Biased & Unbiased Estimators (College Board AP® Statistics): Study Guide

Biased & unbiased estimators

What is an estimator?

• So far, all the population parameters, , or , have been known

  • They are given in the question

    • e.g. the weights of cats in Arizona follow a normal distribution with mean 4.5 kg and standard deviation 0.6 kg

• However, population parameters are often unknown

  • e.g. no one really knows the mean weight of all cats in Arizona

    • and knowing their standard deviation is even more unlikely!

• When this happens, samples can be used to find estimates of population parameters

  • e.g. a sample 50 cats from Arizona gives

    • a sample mean of 4.2 kg (an estimate of )

    • a sample standard deviation is 0.7 kg (an estimate of )

• The type of sample statistic being used to estimate the population parameter is called an estimator

  • e.g. the sample mean is an estimator of the population mean

What is an unbiased estimator?

• It is not always clear if an estimator is a good predictor of a population parameter

  • e.g. does the mode of a sample provide a good estimate for the mode of the population?

    • It turns out, no!

• To know if an estimator is a good predictor

  • all possible estimates from all possible samples of size must be generated

  • then checked to see if, on average, these estimates are centered around the value of the population parameter that is being estimated

    • i.e. the mean of the sampling distribution must be inspected

• An estimator is said to be unbiased if

  • the mean of its sampling distribution is equal to the population parameter being estimated

    • If this is not the case, the estimator is biased

What are the three unbiased estimators that I need to know?

• You need to know that

  • the sample mean, , is an unbiased estimator of the population mean,

    • Its sampling distribution has a mean of

  • the sample proportion, , is an unbiased estimator of the population proportion,

    • Its sampling distribution has a mean of

  • the sample standard deviation, , is an unbiased estimator of the population standard deviation,

• You should not assume any other estimators are unbiased

  • e.g. the sample mode / median are both biased estimators of population mode / median

    • They consistently over or underestimate their population parameter

How does sample size affect an unbiased estimator?

• Varying the sample size, , does not affect the mean of an unbiased estimator

  • The mean of its sampling distribution is always equal to the population parameter being estimated

    • regardless of

• However, varying the sample size, , can have an affect on the standard deviation of an unbiased estimator

  • The best unbiased estimators are ones for which their standard deviation decreases as the sample size, , increases

    • This means larger samples give less variability in estimates

    • Estimates will tend to be closer to the true population parameter

• For example, the sample mean is an unbiased estimator with this desirable property

  • Its sampling distribution has a mean of and a standard deviation of

    • and decreases as increases

    • which 'squashes' the shape of the distribution closer to

    • so sample means from larger samples are usually closer to the population mean