When do the mean and the median differ a lot?

When the data contain an extreme value the mean is pulled towards it while the median only shifts by position, so the two separate.

Why does the standard deviation take a square root?

Squaring removes the signs but also squares the units, so the square root brings the answer back to the original units.

Why is the standard deviation unchanged when every value gains the same number?

The distance of each value from the mean stays exactly the same, so the sum of squared deviations and the standard deviation do not change.

S4 Maths statistics paper 4: error drills on mean, standard deviation and quartiles

2026-11-10

Tutor Prep Guide · S4 Maths Statistics and Dispersion

Ten items target the usual slips: forgetting to divide by the count, skipping the square root, confusing variance with standard deviation, and a mis-set outlier fence.

Direct answers　Items 01 to 03 revisit a missing value, the standard deviation steps and the three averages, 04 to 06 a frequency-table mean, a combined mean and the effect of a shift, 07 to 09 quartiles, the outlier fence and working back from a variance, and 10 variance mistaken for standard deviation.

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)

FoundationFive numbers have mean 12. Four of them are 8, 10, 13 and 17. Find the fifth number.
FoundationFind the standard deviation of 6, 8, 10 and 12, correct to two decimal places.
FoundationFor the data 4, 6, 6, 9, 12 and 15, state the mode, the median and the range.
IntermediateA frequency table gives values 1, 2, 3, 4 with frequencies 3, 5, 7 and 5. Find the mean.
IntermediateGroup A has 15 students averaging 68 marks and group B has 10 averaging 78. Find the combined mean.
IntermediateEach value of 2, 4, 6, 8, 10 is increased by 5. What happens to the mean and the standard deviation?
ChallengeFind the lower quartile, upper quartile and interquartile range of 4, 6, 7, 9, 11, 13, 15, 17, 20.
ChallengeA data set has Q1 = 12 and Q3 = 24. Using the 1.5 IQR rule, find the upper fence and decide whether 45 is an outlier.
ChallengeSix values have variance 4. Find the standard deviation and the sum of squared deviations.
Error redoFor the data 6, 8, 10, 12 and 14 a student writes the variance 8 as the standard deviation. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	12	The total is 5 x 12 = 60 and the four known values add to 48, so the fifth is 60 – 48 = 12.
2	About 2.24	The mean is 9 and the squared deviations total 20, so the variance is 5 and the standard deviation is about 2.24.
3	Mode 6, median 7.5, range 11	6 appears twice so it is the mode; the middle pair 6 and 9 average to 7.5; the range is 15 – 4 = 11.
4	2.7	The frequencies total 20 and the values total 3 + 10 + 21 + 20 = 54, so the mean is 54 / 20 = 2.7.
5	72 marks	Weight by group size: (15 x 68 + 10 x 78) / 25 = 1 800 / 25 = 72.
6	Mean 11, standard deviation unchanged at about 2.83	Adding the same number raises the mean from 6 to 11, while distances from the mean stay the same, so the standard deviation remains about 2.83.
7	Q1 = 6.5, Q3 = 16 and the interquartile range is 9.5	The nine values have median 11, the lower half gives Q1 = 6.5 and the upper half gives Q3 = 16, so the interquartile range is 9.5.
8	Upper fence 42, so 45 is an outlier	The interquartile range is 12, so the fence is 24 + 1.5 x 12 = 42, and 45 lies above it and counts as an outlier.
9	Standard deviation 2 and sum of squared deviations 24	The standard deviation is the square root of 4, which is 2, and the sum of squared deviations is 4 x 6 = 24.
10	About 2.83	After reaching a variance of 8 one more square root is needed, so the answer is about 2.83; quoting the variance overstates the spread.

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.

Book a Maths tutor Tutor Registration →

Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.