range = largest minus smallest = 40 – 12 = 28; it uses only the two extremes and ignores the middle values.
5	12	The total is 5 x 12 = 60 and the four known values add to 48, so the fifth is 60 – 48 = 12.
6	Q1 = 5, Q3 = 13, IQR = 8	Split the eight values into halves: the lower half 2, 4, 6, 8 has median 5 and the upper half has median 13, so IQR = 8.
7	Mean goes from 9 to 12 and the range stays 11	Adding the same number to every value shifts the whole set, so the mean rises by 3 while the gap between the largest and smallest is unchanged.
8	Mean goes from 9 to 18 and the range from 11 to 22	Doubling every value scales the whole set, so the mean and the range both double and the spread grows by the same factor.
9	The set 2, 5, 5, 8 (standard deviation 0 against about 2.12)	The first set has four identical values so its standard deviation is 0; the second is stretched by 2 and 8, giving about 2.12.
10	9	The mean divides the total by the count, so 45 / 5 = 9; forgetting to divide by 5 is the usual slip here.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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