Who Is Sir Francis Galton?

In the previous article we saw how Francis Galton exploited the normal distribution to give a visual as well as a mathematical insight into measurements of naturally varying characters. We will now look at how Galton used a variety of representations to make even clearer the consistent patterns of natural variation.

Galton preferred to work with something called the “Ogive”. This is partly because the ogive representation reflects a physical reality that the frequency distribution (Figure 1) lacks.

The values on the horizontal axis represent the range of heights of Englishmen in the Victorian age. The vertical axis is a relative scale indicating the proportion of Englishmen from the total population who have a particular height within the range. We see that the average height of an Englishmen is 5'6” and the range of heights is between 5'3” and 5'9”.

An Englishmen less than 5'3” would be considered unusually short, and one taller than 5'9” would be unusually tall. You would expect to see only about one in a thousand Englishmen above or below the normal range. Information of this type can be very helpful. Let's imagine that you are in the business of manufacturing frock coats. You would like to know how many to produce of what sizes. Analysing the normal distribution in only a slightly more rigorous way than we have just done, you could easily make the necessary decisions.

Let's look at Figure 2. Here we see a small sample of the population of English men during the Victorian age. 36 English men are represented here. This sample is representative of the entire population of English men. Each interval in which we count the number of men is one half inch wide.

Figure 3 is the true Galton Ogive. Here, the vertical scale is the height, and the horizontal scale is the running count of the population. In other words, the shortest man is at the left, and the tallest man is at the right, with the rest between. This particular graph is scaled to represent the actual height, and of course the total width of the graph is squeezed so as to get all 36 men on a single page.

Here we see the representation of the scene described by Galton earlier when he was extolling the wonders of the “Law of Error”. Here we see a simple, logical, and even feasible arrangement of our Englishmen. They are arranged so that all have found their “Niche”, as Galton put it. Every man has a unique place. If we accept the fact that, in nature, no two things are exactly the same, then such a result is expected. What is perhaps not intuitive, is the shape of a line drawn just above the heads of the sample population. We see a curve, concave to the right as we begin at the left with the shortest man, a curve almost straight as we move through the centre, and then another curve concave to the left as we reach the tallest man. The curve is symmetrical about the average Englishman. I was surprised, when I first learned it, that Francis Galton preferred to use the “Ogive” approach, whenever he was working with data and evaluating its distribution. After some experience, I have also used the Ogive to help me gain a better understanding of some mathematical expressions, which I otherwise might have had to accept on faith. Galton used the Ogive because he always sought the approach that most closely represented the real world, rather than some mathematical abstraction.

In Figure 4, another example of a “real world” representation, Galton asks us to imagine a board on which the height of many Englishmen has been recorded with a mark. We see again the normal distribution in a different way, but of considerable interest is the fact that Galton speaks in terms of “parts per million”, a terminology which has been “reinvented” in today's world of statistical methods. Another example of how far ahead of his time was Francis Galton.

Francis Galton was greatly concerned about the difficult problems facing human society. Population growth, poverty, illness, even the “noise” created by the media of his day; all these things spurred Galton to make a contribution.

Now Francis Galton thought that the ability of mankind to solve problems of its own making was not adequate, even during the Victorian age, when many English people thought that solutions for almost any problem were possible because of the major advances that were being made in science.

It is fascinating to imagine what Sir Francis would say if he could look at today's situation. A small sample of the types of articles he might examine are shown here. They are taken from a large, thick notebook which some friends have called the “Doomsday Book”. And also, what might he think of today's media “noise”?

The pioneers in statistical thinking found numerous examples of populations of characteristics which followed the “Law of Error”, now known as the normal distribution. These populations tended to be made up of physical characteristics which could be measured directly. The height of Englishmen, as we have seen, is a good example.

Francis Galton believed that characteristics, other than physical, such as ability, were also normally distributed within a population. Thus the data in Table 1. This is from Galton's book “Hereditary Genius”, the 1892 edition. Here Galton has arranged the data, not as a histogram, but in tabular form. Because of his skill in arranging the data “geometrically”, we can easily see the behaviour of a normal distribution, even though no curve has been drawn. The same data, arranged in other formats, would not convey the same information. This talent for arranging data in order to obtain visual information was to enable Galton to discover what is now known as Regression Analysis, a method used widely today.

In Figure 5 we see what Galton's data might look like if presented as a frequency distribution. The classes of human ability are along the horizontal axis, while the number in each class is shown vertically as a relative proportion. (No exact numbers should be inferred.)

In Table 1, we are told that class F or f was present in the population as one person in 4300 of the total population. In today's terminology, this would put these persons beyond three standard deviations of the total population.

I want to use this fact to illustrate something about Galton that we might otherwise miss; he was a masterful teacher. He was always using analogies or illustrations using materials with which we are familiar, in very creative ways. For example, he tells us that if you wish to get an idea of how rare one person in 4300 is, then go out on a clear night and look up at the stars. Find the brightest star, and because there are close to 4000 stars in the full sky, you have a real picture of the rarity of the class F/f person.