Superimposing histograms

To compare the distributions in two groups of values (e.g. measurements for males and females), histograms for the two groups can be superimposed on the same axes.

Colour or shading should be used to help distinguish the two histograms — in ordinary black-and-white histograms it can be difficult to tell which lines belong to which histograms.

Relative frequencies to compare two groups

If the number of values in the two groups differ, when two standard histograms are drawn against a common frequency axis, one histogram can be much smaller than the other. This makes the two distributions much harder to compare.

The solution is to make each rectangle height equal to the proportion in that class instead of the class frequency. These proportions are also called the relative frequencies in the classes.

Use relative frequency histograms to compare groups.

An individual relative frequency histogram has the same shape as the corresponding frequency histogram — each bar height is simply divided by the total number of observations which rescales the histogram height. However using relative frequencies allows us to make more meaningful comparisons between the distributions of different groups.

Weights of newborn calves

As part of a study of newborn calves at the author's university, a researcher recorded their birth weights. The calves were of two breeds, Friesian and Angus.

The area of the histogram of Friesian calf weights is about three times that for the Angus calves (since there were about three times the number of Friesian calves in the study) and this makes comparisons a little harder.

Select Relative frequency from the pop-up menu to scale both histograms to have the same area.

It is easier to compare the relative frequency histograms. For example, we can now determine visually that a higher proportion of Angus calves have weights between 34 and 40 kg than the corresponding proportion of Friesian calves.

Population pyramids

When two groups are to be compared, an alternative to superimposition is to draw their two histograms back-to-back (in a similar way to back-to-back stem and leaf plots).

When used to compare age distributions of males and females in a population, these back-to-back histograms are called population pyramids — a common tool in demography.

The population pyramids below show the age distributions of New Zealanders of European and Maori descent in 1989.

Since the two ethnic groups are of different sizes, relative frequency (in the form of a percentage) is used rather than frequency, permitting easier comparison of the groups. Note that...