## Quiz #4

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E

Considering the situation of observing iid random variables, which of the following statements is true concerning the the use of the sample mean and sample median to estimate the mean of the distribution underlying the observations? (Only one statement is true.)
• [ A ] The sample mean is an unbiased estimator of the distribution mean because the law of large numbers gives us that it converges to the true distribution mean.
• [ B ] The sample mean is an unbiased estimator of the distribution mean because with probability 1 it will equal the true distribution mean.
• [ C ] If the distribution underlying the observations is symmetric, the sample median will always be equal to the sample mean.
• [ D ] If the distribution underlying the observations is skewed, the sample median is an unbiased estimator of the distribution mean since it's expected value is equal to the true distribution mean.
• [ E ] If the distribution underlying the observations is symmetric, the sample median can be superior to the sample mean as an estimator of the true distribution mean.
An unbiased estimator is one for which it's expected value is equal to the estimand. While the sample mean is unbiased for the distribution mean, the reasons given by [ A ] and [ B ] are not correct. As for [ D ], the sample median is unbiased for the mean/median of a symmetric distribution, but for a skewed distribution it is typically a biased estimator of both the mean and the median. (It converges to the median as the sample size increases, but can be a rather poor estimator of the mean, even for large sample sizes, because the distribution mean and distribution median can have very different values if the distribution is skewed.)

Although the distribution mean is equal to the distribution median with a symmetric distribution (ignoring wierd cases for which the distribution mean is undefined --- a technicality which I said little about in class), the sample mean and sample median need not be equal (and usually are not equal). (For example, consider a sample consisting of the 5 values: 1.2, 3.6, 0.7, 2.1, and 2.4. The sample median equals 2.1 while the sample mean equals 2.0. The sample mean will equal the sample median if the sample is symmetric about the sample median, but even if the underlying distribution is symmetric, a sample from it is typically not symmetric.) So [ C ] is false.

[ E ] is true. If the underlying distribution is a symmetric distribution having rather heavy tails, the sample median can be a better than the sample mean as an estimator of the mean/median. (Note: If the distribution has only moderately heavy tails, the sample mean will be superior. But the sample median can be better in more extreme cases.)