Reason for considering histograms with mixed bin widths

When drawing histograms, you should usually define bins that all have the same width. However this is not essential. Histograms can be drawn with mixed bin widths — indeed, a histogram can be drawn corresponding to any choice of bins.

Although the details will be of little practical importance to you when drawing histograms, the underlying principles will help you to interpret histograms and, in a later section, normal distribution curves.

Combining histogram bins

To retain the correct visual impression, in a histogram with bins of different widths the vertical axis must not be 'frequency'. Instead, the vertical axis must be labeled 'density'. (We will not give a precise definition here.) The guiding principle is...

In a correctly drawn histogram, each value contributes the same area.
 

The histogram below shows the 25 values in the maths test marks data.

Select Wider classes from the pop-up menu to combine the highlighted bins. Observe that each value is still represented by a rectangle with the same area, but of a different shape. The total highlighted area remains the same.

If the height had been 'frequency', the height of the combined bins would have been doubled, incorrectly distorting the visual impact of the bin. The correct height is the average height of the two bins that have been combined.

Select Narrower classes from the pop-up menu and observe that the areas contributed by each value again remain the same.

Why use mixed bin widths?

When all bin widths are the same, frequencies can be written on the vertical axis, simplifying interpretation. If possible, histograms should therefore be drawn with constant bin widths.

However the goal of smoothness can sometimes be attained better by using narrower bins in regions of high density.

The histogram below shows 100 marks (percentages) from a test where most students performed very well — two thirds got marks of 80 or more.

Although the histogram is fairly smooth at the higher marks, it becomes more jagged at marks of 50 or less. However increasing all bin widths to smooth the lower marks leaves the histogram blocky on the right. (Select All classes wide from the pop-up menu.)

Select Mixed classes from the pop-up menu and observe that it gives a smoother picture of the distribution.

Interpreting histograms

The guiding principle for interpreting all histograms is that area equals relative frequency. For example, if half the area of a histogram is above a particular range of values then half of the data are in that range.

The histogram below shows the above skew distribution of 100 marks using bins of mixed widths.

Drag over the three bins that cover the marks between 0 and 69. The area is 17% of the total histogram, so 17% of the values are 69 or below.