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S4 Maths statistics paper 2: standard deviation, weighted mean and outliers
S4 Maths statistics paper 2: standard deviation, weighted mean and outliers
Ten items move into dispersion: a frequency-table mean, the steps of a standard deviation, a combined mean, the 1.5 IQR rule and comparing box plots.
Direct answers Items 01 to 03 cover a frequency-table mean, computing a standard deviation and quartiles of ten values, 04 to 06 an outlier fence, a new value and a combined mean, 07 to 09 the effect of a shift on dispersion, box plot comparison and median with mode, and 10 variance confused with 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)
- FoundationA frequency table shows value 0 twice, 1 five times, 2 six times and 3 twice. Find the mean.
- FoundationFind the standard deviation of 2, 4, 6, 8.
- FoundationFor the ten integers from 1 to 10, find the lower quartile, upper quartile and interquartile range.
- IntermediateA data set has Q1 = 12 and Q3 = 20. Using the 1.5 x IQR rule, is 35 an outlier?
- IntermediateSix numbers have mean 14. A value of 28 is added. Find the new mean.
- IntermediateGroup A has 10 students averaging 60 marks and group B has 15 students averaging 70. Find the combined mean.
- ChallengeIf 5 is added to every value in a data set, what happens to the standard deviation?
- ChallengeTwo box plots share a median of 60, but the box of A is 10 long and the box of B is 25 long. Which set is more dispersed?
- ChallengeFind the median and the mode of 12, 15, 15, 18, 20, 22, 25.
- Error redoA data set has variance 16 and a student answers that the standard deviation is 16. What is correct?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | About 1.53 | There are 15 values in total and the weighted sum is 0 x 2 + 1 x 5 + 2 x 6 + 3 x 2 = 23, so the mean is 23 / 15, about 1.53. |
| 2 | About 2.24 | The mean is 5 and the squared deviations add to 9 + 1 + 1 + 9 = 20, so the variance is 20 / 4 = 5 and the standard deviation is √5, about 2.24. |
| 3 | Q1 = 3, Q3 = 8, IQR = 5 | The ten values split into two halves of five: the lower half 1 to 5 has median 3 and the upper half 6 to 10 has median 8, so the IQR is 5. |
| 4 | Yes, it is an outlier (upper fence 32) | The IQR is 8 so the upper fence is 20 + 1.5 x 8 = 32, and 35 lies beyond it, so it counts as an outlier. |
| 5 | 16 | The original total is 6 x 14 = 84, and adding 28 gives 112 over seven values, so 112 / 7 = 16. |
| 6 | 66 marks | The combined mean is weighted by group size: (10 x 60 + 15 x 70) / 25 = 1650 / 25 = 66, not the plain average of 60 and 70. |
| 7 | It stays the same | Adding the same constant to every value leaves each deviation from the mean unchanged, so the variance and standard deviation do not change. |
| 8 | Set B | The box length is the interquartile range, and 25 is greater than 10, so with the same median the middle half of B is spread out more. |
| 9 | Median 18, mode 15 | With seven ordered values the fourth is 18, and 15 appears twice, so the only repeated value is the mode. |
| 10 | 4 | The standard deviation is the square root of the variance, so √16 = 4; quoting the variance directly 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.
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