AP Stats 3.1: Estimators

What topic 3.1 covers

Topic 3.1 introduces estimators, the tools you use to guess an unknown population value from sample data. Almost every method in the rest of Unit 3 starts here, so the idea is worth getting right the first time.

A statistic computed from a sample, such as the sample proportion, is a point estimator of the matching population parameter. The single number it produces for one sample is the point estimate. For a population proportion $p$, the sample proportion $\hat{p}$ (read "p-hat") is the point estimator you use throughout the unit.

What makes an estimator unbiased

An estimator is unbiased when, on average over all possible samples of the same size, it does not systematically overestimate or underestimate the parameter. Imagine taking every possible sample and computing the estimate each time. Those estimates form a sampling distribution, and the estimator is unbiased when the mean of that distribution equals the true parameter.

The sample proportion is unbiased because the mean of its sampling distribution is exactly $p$, a fact you build on in topic 3.2. Being unbiased does not mean a single estimate equals $p$; it means the estimates are centered on $p$ rather than leaning high or low.

Bias is not the same as variability

Bias describes where the estimates center, while variability describes how spread out they are. A good estimator is both unbiased and low in variability, so its estimates cluster tightly around the true value. Increasing the sample size does not remove bias, but it does shrink variability, which is why larger random samples give more reliable estimates. Bias in real studies usually comes from a flawed sampling method, such as voluntary response, rather than from the arithmetic of the formula itself.

Computing a point estimate

To estimate a population proportion, count the successes $x$ in a sample of size $n$ and divide: $\hat{p} = \dfrac{x}{n}$. This one number is your best single guess for $p$, but it varies from sample to sample, which is why later topics attach a margin of error or a p-value. A point estimate on its own says nothing about how far it might sit from the truth, so it is always a starting point rather than a final answer.

Topic 3.2 studies how $\hat{p}$ varies, topics 3.3 and 3.4 wrap a confidence interval around it, and topics 3.5 through 3.7 test a claim about $p$. Recognizing $\hat{p}$ as an unbiased estimator is the foundation for trusting all of those methods.