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What is normal distribution?

The normal distribution is a symmetric, bell-shaped continuous probability distribution defined by its mean and standard deviation.

Normal distribution explained

About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three — the figures examiners expect you to quote.
Standardising with z=x−μσz = \frac{x - \mu}{\sigma} converts any normal variable to the standard normal, which is what calculator functions and tables use.

Key formula

X∼N(μ,σ2),z=x−μσX \sim N(\mu, \sigma^{2}), \quad z = \dfrac{x - \mu}{\sigma}

Worked example

If X∼N(50,42)X \sim N(50, 4^{2}) then P(X<58)=P(z<2)≈0.977P(X < 58) = P(z < 2) \approx 0.977.

Examiner tip

Sketch the bell curve and shade the region you want before touching the calculator. It stops you finding the complement by mistake.

Normal distribution: common questions

When can I model data as normal?
When the data is continuous, roughly symmetric about a single peak, and extreme values are rare — check with a histogram or box plot.
What is a z-score?
It is the number of standard deviations a value lies from the mean, which allows comparison across different normal distributions.

Related terms

Standard deviation

Standard deviation measures the average spread of data values about the mean — a small value means tightly clustered data.

Probability

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain).

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