The normal distribution
A continuous random variable X that follows a normal distribution is written X ~ N(μ, σ²), where μ is the mean and σ is the standard deviation. The curve is a symmetric bell shape about μ, and the total area under it equals 1.
Standardising
To find probabilities we convert X to the standard normal variable Z ~ N(0, 1) using the Z-score. Then we read areas from the standard normal table, where P(Z > 0) = 0.5.
Key formula
Z = (X − μ)/σ. Common values: P(Z > 1) = 0.1587, P(Z > 2) = 0.0228, P(Z < 1) = 0.8413, P(−1 < Z < 1) = 0.6826.
Worked example
Worked example
X ~ N(50, 5²). Find P(X > 55). Standardise: Z = (55 − 50)/5 = 1. So P(X > 55) = P(Z > 1) = 0.1587.
For X ~ N(100, 10²), the value X = 120 gives Z = (120 − 100)/10 = 2, so P(X > 120) = P(Z > 2) = 0.0228.
Remember
- The curve is symmetric: P(Z > a) = P(Z < −a).
- σ is the standard deviation, so σ² is the variance.
- Always standardise before reading the table.