The normal distribution

The normal model N(μ, σ²) for continuous data — its symmetric bell shape, mean = median = mode = μ, spread set by σ, and standardising any normal variable to the standard normal Z = (X − μ)/σ.

Statistics · Statistical distributions

The normal distribution

Why it works

Heights, masses, exam marks, measurement errors — vast amounts of continuous, natural data pile up in the same shape: a symmetric bell curve. The normal distribution

N(\mu, \sigma^2)

is the model for it, fixed by just two numbers: the mean $\mu$ (where the peak sits) and the standard deviation $\sigma$ (how spread out it is).Its key features:

Symmetric about $\mu$ , so the mean, median and mode are all $\mu$ . Exactly

half the area lies on each side:

P(X < \mu) = 0.5

The total area under the curve is $1$ (it's a probability density), and a

probability is an area under the curve between two values.

The spread is governed by $\sigma$ : roughly $68\%$ of the data lies within $1$

standard deviation of the mean, $95\%$ within

2

, and $99.7\%$ within

3

. The curve has points of inflection at

\mu \pm \sigma

.

Standardising. Every normal distribution is the same bell, just shifted and stretched — so any

X \sim N(\mu, \sigma^2)

can be converted to the one standard normal distribution

Z \sim N(0, 1)

by measuring how many standard deviations

X

is from the mean:

Z = \frac{X - \mu}{\sigma}.

Z

-value (or

z

-score) of

1.5

means "

1.5

standard deviations above the mean". Standardising is what lets you compare values from different normal distributions and look probabilities up in one place.