Biased and unbiased estimators from sampling distributions examples

Alejandro was curious if sample median was an unbiased estimator of population median. He placed ping-pong balls numbered from zero to 32 in a drum and mixed them well. Note that the median of the population is 16. He then took a random sample of five balls and calculated the median of the sample. 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 this, right over here, is a sampling distribution of the sample median. You have a population of balls. We know a parameter; we know that the median of the population is 16. This, right over here, is the population.

Then, he keeps taking samples of five balls. So that is a five-ball sample right over there. He calculates the statistic, the median of the sample. He calculates the median there, then he does it again, he calculates the median, then he does it again, he calculates the median, and then he does this 50 times. He does this 50 times, and so what you see plotted right over here, you see 50 dots. Every time he took a sample of five balls and calculated the median, we see that is one of the dots here.

For example, four times he calculated a median of 20. Two times he calculated a median of 19. One time he calculated a median of four. Now they ask us, based on these results, does the sample median appear to be a biased or unbiased estimator of the population median? So pause this video and see if you can come up with an answer to that.

In order to be an unbiased estimator, and that is what you want your estimator to be, you want it to be unbiased. The sampling distribution for that statistic has to be evenly distributed about the true parameter for the population. For example, in this situation, the true parameter for the population—they say note that the median of the population is 16. So that's the thing that you're trying to estimate.

This is where the median of the population is. If we look at the sampling distribution, we see that it is roughly balanced to the left and to the right of that true parameter. So I would say that this is looking pretty unbiased because the sampling distribution is evenly balanced to the left and the right of the true median of the population, that true parameter.

If for some reason, the distribution looked something like this, then it would not be unbiased. Or if the distribution looked something like this, if the distribution if the distribution looked like this, if our sampling distribution was like this, this would mean that we're consistently underestimating the true parameter. And if our distribution looked like this, this means that our statistic is overestimating the true parameter.

Let's do another example. The dot plots below show an approximation to the sampling distribution for three different estimators of the same population parameter. If the actual value of the population parameter is 5, which dot plot displays the estimator with both low bias and low variability? Pause this video and see if you can come up with the answer.

So the population parameter that we're trying to estimate is five. So let's show that on each of these distributions, these approximations of the sampling distributions. So well, over here we have our scale, but this should be one, two, three, four, five, six, seven, and one, two, three, four, five, six, seven. And so our population parameter is five, so that is right over there; that is right over there, and this is right over here.

So which of these statistics seems to be biased? Let's just start there. Pause this video. Which of these look biased to you? Well, the one that looks clearly biased is statistic C. When you look at its sampling distribution, it is consistently to the left of our true parameter, so it's consistently underestimating our parameters. So this one right over here is biased.

Now statistic A and statistic B both look reasonably unbiased. Remember, these are approximations of the sampling distribution. If you want to get closer and closer to the sampling distribution, the true sampling distribution, you would keep taking samples and keep calculating the statistic and keep plotting them on this distribution right over here.

But for statistic A, it looks reasonably balanced to the left and the right. It's not perfectly balanced, but it's reasonable. Remember, these are approximations, and statistic B is also reasonably balanced to the left and the right. Now they don't just say which one has a low bias, but they also say low variability.

I would say both statistic A and statistic B have relatively low bias, but if we talk about low variability, that's how much spread do they have, especially around the true parameter. It looks like statistic A has much lower spread than statistic B right over here.

Statistic B, you've calculated statistics that are two away from the true parameter, while in statistic A, you've calculated a statistic of certain samples that are roughly half away from the true parameter. So the statistic that has both low bias and low variability, that would be statistic A.