Chapter 1 Sampling Distribution: How Different Could My Sample Have Been?

Key concepts: inferential statistics, generalization, population, random sample, sample statistic, sampling space, random variable, sampling distribution, probability, probability distribution, discrete probability distribution, expected value/expectation, unbiased estimator, parameter, (downward) biased, representative sample, continuous variable, continuous probability distribution, probability density, (left-hand/right-hand) probability.

Watch this micro lecture on sampling distributions for an overview of the chapter.

Summary

What does our sample tell us about the population from which it was drawn?

Statistical inference is about estimation and null hypothesis testing. We have collected data on a random sample and we want to draw conclusions (make inferences) about the population from which the sample was drawn. From the proportion of yellow candies in our sample bag, for instance, we want to estimate a plausible range of values for the proportion of yellow candies in a factory’s stock (confidence interval). Alternatively, we may want to test the null hypothesis that one fifth of the candies in a factory’s stock is yellow.

The sample does not offer a perfect miniature image of the population. If we would draw another random sample, it would have different characteristics. For instance, it would contain more or fewer yellow candies than the previous sample. To make an informed decision on the confidence interval or null hypothesis, we must compare the characteristic of the sample that we have drawn to the characteristics of the samples that we could have drawn.

The characteristics of the samples that we could have drawn constitute a sampling distribution. Sampling distributions are the central element in estimation and null hypothesis testing. In this chapter, we simulate sampling distributions to understand what they are. Here, simulation means that we let a computer draw many random samples from a population.

In Communication Science, we usually work with samples of human beings, for instance, users of social media, people looking for health information or entertainment, citizens preparing to cast a political vote, an organization’s stakeholders, or samples of media content such as tweets, tv advertisements, or newspaper articles. In the current and two subsequent chapters, however, we avoid the complexities of these samples.

We focus on a very tangible kind of sample, namely a bag of candies, which helps us understand the basic concepts of statistical inference: sampling distributions (the current chapter), probability distributions (Chapter 2), and estimation (Chapter 3). Once we thoroughly understand these concepts, we turn to Communication Science examples.