Estimators

Estimators and estimates

You almost never know a parameter. What you have is a sample, and you use a statistic from that sample to estimate the parameter. A statistic used this way is a point estimator. The number it produces from your data is the point estimate.

The sample proportion $\hat{p}$ is the point estimator of the population proportion $p$. If 84 of 200 randomly chosen voters favor a new library tax, the point estimate of the proportion of all voters who favor it is $\hat{p}=\frac{84}{200}=0.42$. It is called a point estimate because it is a single number. In 3.3 Constructing a Confidence Interval for a Population Proportion you will put a margin of error around it.

Every parameter you meet in this course has a matching statistic, and the notation from 1.2 Variables keeps the pairs straight.

Unbiased means right on average

Nearly every estimate misses the parameter by a little, so missing is not what makes an estimator bad. What matters is whether the misses lean one way. An estimator is unbiased if, on average, its value does not underestimate or overestimate the parameter. An estimator whose values tend to land on one side of the parameter is biased.

To judge this, imagine every possible random sample of the same size, with the estimate computed from each one. If those values center on the parameter, the estimator is unbiased. You rarely see every sample, but a simulation gets close: start with a population whose parameter you know, draw many random samples, and look at where the estimates pile up.

Here is that simulation for the ride-share company. The computer drew 500 random samples of 30 rides and computed two estimators from each sample.

Reading the evidence

The sample mean is unbiased. Its 500 values average 5.6 minutes, the same as the mean of all 12,000 rides. The sample maximum is biased. Its values center around 13.7 minutes when the truth is 31.0, and none of them can land above the truth.

The sample proportion behaves like the sample mean. When the data come from a random sample, the values of $\hat{p}$ center on $p$. That is why $\hat{p}$ is the estimator behind the intervals and tests for proportions in this unit, and 3.2 Sampling Distributions for Sample Proportions shows exactly where its values center and how far they spread.

Bias can come from how the sample was chosen

A good formula is not enough on its own. You can count on the sample proportion being unbiased only when the sample is random. Suppose a radio host asks listeners to call in and uses the proportion of callers who oppose a new stadium to estimate the proportion of all listeners who oppose it. Listeners who are angry about the stadium are the ones most likely to pick up the phone, so sample after sample this estimate would come out too high. That is the systematic error from 1.12 Potential Problems with Sampling, seen from the estimator's side.