Using an activity to teach about unbiased estimators

Today in AP Statistics, we had one of our better classes, so I wanted to share it with you. We did an activity to better understand the differences between parameters and statistics and what makes for an unbiased estimator. I based it off of an activity in our textbook and it worked beautifully.

Each of my 12 students wrote their heights in centimeters on a slip of paper and put that paper in a box. Each group then took turns randomly selecting a sample of size three with replacement. The groups recorded the data and calculated the sample mean and the sample range. Each group then wrote up their data and their statistics on the board. I then had each group take another SRS of size three and report their findings. Meanwhile, two students were tasked with finding the population mean and population range, but they were asked to not share these numbers with the class.

So, now we had eight samples, eight sample means, and eight sample ranges. I briefly talked about how statistics were used to estimate parameters.

I then asked the students which sample mean was the best estimate of the population mean and to explain their thinking. Four students were willing to offer their opinions and they gave reasons such as it seemed like the number in the middle. Interestingly, two students selected 172 cm and two others selected 168 cm.

I then asked the same question regarding the sample ranges. This discussion went differently. One student immediately realized that the largest value, 30 cm, was the closest to the population range. The same student continued and argued effectively that the range was at least 33 cm and that no sample would ever have a range greater than the population range.

We observed and discussed how the various sample statistics deviated from each other. We discussed the centers of the sample distributions. And then I asked the class which of the two statistics they thought was unbiased. One student bravely spoke up and said the sample ranges were unbiased and offered up his reasoning. The others listened and then, respectfully, argued against his claim and presented their own reasoning.

The population parameters were then shared. We then drew dotplots of the sampling distributions and added in the population values to the dotplots so the students could visualize how the sample ranges were all less than the population range and how the sample means fell on either side of the population mean. We also discussed how the sample ranges deviated on average more from the population range than the sample means deviated from the population mean.

The dotplots were very effective. We talked about how the population parameter is the target and the sample statistics are our attempts to hit that target. Using this lens, it was easy to see how the sample mean was a better estimator than the sample range.

This was only a 35-minute class, so we did not get into how we could improve the accuracy of our estimates (so how to affect variability), but that would be an easy extension. Again, a dotplot could be used to show various levels of accuracy dependent on sample size.