16.03 Normal distributions

When we have a set of univariate data, it often happens that most of the measurements will be clustered close to the mean value with the density of the observations falling off with distance away from the mean. Results of this kind displayed in a histogram show a central peak with columns of decreasing height on each side of the mean.

A data set that is symmetrical and bell-shaped about the mean, and which meets certain more precise shape requirements, is said to have a normal distribution. 

The shape of the normal distribution will depend on its mean ($\mu$μ) and standard deviation $(\sigma)$(σ). The mean is where the graph peaks

The lower the standard deviation, the more clustered the scores will be around the mean. Conversely, the higher the standard deviation, the more spread the scores will be.

For example, this distribution curve comes from a data set that has a very small standard deviation, $\sigma=0.2$σ=0.2,  and hence is clustered tightly around the mean:

whereas this normal distribution has a larger standard deviation, $\sigma=0.9$σ=0.9,  and hence is quite spread.

 

To see how a change in the mean and standard deviation affects the shape of the normal distribution curve, check out this graph. By moving the sliders around you can vary the mean and standard deviation, and see how the bell curve changes as a result.

 

Practice questions

QUESTION 1

QUESTION 2

QUESTION 3