The normal distribution · Year 13

In Year 13 you learn: the normal distribution.

Many measurements, such as heights, masses and exam marks, follow a symmetrical bell-shaped curve called the normal distribution. It is described by its mean μ and standard deviation σ, written X ~ N(μ, σ²). Probabilities are areas under the curve.

Key facts

• The curve is symmetrical about the mean, so mean = median = mode.
• The total area under the curve is 1.
• About 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ.
• P(X > x) = 1 − P(X < x) and P(a < X < b) = P(X < b) − P(X < a).
• For a continuous variable, P(X = x) = 0. So < and ≤ give the same answer.