Exploring data: graphs and numerical summaries
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# 4.1 Histograms

It is a fundamental principle in modern practical data analysis that all investigations should begin, wherever possible, with one or more suitable diagrams of the data. Such displays should certainly show overall patterns or trends, and should also be capable of isolating unexpected features that might otherwise be missed. The histogram is a commonly-used display, which is useful for identifying characteristics of a data set. To illustrate its use, we return to the data set on infants with SIRDS that we looked at briefly in Section 1.4.

The birth weights of 50 infants with severe idiopathic respiratory distress syndrome were given in Table 3. The list of weights is in itself not very informative, partly because there are so many weights listed. Suppose, however, that the weights are grouped as shown in Table 8.

## Table 8 Birth weights (kg)

Group | Birth weight (kg) | Frequency
--- | --- | ---
1 | 1.0–1.2 | 6
2 | 1.2–1.4 | 6
3 | 1.4–1.6 | 4
4 | 1.6–1.8 | 8
5 | 1.8–2.0 | 4
6 | 2.0–2.2 | 3
7 | 2.2–2.4 | 4
8 | 2.4–2.6 | 6
| 2.6–2.8 | 3
10 | 2.8–3.0 | 2
11 | 3.0–3.2 | 2
12 | 3.2–3.4 | 0
13 | 3.4–3.6 | 1
14 | 3.6–3.8 | 1

Such a table is called a grouped frequency table. Each listed frequency gives the number of individuals falling into a particular group: for instance, there were six children with birth weights between 1.0 and 1.2 kilograms. It may occur to you that there is an ambiguity over borderlines, or cutpoints, between the groups. Into which group, for example, should a value of 2.2 go? Should it be included in Group 6 or Group 7? Providing you are consistent with your rule over such borderlines, it really does not matter.

In fact, among the 50 infants there were two with a recorded birth weight of 2.2 kg and both have been allocated to Group 7. The infant weighing 2.4 kg has been allocated to Group 8. The rule followed here was that borderline cases were allocated to the higher of the two possible groups.

With the data structured like this, certain characteristics can be seen even though some information has been lost. There seems to be an indication that there are two groupings divided somewhere around 2 kg or, perhaps, three groupings divided somewhere around 1.5 kg and 2 kg. But the pattern is far from clear and needs a helpful picture, such as a bar chart. The categories are ordered, and notice also that the groups are contiguous (1.0–1.2, 1.2–1.4, and so on). This reflects the fact that here the variable of interest (birth weight) is not a count but a measurement.

The distinction between ‘counting’ and ‘measuring’ is quite an important one. In later units we shall be concerned with formulating different models to express the sort of variation that occurs in different sampling contexts, and it matters that the model should be appropriate to the type of data. Data arising from measurements (height, weight, temperature, and so on) are called continuous data. Those arising from counts (family size, hospital admissions, nuclear power stations) are called discrete.

In this situation, where we have a grouped frequency table of continuous data, the bars of the bar chart are drawn without gaps between them, as in Figure 10.

This kind of bar chart, of continuous data which have been put into a limited number of distinct groups or classes, is called a histogram. In this example, the 50 data items were allocated to groups of width 0.2 kg: there were 14 groups. The classification was quite arbitrary. If the group classifications had been narrower, there would have been more groups each containing fewer observations; if the classifications had been wider, there would have been fewer groups with more observations in each group. The question of an optimal classification is an interesting one, and surprisingly complex.

How many groups should you choose for a histogram? If you choose too many, the display will be too fragmented to show an overall shape. But if you choose too few, you will not have a picture of the shape: too much of the information in the data will be lost.

Figure 10 Birth weights (kg) of infants with SIRDS
Figure 10

When these data were introduced in section 1.4, the questions posed were as follows. Do the children split into two identifiable groups? And is it possible to relate the chances of survival to birth weight? We are not, as yet, in a position to answer these questions, but we can see that the birth weights might split into two or even three ‘clumps’. On the other hand, can we be sure that this is no more than a consequence of the way in which the borderlines for the groups were chosen? Suppose, for example, we had decided to make the intervals of width 0.3 kg instead of 0.2 kg. We would have had fewer groups, with Group 1 containing birth weights from 1.0 to 1.3 kg, Group 2 containing birth weights from 1.3 to 1.6 kg, and so on, producing the histogram in Figure 11.

Figure 11 Birth weights, 0.3 kg interval widths
Figure 11

The histogram in Figure 11 looks quite different to that in Figure 10, but then this is not surprising as the whole display has been compressed into fewer bars. The basic shape remains similar, so you might be tempted to conclude that the choice of grouping does not really matter. But suppose we retain groupings of width 0.3 kg and choose a different starting point. Suppose we make Group 1 go from 0.8 to 1.1kg, Group 2 from 1.1 to 1.4kg, and so on. The resulting histogram is shown in Figure 12(a). In Figure 12(b), the groups again have width 0.3kg, but this time the first group starts at 0.9 kg.

Figure 12 Birth weights, 0.3 kg interval widths

## Activity 5: Comparing histograms

What information do the histograms in Figures 10, 11 and 12 give about the possibility that the children are split into two (or more) identifiable groups on the basis of birth weight?

## Discussion

You might have felt that only Figures 10 and 12(b) give a really clear indication that the data are split into two ‘clumps’. Figures 10, 11 and 12(a) all give, to varying degrees, the impression that there is perhaps an identifiable group of babies with particularly low birth weights.

What you have seen in Figures 10 to 12 is a series of visual displays of a data set which warn you against trying to reach firm conclusions from histograms. It is important to realise that histograms often produce only a vague impression of the data – nothing more. One of the problems here is that we have only 50 data values, which is not really enough for a clear pattern to be evident. However, the histograms all convey one very important message: the data do not appear in a single, concentrated clump. Clearly it is a good idea to look at the way frequencies of data, such as the birth weights, are distributed and, given that a statistical computer package will quickly produce a histogram for you, comparatively little effort is required. This makes the histogram a valuable analytic tool and, in spite of some disadvantages, you will find that you use it a great deal.

It is, of course, quite feasible to produce grouped frequency tables and draw histograms by hand. However, the process can be very long-winded, and in practice statisticians almost always use a computer to produce them.

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