Measures of Dispersion for Ungrouped Data · Form 4
Measures of Dispersion for Ungrouped Data: Paper 2 Answering Guide
How Measures of Dispersion for Ungrouped Data appears in Mathematics Paper 2 (Subjective), and how to lay out your working so you earn every method mark.
How it is examined
Paper 2 gives a list of values and asks for the mean, then a measure of spread, commonly the standard deviation using the formula the paper provides. The arithmetic is where marks slip, so setting out a clear table pays off.
Showing your working
Mathematics Paper 2 (Subjective) is worth 100 marks and gives marks for the steps, not only the answer. Write each line clearly: state the formula or rule, substitute the numbers, then simplify.
If the question carries units, carry them through to the final line.
Worked Paper 2 example: mean, variance and standard deviation
Eight pupils recorded the number of sit-ups they completed in one minute: 5, 7, 8, 10, 11, 13, 14, 16. Calculate (a) the mean number of sit-ups, (b) the variance, and (c) the standard deviation of the data.
- (a) State the rule: mean x̄ = Σx / N.
- Substitute: Σx = 5 + 7 + 8 + 10 + 11 + 13 + 14 + 16 = 84 and N = 8.
- Simplify: x̄ = 84 / 8 = 10.5. Answer: the mean is 10.5 sit-ups.
- (b) State the rule: variance σ² = Σx²/N − x̄².
- Substitute Σx²: 5² + 7² + 8² + 10² + 11² + 13² + 14² + 16² = 25 + 49 + 64 + 100 + 121 + 169 + 196 + 256 = 980.
- Simplify: σ² = 980/8 − (10.5)² = 122.5 − 110.25 = 12.25. Answer: the variance is 12.25 sit-ups².
- (c) State the rule: standard deviation σ = √variance.
- Substitute and simplify: σ = √12.25 = 3.5. Answer: the standard deviation is 3.5 sit-ups.
Where the method marks are won or lost
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
If I use the deviation formula Σ(x − x̄)²/N instead, will I still get full marks?
Yes. Both variance formulae are on the sheet and both earn full marks.
For this data the deviation form gives the same 12.25, but it needs eight subtractions and squarings, so it is slower and more error-prone. Use it if the mean is a whole number; otherwise the Σx²/N − x̄² form is usually cleaner.
The question only asks for the standard deviation, must I still show the variance?
Show it. The standard deviation is defined as the square root of the variance, so the variance line is a natural method step the marker expects to see.
Write σ² = 12.25 first, then σ = √12.25 = 3.5. It also lets you catch an error before the final root, when it is still easy to fix.
How many decimal places should the final standard deviation have?
Follow the question's instruction; if it says 'to two decimal places', give 3.50. When it says nothing, a clean value like 3.5 is fine as it is, and otherwise four significant figures is a safe default.
Never round earlier than the final line, and always keep the exact figure in your calculator until then.