Title: The normal curve
Course: UGQuants (SSPS) > Section: Probability

The normal curve or Gaussian distribution

These crucial questions (how many is many, how close is closer?) have a well established answer. The normal curve - or to give it its scientific name: a ‘Gaussian Distribution’ (named after Johann Gauss, 1777 – 1855) - has all kinds of useful applications. We can also think of the normal curve as the shape of chance, able to describe the probability of events in many social and natural processes.

Let us return to our coin tossing analogy. Imagine a class of two hundred students. Each is given a coin to toss it repeatedly, recording the result each time. We know that, on average, or in the long run, each student would eventually obtain an equal number of heads and tails. However, within any limited number of tosses, no matter how large, all kinds of different results are possible. In the first hundred tosses, some students might get exactly 50 heads and 50 tails. Others might get results like, say, 47 tails and 53 heads. Some, though very few, might get an even greater preponderance of heads or tails: say 77 tails and 23 heads. There might even be one student who obtains 100 heads or 100 tails. Now we can approach the analysis of this situation either theoretically or empirically, as we did before:

Theoretically it is possible to calculate the chances of each possible outcome of heads and tails occurring (from all tails and no heads through to no tails and all heads) by working out all the possible sequences of 100 successive toss results that could occur, as we have done for a smaller set of tosses previously.

Empirically it is possible to run an experiment whereby people are invited to toss a coin 100 times and record the results. The greater the number participating in the experiment, the clearer the shape that the distribution of the results takes will be.

In both cases, the results will look like this: a normal curve, or Gaussian distribution. Along the horizontal, or X axis, of the graph lies the different possible numbers of 'heads' obtained in a run of 100 tosses from 0 heads (100 tails) to 100 heads (0 tails). Along the vertical or Y axis of the graph we plot the number of 100-toss series (trials) that obtained this particular result.

We can think of this distribution, or curve, as the 'shape' of chance. Tossing a coin is analogous to pure chance. No matter how much knowledge we were able to gather - about the coin tosser (male or female, tall or short), the characteristics of the coin itself (Spanish doubloon or £2), climatic conditions and so on - we would find it very difficult to predict the result of any individual toss as there are so many different factors that might determine its outcome.

However, the very fact that any individual toss is impossible to predict, make it possible for us to predict very accurately not only the share of tosses resulting in heads or tails in the long run (half and half), but also how likely or unlikely it is that any given number of tosses (e.g. one hundred) will result in a balance of heads and tails different from this. The greater the difference of any particular result from 50/50, the lower the probability of such a result occurring.

If we organised a very large number of 100 toss series, we would find that the most frequent or probable result was to obtain 50 heads and 50 tails. Slightly fewer series would have 49 or 51 heads. Slightly fewer still would obtain 48 or 52 heads, and so on. Very few series indeed will result in as many as 95 or as few as 5 heads.

Normal curves are the result of any process subject to a number of different random influences. Many social and natural phenomena approximate to a normal distribution, because they are the result of a large number of different factors that come together in individual, unpredictable, or random ways. We simulated this by using series of coin tosses, within which the outcome of each individual toss is unpredictable. Another example would be the height of a large group of people of a given sex and age. Many different factors (genetics, diet etc) will have influenced the precise height of any individual.

So far we have thought in terms of how improbable it is to obtain results that are far from our empirically or theoretically established limit. For example in a series of 100 coin tosses the probability of getting 1 head and 99 tails (instead of the 'limit' of 50 heads) is about one in a nonillion, which is about ten billion, billion, billion, or a one followed by thirty zeros. Now turn this logic the other way round. If a pattern like this (e.g. obtaining 1 head and 99 tails) is highly unlikely to occur randomly or by 'chance', then we can pretty safely conclude that when it does occur, it is evidence of something interfering with chance: that it is evidence of a pattern or structure.