Statistical inference is the process whereby we seek to draw conclusions from samples that apply to a finite or infinite population from which these samples are assumed to have been drawn (see also, the closely related topic of estimation). As such inferential statistics is essentially analytical, requiring a representative (random) sample from a larger population, together with some model of how data obtained from such samples relate to the population as a whole. This approach is distinct from descriptive statistics, which seek to measure and report (numerically or graphically) key features of the data.
Statistical inference typically leads to some form of proposition regarding the population, for example that the population mean has an estimated value of x with a confidence or belief interval of [a,b]. In the frequentist view of statistics it is imagined that the particular sample selected is one of many possible such samples that could be obtained by repeated random sampling (or simulated repeated sampling). The particular sample in question can thus be identified with respect to its likelihood of occurring and from this inferences can be made regarding the population as a whole (for example, obtaining a confidence interval for the mean). In Bayesian inference samples are regarded as evidence that affects the degree of belief that is held in a given hypothesis. As more evidence is gathered, i.e. as more information becomes available and as more samples are taken, the degree of belief in the hypothesis should be amended. In some instances, particularly in many areas of medical research, increasing emphasis has been placed on this latter approach and some authors regard this as the only meaningful form of inference (see further, our earlier discussion of medical testing and the impact of false positives). Others accept its use in some instances, but are wary of possible bias in the initial hypothesis and the risks this then poses.