The normal distribution

Many measurements, like heights or exam marks, follow a bell-shaped normal distribution, symmetric about the mean. The standard deviation sets how wide the bell is.

Key facts

  • X ~ N(μ, σ²): mean μ, standard deviation σ.
  • About 68% lie within 1σ of the mean, 95% within 2σ, 99.7% within 3σ.
  • Standardise with z = (x − μ)/σ.
  • Probabilities are areas under the curve.

Worked examples

Example 1

Heights are N(165, 8²). Roughly what percentage are between 149 cm and 181 cm?

  1. 149 = 165 − 2 × 8 and 181 = 165 + 2 × 8.
  2. Within 2σ: about 95%.

Answer: 95%

Example 2

X ~ N(50, 10²). Find P(X < 65).

  1. z = (65 − 50)/10 = 1.5
  2. P(Z < 1.5) = 0.9332

Answer: 0.9332