Unit 2 Estimation: Why the Sample Mean?

Every empirical study begins with a population and ends with a sample.

We would like to know the average income of every household in a country, the average yield of every farm, or the average height of every adult. Measuring an entire population is almost never possible. Instead we observe a sample and use it to learn about the population it came from.

That raises an immediate question.

If the population average is unknown, how should we estimate it from the sample?

Many answers are possible. We could use the sample mean. We could use the median. We could report the largest observation, or pick one household at random and use its income.

Clearly some of these are better than others. This unit is about how we judge the difference.

The sample mean turns out to have remarkable properties. Across repeated samples it gets the right answer on average. As the sample grows it becomes more precise. And its behaviour can be described mathematically, which lets us say how uncertain any particular estimate is.

Those ideas extend far beyond the sample mean. Every estimator in this book — the sample proportion, a regression slope, a difference between two groups — is judged by exactly the same questions.

By the end of this unit you should be able to ask three things of any estimator.

1. Does it get the right answer on average?
2. How much does it vary from sample to sample?
3. How does that variability change as the sample grows?

The sample mean is simply our first estimator.