Sample statistic bias worked example | Sampling distributions | AP Statistics | Khan Academy

We're told Alejandro was curious if sample median was an unbiased estimator of population median. He placed ping pong balls numbered from 0 to 32, so I guess that would be what, 33 ping pong balls in a drum and mixed them well.

Note that the median of the population is 16, right? The median number, of course, yes in that population is 16. He then took a random sample of five balls and calculated the median of the sample.

So we have this population of balls; he takes a sample. We know the population parameter; we know that the population median is 16. But then he starts taking a sample of five balls, so n equals five, and he calculates a sample median.

Then he replaced the balls and repeated this process for a total of 50 trials. His results are summarized in the dot plot below where each dot represents the sample median from a sample of five balls.

So he does this; he takes these five balls, puts them back in, then he does it again, then he does it again. Every time he calculates the sample median for that sample and he plots that on the dot plot. So, and he'll do this for 50 samples, and each dot here represents that sample statistic.

So it shows that four times we got a sample median in four of those 50 samples; we got a sample median of 20. In five of those sample medians, we got a sample median of 10.

And so what he ends up creating with these dots is really an approximation of the sampling distribution of the sample medians. Now, to judge whether it is a biased or unbiased estimator for the population median, well, actually pause the video, see if you can figure that out.

All right, now let's do this together. Now to judge it, let's think about where the true population parameter is—the population median. It's 16. We know that, and so that is right over here, the true population parameter.

So if we were dealing with a biased estimator for the population parameter, then as we get our approximation of the sampling distribution, we would expect it to be somewhat skewed.

So for example, if this approximation of the sampling distribution looks something like that, then we would say, "Okay, that looks like a biased estimator." Or if it was looking something like that, we'd say, "Okay, that looks like a biased estimator."

But if this approximation for our sampling distribution that Alejandro is constructing shows that roughly the same proportion of the sample statistics came out below as came out above the true parameter, and it doesn't have to be exact, but it seems roughly the case, this seems pretty unbiased.

So to answer the question, based on these results, it does appear that the sample median is an unbiased estimator of the population median.