Family of normal distributions

One particular type of continuous distribution called a normal distribution is particularly important in statistics. This is actually a family of distributions with similar shapes. A particular normal distribution from this family is specified by the values of two constants called parameters of the distribution, and these are usually denoted by \(\mu\) and \(\sigma\).

\[ X \;\; \sim \; \; \NormalDistn(\mu,\; \sigma^2) \]

Confusingly, the second parameter of the normal distribution may be written as either "\(\sigma\)" or as "\(\sigma^2\)". We will try to avoid this ambiguity in specific normal distributions by including the parameter symbols, such as

\[ X \;\; \sim \; \; \NormalDistn(\mu=12,\; \sigma^2=5) \]

Mathematical formula for pdf

The probability density function of a normal distribution can be written as a mathematical function, but its pdf is relatively complex,

\[ f(x) = \frac 1 {\sqrt{2\pi}\;\sigma} e^{\large -\frac 1{2\sigma^2} (x-\mu)^2 } \quad\quad \text{for } -\infty \lt x \lt +\infty \]

Normal distributions will be described more fully in a later section. At this point, we simply state some of their properties without proof.

Shape of normal distributions

The exact shape of a normal distribution's pdf is determined by the values of the two parameters, \(\mu\) and \(\sigma\), but all normal distribution's share the following properties:

Symmetry
All normal distributions are symmetric with their maximum pdf at \(\mu\).
Mean and median
For any symmetric distribution, both the mean and median are at the point of symmetry. This holds for normal distributions so, in particular,
\[E[X] = \mu\]
Variance
The variance of the normal distribution is
\[\Var(X) = \sigma^2\]
Shape
Other than their centre and spread of values, all normal distributions have pdfs with the same basic shape. Many authors call this "bell-shaped" though the peak at \(\mu\) is sharper than most bells!

We will prove some of these results later.

Normal distribution shape

The diagram below shows the shapes of normal distributions for various different values of the two parameters, \(\mu\) and \(\sigma\).

Drag the two sliders to adjust the two parameter values. Observe that

To illustrate the fact that all normal distributions have the same basic 'shape', we will now repeat the above diagram, but will rescale the axes when the parameters are adjusted.

Note that the basic 'shape' of the pdf is the same for all parameter values.