Title: Probability, populations and samples
Course: UGQuants (SSPS) > Section: Probability

Probability, populations and samples

The normal curve has all sorts of useful applications. In particular the properties of the normal curve allow us to understand the relationship between samples and populations. The normal curve allows us to precisely calculate the likelihood of  drawing small samples that have characteristics very different to the larger population from which they have been drawn. This is extremely helpful, because it allows us to investigate society by looking only at very small parts of it. Instead of having to study everyone that we are interested in (at best an extremely lengthy, costly and laborious process, but more often practically impossible), we can, as long as we stick carefully to some simple rules, draw conclusions about the behaviour of millions of people by studying samples of just a few hundred. This process is known as inferential statistics (since we infer things about populations from studying samples drawn from them).

More coin tossing …

Imagine we carry out a very large number of coin tosses (say fifty million). We know that (assuming the coin is unbiased) the share of 'heads' in all these tosses will be very, very close to 0.5.

Now think of any individual series of 100 tosses, extracted from these fifty million tosses. Most of the selections will have a share of heads of 0.5 or very close to it (0.49, 0.51, 0.48, 0.52 and so on), since we know that this is most probable.

Now think of the original fifty million tosses as a population, and of each series of 100 tosses as a sample randomly drawn from the population. We can think of the proportion of ‘heads’ as a characteristic (or parameter) of the population that we wish to measure. If we take a 'sample' of 100 tosses, the most likely result is exactly the same as the one we would obtain in the population: 0.5. However, whilst this is the most likely result, it is certainly not the only result we might get. We can use the characteristics of  normal curve to calculate exactly what the chances are of getting a result a certain distance away from that in the population as a whole (here, e.g. a proportion of 0.48 heads instead of 0.5 heads).

We’ll return to that shortly. For now consider the logic the other way around. So far we have been considering a case where we know the value of the population characteristic. However in most real life situations we won’t know this. So consider a situation where we have a result from a sample but don’t know what the result is in the broader population. Just as before, we can calculate the probability of getting a result in any one sample which is a certain distance away from the true (but unknown) population parameter. Now we can use this to calculate the probability that the (‘real’) population value lies within a specified distance above or below the value we found in our sample.

This is an invaluable procedure. It means that instead of trying to measure a characteristic in a population - time consuming and expensive - we can measure a characteristic in a sample drawn from that population (relatively quick and relatively cheap), and then calculate the probability that the population characteristic lies within a certain distance from our sample measurement.

We can do this as long as every of the 'original' fifty million tosses has an absolutely equal chance of being selected in our sample of 100 tosses. If this were not the case, we could not calculate the probability, because we would not be able to correctly predict all the possible sequences of tosses on the basis of their equal likelihood of occurrence. This is the basis of random sampling.