Form 4 · Statistics and Probability
Measures of Dispersion for Ungrouped Data
Averages tell you the centre; dispersion tells you the spread. This chapter measures how scattered a set of data is.
What is Measures of Dispersion for Ungrouped Data?
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.
Content standards (DSKP)
The DSKP KSSM sets these content standards for this chapter:
- 8.1 Dispersion
- 8.2 Measures of Dispersion
The key ideas
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.
How this chapter 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.
Formulae given in the exam for this chapter
Common mistakes to avoid
- Forgetting to take the square root for standard deviation
- Dividing by the wrong N
- Rounding too early and carrying an error through the calculation
Why standard deviation beats the range
The range looks at only two numbers, the largest and the smallest, so one unusual value can blow it up while telling you nothing about the rest of the list. Standard deviation uses every value: it measures, on average, how far each value sits from the mean.
That is why the exam leans on it. Picture two classes with the same mean mark.
The class whose marks cluster tightly near the mean has a small standard deviation, while the class with marks scattered from very low to very high has a large one. Same centre, very different stories, and only the standard deviation (or the interquartile range) captures that difference.
What adding or scaling every value does to the spread
A favourite exam twist is to change every value in the same way and ask what happens to the mean and the standard deviation. Two rules cover it.
First, if you add a constant c to every value, the whole set slides along the number line: the mean goes up by c, but the spread does not change at all, so the standard deviation stays the same. Second, if you multiply every value by a constant k, the mean is multiplied by k and the standard deviation is multiplied by the size of k (its absolute value), while the variance is multiplied by k².
So adding 5 marks to everyone leaves the standard deviation untouched; doubling everyone's marks doubles it. Spotting this saves you from recalculating from scratch.
Reading the question: which measure of spread do they want?
Marks are easy to lose by computing the wrong measure, so read the command word. If the question says 'range', subtract the smallest value from the largest, one line.
If it says 'interquartile range', you need the first and third quartiles, so order the data, find the median, then the middle of each half; the answer is Q3 − Q1. If it says 'variance', stop at the squared figure and do not take the root; if it says 'standard deviation', you must take the square root at the very end.
When a Paper 2 question hands you Σx and Σx² already, that is a strong hint it wants the mean and standard deviation through the σ² = Σx²/N − x̄² formula, not a deviation-by-deviation slog.
A worked exam-style example
A short list of marks, the classic 'find the mean then the standard deviation' pair.
- Count the values: N = 6. Add them: Σx = 3 + 5 + 6 + 8 + 9 + 11 = 42.
- Mean x̄ = Σx ÷ N = 42 ÷ 6 = 7.
- Square each value and add: Σx² = 3² + 5² + 6² + 8² + 9² + 11² = 9 + 25 + 36 + 64 + 81 + 121 = 336.
- Variance σ² = Σx²/N − x̄² = 336 ÷ 6 − 7² = 56 − 49 = 7.
- Standard deviation σ = √(variance) = √7 = 2.65 (3 s.f.).
How to study this chapter
Study Measures of Dispersion for Ungrouped Data
Frequently asked questions
How this chapter is examined
SPM Mathematics assesses this chapter across Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective), drawing on the DSKP content standards above. Paper 2 gives marks for working, so showing every step matters.
Common mistakes to avoid
Forgetting to take the square root for standard deviation; Dividing by the wrong N; Rounding too early and carrying an error through the calculation.
Formulae given in the exam for this chapter
Yes, Mean (ungrouped), Variance (ungrouped), Standard deviation (ungrouped) appear on the formula sheet the exam provides. Anything else in this chapter you are expected to know.
Do I divide by N or by N − 1 for the standard deviation?
For SPM 1449 you always divide by N, the number of values, for both the variance and the standard deviation. The n − 1 version you may have seen elsewhere is for samples in higher statistics and is not used here.
Use the formulae exactly as the SPM formula sheet gives them, with N on the bottom.
Which variance formula should I use in the exam?
Both formulae give the same answer, so use whichever is faster. The Σx²/N − x̄² form is usually quicker because you only need two column totals and never subtract the mean from each value.
Keep the deviation form, Σ(x − x̄)²/N, for when the mean is a whole number and the list is very short.
Can the standard deviation ever be negative?
No. It is the square root of a sum of squared quantities divided by N, and squares are never negative, so the standard deviation is always zero or positive.
If your working ever produces a negative value, you have made an arithmetic slip, most often forgetting to subtract x̄² correctly. It is zero only when every value is identical.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)· SPM: Format Pentaksiran mulai 2021, Matematik (1449)
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