## 18.1 Introduction

The last three chapters introduced tools to apply the decision-making process (Sect. 15.4) used in research:

1. Make an assumption about the population parameter.
2. Based on this assumption, describe what values the sample statistic might reasonably be expected from all possible samples.
3. Observe the sample data, and see if it seems consistent with the expectation, or if it contradicts the expectation.

One key observation is that, under certain conditions, the variation of many sample statistics (such as the sample mean, etc.) from sample to sample can be described approximately by a normal distribution. As a result, the expected behaviour of these statistics can be described, so we know what to expect from the sample statistic.

This has been alluded to before. In Sect. 15.4, the sample proportion of red cards in a sample of 15 varied from hand to hand, and was approximately distributed as a normal distribution. This is no accident: Many sample statistics vary from sample to sample with an approximate normal distribution if certain conditions are met. This is called the Central Limit Theorem.

A sampling distribution describes the distribution of the sample statistic: How the value of the sample statistic varies from sample to sample for many samples. The sampling distribution here is a normal distribution.

Definition 18.1 (Sampling distribution) A sampling distribution is the distribution of some sample statistic, showing how its value varies from sample to sample.