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S4 Maths statistics paper 3: reverse problems, weighting and extreme values
S4 Maths statistics paper 3: reverse problems, weighting and extreme values
Ten items push into analysis: a missing value, a weighted mean, the effect of changing one value, removing an outlier, and the stability of the median.
Direct answers Items 01 to 03 cover a missing value, a standard deviation and a weighted mean, 04 to 06 changing a value, removing an outlier and comparing stability, 07 to 09 quartiles of nine values, a sum of squares and adding an extreme value, and 10 a missing square root.
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 9. Four of them are 6, 8, 10 and 13. Find the fifth.
- FoundationFind the standard deviation of 3, 5, 7, 9.
- FoundationTwenty boys average 165 cm and thirty girls average 155 cm. Find the overall mean height.
- IntermediateIn 4, 6, 8, 10, 12 the value 12 is replaced by 22. How do the mean and the range change?
- IntermediateThe data 5, 6, 7, 8, 9, 10, 40 lose the value 40. What are the new mean and median?
- IntermediateTwo classes both average 65 marks, with standard deviations 5 and 12. Which class is more consistent?
- ChallengeFind the lower quartile, upper quartile and interquartile range of 3, 5, 7, 8, 10, 12, 14, 15, 18.
- ChallengeFive values have sum of squared deviations Σ(x – mean)² = 60. Find the variance and the standard deviation.
- ChallengeThe data 10, 12, 15, 18, 20 gain the value 90. What are the new mean and median?
- Error redoWhen finding the standard deviation of 2, 4, 6, 8 a student gives the variance 5 as the answer. What is correct?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 8 | The total is 5 x 9 = 45 and the four known values add to 37, so the fifth is 45 – 37 = 8. |
| 2 | About 2.24 | The mean is 6 and the squared deviations total 20, so the variance is 5 and the standard deviation is about 2.24. |
| 3 | 159 cm | Weight by group size: (20 x 165 + 30 x 155) / 50 = 7950 / 50 = 159 cm. |
| 4 | Mean goes from 8 to 10 and the range from 8 to 18 | The total rises from 40 to 50, so the mean becomes 10, while the smallest stays 4 and the largest becomes 22, giving a range of 18. |
| 5 | Mean 7.5, median 7.5 | After removing 40 the six values 5 to 10 total 45, so the mean is 7.5 and the median is the average of 7 and 8, also 7.5. |
| 6 | The class with standard deviation 5 | With equal means the smaller standard deviation means the marks cluster more tightly, so the class with 5 is more consistent. |
| 7 | Q1 = 6, Q3 = 14.5, IQR = 8.5 | The nine values have median 10, the lower four give Q1 = 6 and the upper four give Q3 = 14.5, so the IQR is 8.5. |
| 8 | Variance 12, standard deviation about 3.46 | The variance is 60 / 5 = 12 and the standard deviation is √12, about 3.46, dividing by the number of values rather than one less. |
| 9 | Mean 27.5, median 16.5 | The total rises from 75 to 165 so the mean jumps to 27.5, while the median moves only to the average of 15 and 18, which is 16.5. |
| 10 | About 2.24 | After the variance of 5 one more square root is needed, so √5 is about 2.24; skipping it is the usual slip. |
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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