Having considered the normal distribution, we will now examine the relationship between the shape of any distribution and the proportion of scores that fall within a particular area within it. To explain this idea, it will help to take a concrete example. Imagine a large school with a playground that is exactly rectangular in shape. The teachers at the school ask all of the teenagers at the school to stand in the playground. The smallest teenagers are positioned to the extreme left of the playground, and the taller the teenagers are the further to the right they are asked to stand, with the very tallest teenagers at the extreme right of the playground. The distribution of the teenagers within the playground is shown in Figure 3.3a. Imagine that a chalk line is drawn on the playground, three-quarters of the way from the left to the right as in Figure 3.3b. If a teenager called John is standing directly over the line, what proportion of the teenagers are short, or shorter, than he is?
1111 2 3 4 5 6 7 8 9 10 1 2 3 411 5 6 7 8 9 20111 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 3 4111
THE MECHANICS OF INFERENTIAL STATISTICS
Frequency
Score on the variable under consideration Figure 3.2 The normal distribution
The answer, as you may have worked out for yourself, is that 75 per cent of the teenagers to the left of the chalk line would be as short or shorter than John. If the teenagers gradu-ally increase in height from left to right, and the line bisects a point 75 per cent of the way from left to right of the rectangle, then it follows that 75 per cent would be as short, or shorter than the teenagers to the right of the chalk line. In the same way, if the chalk line was drawn one-quarter of the way from the left of the playground, 25 per cent of the teenagers would be shorter than one standing on this new chalk line.
We can work this out because we know two things. The first is that the teenagers in the playground are distributed in such a way that their height systematically increases from left to right. The second is that we know that if we draw a line 75 per cent of the way to the right of a rectangle as in Figure 3.3b, exactly 75 per cent of the area of the rectangle will be to the left of the line, and exactly 25 per cent will be to the right. If we were not dealing with a regular shape like a rectangle, this would no longer follow. For example, imagine that the playground was laid out not in the regular shape of a rectangle, but in the shape of a bell as shown in Figure 3.4. Let’s say a chalk line was drawn 75 per cent of the way from the left to the right of this bell-shaped playground. What proportion of the teenagers would be as short or shorter than Wendy, a girl standing on this new line? To try to answer this question, it may help you to draw the line on Figure 3.4.
The answer is that you can’t tell what proportion of the teenagers standing in the bell-shaped playground would be as short or shorter than those standing to the right of the new line, and the reason you can’t is that you don’t know the precise percentage of the area of this playground that lies to the right of the chalk line. With the rectangular playground you know that 75 per cent of the playground was to the left of the line, and that therefore 75 per cent of the teenagers would be as short, or shorter, than those standing on the line.
Figure 3.3 (a) A playground in which teenagers are positioned so that the shortest are to the left and the tallest to the right; (b) As (a) but with a chalk line drawn three-quarters of the way from the left to the right
(a)
(b)
But in the case of the bell-shaped playground you don’t know the proportion of the play-ground that is to the left of the chalk line, and so you don’t know the proportion of the teenagers who would be shorter than Wendy, the girl standing on it.
Ok, but let’s say that a teacher asked a mathematician to calculate the proportion of the playground to the right of the chalk line drawn on the bell-shaped playground and that she worked out that 85.3 per cent of the playground area lies to the left of it. Are you now in a position to work out what proportion of teenagers would be shorter than those to the right of the chalk line? The answer this time is, yes. Because you know the teenagers are distrib-uted according to their heights, with taller and taller teenagers standing further and further towards the right, and because you know that 85.3 per cent of the playground lies to the left of the line, you can rightly conclude that 85.3 per cent of the teenagers in the bell-shaped playground are as short or shorter than Wendy.
What has all this got to do with the normal distribution? Well, the answer is, quite a lot. We can talk of the bell-shaped playground as having a distribution and the teenagers standing in it are distributed according to two factors. The first factor is that the taller they are the further to the right they stand. The second factor is the shape of the playground. It is the combination of these two factors that distributes the teenagers in a particular posi-tion from left to right and, in turn, it is this distribuposi-tion of the teenagers which allows us to work out what proportion of them are shorter than those standing to the right of the chalk line. If we take the normal distribution (which has a shape just like the bell-shaped playground) we can use the same principle to work out the proportion of that distribution that is to the right (or the left) of a line drawn through it in exactly the same way that we did with the line drawn through the bell-shaped playground. In fact, the chalk line drawn on the playground in Figure 3.4 is just like a line drawn on a normal distribution.
A knowledge of the area of the playground to the left of the chalk line in Figure 3.4 pro-vided us with information about the proportion of teenagers as short, or shorter, than one standing on the line. In just the same way, a knowledge of the area of a normal distribution 1111
THE MECHANICS OF INFERENTIAL STATISTICS
Figure 3.4 A bell-shaped playground in which the height of teenagers increases from left to right
to the left of any vertical line drawn through it tells us the proportion of scores which are the same or smaller than those on which that line falls. This is shown in Figure 3.5.