How to use standard deviation
What standard deviation actually tells you, how it behaves when data is transformed, and the comparison questions that carry the marks.
What the examiner is testing
Not arithmetic — your calculator does that. The marks are for what the number means and for how it behaves when the data changes. Comparison questions are where they are won and lost.
Standard deviation measures spread about the mean. A small value means the data clusters tightly; a large value means it is scattered.
Getting the value
Enter the data into your GDC's statistics list and read \( \sigma_x \) (population standard deviation), which is what the IB uses unless a question explicitly says otherwise.
Write down what you entered. If the final value is wrong but the method is visible, method marks still stand.
Worked example: a comparison question
Two machines fill bottles. Ten bottles from each are measured, in millilitres.
- Machine A: mean 500.0, standard deviation 1.2
- Machine B: mean 500.0, standard deviation 4.5
Which machine should the factory use, and why?
A full-mark answer has three parts:
- Compare the means. Both average 500.0 ml, so neither is systematically over- or under-filling.
- Compare the spreads. Machine A's standard deviation is much smaller, so its output clusters far more tightly around 500 ml.
- Answer in context. Machine A, because consistency matters — a bottle from B could be several millilitres short, which risks under-filling below the labelled volume.
Point 3 is the one students skip, and it is usually worth a mark on its own. "A has a smaller standard deviation" states the number; it does not answer the question that was asked.
How transformations affect it
This appears every session and is quick marks if you know it.
| Change to every value | Mean | Standard deviation |
|---|---|---|
| Add \( c \) | Increases by \( c \) | Unchanged |
| Subtract \( c \) | Decreases by \( c \) | Unchanged |
| Multiply by \( k \) | Multiplied by \( k \) | Multiplied by \( \lvert k \rvert \) |
Example. A dataset has mean 20 and standard deviation 3. Every value is doubled and then increased by 5.
- New mean: \( 20 \times 2 + 5 = 45 \).
- New standard deviation: \( 3 \times 2 = 6 \). The \( +5 \) does nothing to it.
Shifting every value moves the whole distribution without changing how spread out it is — which is exactly why adding a constant leaves the standard deviation alone.
The three mistakes that lose marks
1. Comparing only the means. If two means are equal, the whole question is about the spread. Say so explicitly.
2. Applying the added constant to the standard deviation. Only the multiplier affects spread.
3. Not answering in context. "Smaller standard deviation" is a description. The mark is for saying what that means for the bottles, the students, or whatever the question is actually about.
30-second recap
Standard deviation is spread about the mean. Read it off the GDC and show your list. Adding shifts the mean only; multiplying scales both. In a comparison, address the mean, then the spread, then what it means in context — three parts, three marks.