Measures of Dispersion for Ungrouped Data · Form 4
Measures of Dispersion for Ungrouped Data: Revision Notes
A tight revision summary of Measures of Dispersion for Ungrouped Data for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
Two classes can have the same average mark yet feel completely different, one bunched, one spread out. This chapter measures that spread for ungrouped data using range, interquartile range, variance and standard deviation, and shows what each one is good for.
Key ideas to revise
- Range and interquartile range. Range is simplest but sensitive to outliers; the interquartile range looks at the middle half and is steadier.
- Variance and standard deviation. These measure spread around the mean. The standard deviation is the square root of the variance and shares the data’s units.
- Two formulae, one answer. The variance can be found from Σ(x − x̄)² / N or the quicker Σx²/N − x̄², both are given in the exam.
One worked line per key idea
- Range (spread from lowest to highest): for 3, 5, 8, 12, 20 the range = highest − lowest = 20 − 3 = 17.
- Interquartile range (spread of the middle half): order 4, 6, 7, 9, 10, 12, 15. Median Q2 = 9, lower-half median Q1 = 6, upper-half median Q3 = 12, so IQR = Q3 − Q1 = 12 − 6 = 6.
- Variance from the deviation formula Σ(x − x̄)²/N: for 2, 4, 6 the mean x̄ = 4, so Σ(x − x̄)² = 4 + 0 + 4 = 8 and variance = 8/3 = 2.667.
- Standard deviation is the square root of the variance and carries the data's own units: σ = √2.667 = 1.633.
- The short formula Σx²/N − x̄² must land on the same number: for 2, 4, 6, Σx² = 56, so 56/3 − 4² = 18.667 − 16 = 2.667. Identical to the deviation formula.
A pre-paper checklist for this chapter
- Re-derive both variance forms on a three-number set the night before, so you trust that Σ(x − x̄)²/N and Σx²/N − x̄² agree, then use the shorter one under time pressure.
- The exam gives you both variance formulae and σ = √variance on the formula sheet; it does NOT give you the mean, quartile positions, or a ready-made table, so budget time to build those.
- For range and IQR, order the data first every single time; an unordered list is the most common reason a quartile comes out wrong.
- Keep the mean as an exact fraction or a long decimal in your calculator memory; rounding x̄ before squaring it is what quietly poisons the whole variance.
- The one habit that saves marks: set out an x column and an x² column with their totals Σx and Σx² before touching any formula, the answer then reads straight off two sums.
- Read the last line of the question again: circle whether they asked for variance or for standard deviation, and take the square root only if it is the standard deviation.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
The two variance formulae give me slightly different numbers, which is the real one?
They are algebraically identical, so a real difference means a rounding slip, almost always in the mean. Keep x̄ exact in your calculator instead of rounding it, then square that stored value.
If you still see a gap, recheck Σx², you may have squared each value wrongly. Done cleanly, both land on the same figure.
How much working do I need to show in revision, can I just write the final standard deviation?
Practise showing the full chain, because Paper 2 pays for it: the mean, the substitution into the variance formula, and the square-root step each carry marks. If you only rehearse the final number, you will not have the layout ready under pressure.
Write it out fully now so the structure becomes automatic in the exam.
Should I memorise the range and IQR steps or the standard deviation formula first?
The standard deviation formula is on the sheet, so drill the process, not the letters: build the x and x² table, then substitute. Range and IQR carry no formula sheet help, so memorise the ordering-and-quartile routine instead.
In short, rehearse the table method for variance and the ordering method for the quartile measures.