Standard deviation measures how spread out a set of data is from its mean — a small SD means the data is clustered close to the mean; a large SD means it's spread widely. In O-Level E Maths, the formula is SD = √(Σfx²/Σf − (Σfx/Σf)²), which you calculate directly using your calculator's statistics mode rather than by hand for large data sets.
Two data sets can have exactly the same mean but look completely different. Standard deviation is the number that captures this difference — it tells you whether the data is tightly bunched around the average or scattered widely.
Small SD
Data values are close to the mean. Consistent, predictable results. E.g. test scores of 48, 50, 49, 51, 52 (mean 50).
Large SD
Data values are spread widely from the mean. Inconsistent results. E.g. test scores of 20, 50, 80, 45, 55 (mean 50).
Both example sets above have a mean of 50 — but the first is clearly more consistent. Standard deviation is the number that makes this difference precise and comparable.
Breaking this down: Σfx/Σf is just the mean (the sum of fx divided by the total frequency). Σfx²/Σf is the mean of the squared values. Standard deviation is the square root of the difference between these two quantities.
The table shows the number of siblings for 20 students. Calculate the mean and standard deviation.
| Number of siblings (x) | Frequency (f) | fx | fx² |
|---|---|---|---|
| 2 | 3 | 6 | 12 |
| 4 | 5 | 20 | 80 |
| 6 | 8 | 48 | 288 |
| 8 | 3 | 24 | 192 |
| 10 | 1 | 10 | 100 |
| Total | Σf = 20 | Σfx = 108 | Σfx² = 672 |
Solution
Mean = Σfx / Σf = 108 / 20 = 5.4SD = √(Σfx²/Σf − (Σfx/Σf)²)SD = √(672/20 − (108/20)²)672/20 = 33.6(108/20)² = 5.4² = 29.16SD = √(33.6 − 29.16) = √4.44 = 2.11 (3 s.f.)Use your calculator's statistics mode for the real exam. The table method above shows what's happening underneath, but for actual O-Level questions, enter the data into Statistics (STAT) mode on your fx-97SG and let the calculator compute Σfx, Σfx², the mean, and the standard deviation directly — this is faster and avoids arithmetic slips.
Common mistake
Calculators often display two versions of standard deviation: σx (population SD, divides by n) and sx (sample SD, divides by n−1). O-Level E Maths always uses the population version, σx. Using sx by mistake gives a slightly different — and incorrect — answer.
A very common O-Level question gives you the mean and SD for two data sets (e.g. two classes' test scores) and asks you to compare them, the same skill tested in comparing two data sets questions more broadly. There's a fixed 2-part structure examiners expect.
Example — Class A: mean 65, SD 4.2. Class B: mean 65, SD 9.8. Compare the two classes' performance.
Always address both mean and SD in a comparison — comparing only the mean, or only the SD, typically earns half the available marks. A complete answer states which group performed better on average (from the means) AND which group was more consistent (from the SDs).
A common point of confusion: why calculate Σfx²/Σf first (the mean of the squares) and separately calculate (Σfx/Σf)² (the square of the mean), instead of some other order? These are two genuinely different quantities, and the formula specifically wants the difference between them.
Order matters here
Σfx²/Σf ("mean of the squares") and (Σfx/Σf)² ("square of the mean") are not the same number — mixing up which one you square, and when, is one of the most common calculation errors in this topic. Always compute the mean first, square it separately, and keep it distinct from Σfx²/Σf until the final subtraction step.
Do I need to memorise the standard deviation formula?
The formula is provided in the O-Level formula sheet, so you don't need to memorise it exactly — but you do need to know how to apply it correctly, and how to extract Σfx and Σfx² from a frequency table (or via calculator statistics mode) without errors.
What does it mean if the standard deviation is 0?
A standard deviation of 0 means every value in the data set is identical — there's no spread at all. This is rare in real data but can appear in constructed exam questions to test whether you understand what SD represents.
Can standard deviation be negative?
No — standard deviation is always zero or positive, since it comes from a square root of a value that is mathematically guaranteed to be non-negative. If your calculation produces a negative number under the square root, you've made an arithmetic error and should recheck your working.
How is standard deviation different from range or interquartile range?
Range and interquartile range only use specific data points (the minimum/maximum, or the quartiles, read from a cumulative frequency curve or box plot) to describe spread. Standard deviation uses every single value in the data set, weighted by how far each one is from the mean — making it a more complete, but more calculation-heavy, measure of spread.
— Mr Gan Math Tuition
Mr. Gan teaches the calculator workflow and the exam structure together, so data questions stop costing easy marks.
Chat with Mr. Gan