SUMMARISING AND PRESENTING DATA

NORMAL DISTRIBUTIONS

Modelling

The normal distribution is used to model continuous data with a distribution symmetric about the mean. It depends on two parameters, the mean µ and the standard deviation s. If X follows the normal distribution with mean µ and standard deviation s, this is by convention written X~N(µ, s ²), because the variance s ² is a more fundamental quantity in statistical theory.

Density Function

The density function f(x) is given by

This function is also called the "pdf" for probability density function. All normal pdf's have the same shape. The curve is symmetric about the mean µ = E(X), approaches zero at ± infinity and has points of inflection (changes between 'concave down' and 'concave up') at µ±s.

Note:

Some authors prefer to denote a Normal distribution as N(µ, s) rather than N(µ, s ²). The textbook by Moore and McCabe refers to a Normal distribution according to µ and s. Minitab requires values for µ and s to define a Normal distribution.

In keeping with the more general convention in the statistical literature, this course will assume that the Normal curve is defined according to µ and s ². That is, X~N(µ, s ²).

For example, if X~N(10, 16) then this implies that µ = 10 and s ² = 16 (hence s = 4).

Relative Frequency

The area under a density curve represents relative frequency so any question about relative frequency can be answered by calculating areas under the curve.

Standard Normal Distribution

This is called the standard normal distribution.

Most introductory statistics textbooks, including this, include a table of the standard Normal distribution. It tabulates the area under the curve of the N(0,1) density function.

Tables in other normal tables may give other areas, e.g.