Normal distribution formula: z = (x − μ) / σ
The normal distribution formula is z = (x − μ) / σ, which turns X ~ N(μ, σ²) into Z ~ N(0, 1).
What each letter means
- \(X\) a normally distributed variable, X ~ N(μ, σ²)
- \(\mu\) the mean
- \(\sigma\) the standard deviation (the square root of the variance σ²)
- \(Z\) the standard normal variable, mean 0 and standard deviation 1
When to use it
To turn a normal probability into a standard one, and to find an unknown mean or standard deviation from a given probability.
Worked example
Heights are modelled by \(X\sim N(170,\,8^2)\) cm. Find \(P(X>180)\).
- \(z=\dfrac{180-170}{8}=1.25\)
- \(P(Z>1.25)=1-P(Z<1.25)\)
Answer: \(P(X>180)=0.106\) (3 s.f.)
Common mistake
Dividing by the variance. X ~ N(50, 16) has σ = 4, not 16.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| 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 normal distribution formula?
The normal distribution formula is z = (x − μ) / σ, which turns X ~ N(μ, σ²) into Z ~ N(0, 1). X: a normally distributed variable, X ~ N(μ, σ²); μ: the mean; σ: the standard deviation (the square root of the variance σ²); Z: the standard normal variable, mean 0 and standard deviation 1.
Is the normal distribution (z-score) given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. 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.
What does a z-score mean?
It says how many standard deviations a value lies above (positive) or below (negative) the mean.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.