Standard deviation formula: σ = √(Σx²/n − x̄²)
The standard deviation formula is σ = √(Sxx / n) = √(Σx²/n − x̄²).
What each letter means
- \(\sigma\) the standard deviation
- \(\bar x\) the mean
- \(S_{xx}\) Σ(x − x̄)² = Σx² − (Σx)²/n
- \(n\) the number of values
When to use it
To measure how spread out data are around the mean. The second form is quicker when you are given Σx and Σx².
Worked example
For 20 values, \(\sum x=340\) and \(\sum x^2=6100\). Find the mean and the standard deviation.
- \(\bar x=\dfrac{340}{20}=17\)
- \(\sigma^2=\dfrac{6100}{20}-17^2=305-289=16\)
Answer: mean 17, standard deviation 4
Common mistake
Forgetting the square root (that gives the variance), or squaring the mean of x instead of subtracting μ².
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | In the Edexcel formulae booklet |
| Edexcel International A Level | Not in the booklet: learn it |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
Questions
What is the standard deviation formula?
The standard deviation formula is σ = √(Sxx / n) = √(Σx²/n − x̄²). σ: the standard deviation; x̄: the mean; Sxx: Σ(x − x̄)² = Σx² − (Σx)²/n; n: the number of values.
Is the standard deviation given in the exam?
Edexcel A Level (9MA0): in the Edexcel formulae booklet. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
Should I divide by n or n − 1?
This page divides by n: the standard deviation of the data set itself (σx on a calculator). Dividing by n − 1 (sx) estimates a population's standard deviation from a sample; use it only when the question asks for that.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.