Estimating a Population from a Sample

From Sample to Population

The central problem of inference

We can almost never survey an entire population: too many people, too expensive, too slow. So we observe a sample (a randomly drawn subset) and try to deduce something about the complete population. That is the whole project of inferential statistics: moving from the particular (what we measured) to the general (what we want to know), while keeping control of the uncertainty that this leap involves.

Parameter and estimator

A quantity that describes the population is called a parameter: for example the true proportion p of voters in favor, or the true mean μ of a height. We do not know it. What we compute on the sample is an estimator: the observed proportion (often written p hat) or the sample mean (x bar). The estimator is an approximation of the parameter.

Why the randomness of the sample matters

Population (true proportion p = 0.50, unknown)
        |
        |  we draw a sample at random
        v
Sample 1 -> p_hat = 0.47
Sample 2 -> p_hat = 0.52
Sample 3 -> p_hat = 0.49

(each sample gives a slightly different value: this is sampling fluctuation)

Two different samples give different estimates: this is sampling fluctuation. The estimator is therefore not a fixed value but a random variable, with its own spread.

The larger the sample, the more precise

A fundamental result: the spread of the estimator decreases as the sample size n increases, roughly like 1 over the square root of n. To halve the uncertainty, you therefore have to multiply the sample size by four. That is why large polls are more reliable, but also why gaining precision becomes more and more expensive.

Classic pitfall