## 17.2 Distributions: An example

To begin,
consider the heights of *all* Australian adult males.
Clearly,
the height of *all* Australian adult males is unknown:
no-one has ever, or could ever realistically,
measure the height of all Australian adult males.
The
Australian Bureau of Statistics (ABS),
however,
takes samples of Australians to compute estimates of the
heights
and other measurements.

A model could be **assumed** for the heights of all Australian adult males.
This is a *theoretical* idea
that might be a useful description of the heights of Australian adult males in the *population*.
Suppose a *model* for the heights of Australian adult males
is adopted that has:

- a symmetric distribution,
- with a
*mean height*of 175 cm, and - a
*standard deviation*of 7 cm.

Then,
the *distribution* of the heights of Australian adult males may look like
Fig. 17.1.
That is,
most Australian adult males are between about 168 and 182cm,
and very few are taller than 196cm or shorter than 154cm.

This model represents an idealised, or *assumed*,
picture of the histogram of the heights
of all Australian adult males in the *population*.
If this model is a accurate,
the distribution of heights in any *sample*,
may be shaped a bit like this,
but *sampling variation* will exist.

Any one sample will look a bit different than this model, but this model captures the general feel of the histogram from many of these samples. For example, see the animation below, where many samples of \(n=100\) men are taken.

The model of heights
has approximately a bell-shape:
that is,
most values are near the average height,
but a small number of men are very tall or very short.
A bell-shaped distribution is formally called a
**normal distribution**
or a
**normal model**.
A normal distribution is a way of *modelling* the population.

A *model* is a theoretical or ideal concept.
In the same way that a model skeleton isn’t 100% accurate (wire joins?)
and certainly not exactly like *your* skeleton,
it suitably approximates reality.
None of us probably have a skeleton *exactly* like the model,
but the model is still useful and helpful.

Likewise,
no variable has *exactly* a normal distribution,
but the model is still useful and helpful.
The model is a *theoretical* way of describing
the distribution in the population.