Inferential statistics is concerned with applying conclusions to something wider than the observation at hand due to some properties of that observation. For example, if we met a group of people – men and women – and the women earned more than the men, we could infer that women, generally, earned more than men. What inferential statistics allows us to do is make that inference (or not) with some degree of certainty that what we are inferring from the sample applies to a wider group of men and women, i.e. the population of men and women (Swinscow & Campbell 2002).

One of the first concepts to understand in inferential statistics is that of confidence, which means the confidence with which we can make an inference about a population based on a sample (Gardner & Altman 2000). For example, if we wished to study the patients on a medical ward, all of whom were admitted with a diagnosis of either heart disease or another diagnosis, and to find out how many of each there were, then this can be used to illustrate confidence. In a ward with a limited number of beds we can easily check the patient records or ask the nurse in charge and we will have an answer. However, suppose that you wish to know how many people in the general population with a medical diagnosis have either heart disease or another diagnosis, then we cannot simply ask someone to check all their records – there are too many of them to make this practical. If we consider all the patients on our ward to be the population and there are 20 beds on the ward, then we can start by finding out the diagnosis of one patient; if this patient’s diagnosis is not heart disease then we know that the number of patients with heart disease lies between 0 and 19. If we find out the diagnosis of a second patient is heart disease then we know, with a greater degree of confidence, that the number of patients with heart disease lies between one and 18. The more patients whose diagnosis we discover, the nearer we get to the true figure for those with heart disease until we sample them all. In reality, it is very rare to be able to obtain the whole population for any study; we almost always work with samples and inferential statistics is concerned with drawing conclusions about the population based on a sample.

In statistical terms, to say that something is significant has a particular meaning. For instance, if the outcome of a clinical trial is that one drug is statistically significantly better than another, this means there is a difference in their performance and that this is known at a particular level of statistical significance, usually p < 0.05. What we are doing is ascribing a level of probability to the outcome, which means the extent to which that outcome has happened by chance; in the case of p < 0.05 this means that there is a probability of less than one in 20 that the observed result has happened by chance (Hinton 1995). In other words, the result that we are observing is likely to have happened as a result of the phenomenon that we are measuring. In the example of the drug trial above we are stating how likely we think it is that the difference in the performance of one drug over another happened by chance and not as a result of some property of the drug. If the result were significant at p < 0.1 it would mean that there was a 1 in 10 probability that this would have happened by chance and if the result were significant at p < 0.01 it would be one in 100. The application of the concept of statistical significance will be exemplified in the inferential statistical tests described below.

The use of statistical significance enables us to avoid something called type I error and this means that we do not ascribe statistical significance to a result too readily (Watson et al 2006). We will meet another concept, type II error, towards the end of the chapter.

Statistical significance does not, necessarily, indicate that a result is clinically significant. It is possible to obtain statistical significance, for example, in a study of the action of two drugs, where the actual difference between the effectiveness of the drugs is very small and would, in fact, not confer any clinical advantage. Therefore, the actual size of the difference needs to be calculated. In the case of tests of association between two variables – to be described below – the same thing applies and how to do this will be described. Confidence intervals for the difference between two means is also useful in showing how big a difference really is (Gardner & Altman 2000). Clinical judgement is always required in conjunction with statistical tests.

One of the common uses of inferential statistics is in the study of differences between means as exemplified by the study of the effect of drugs in a clinical trial described above. There are three ways in which the difference between means can be tested and these involve:

The statistical test commonly used in such studies is the t-test, so called because statistical significance is obtained by calculating a value which, arbitrarily, has been labelled ‘t’.