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.

Retail store buyers

Merchandise buyers are critically important to the success or failure of retail ventures, both through negotiations over pricing and decisions about the types of product that will be stocked. Because of high turnover of staff in this area (estimated to be over 25 percent per year), research was carried out to determine the factors related to turnover. The researchers contacted merchandise buyers in several stores and asked them to complete a questionnaire. After 9 months, the researchers determined whether the buyers were in the same job — a classification into 'leavers' and 'stayers'.

The two histograms below describe the job satisfaction ratings of the two groups at the time of the questionnaire, on a scale of 0 (worst) to 30 (best).

The area of the histogram for the stayers is nearly five times that of the movers (since there were nearly five the number of stayers as movers) 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 the distribution of job satisfaction is higher for stayers than for leavers, even though the two distributions overlap considerably.

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...