Click the "Draw 4 numbers" button. Four numbers will be selected from the population. They will be shown in red in the population. They will also be shown in red below the "Draw 4 numbers button." The mean of the 4 numbers is also presented. The population mean is 3.0. See how the sample mean compares to the population mean.

Two formulas for the variance are shown. In the first, the average squared deviation of the four numbers from the sample mean is computed. In the second, the average squared deviation from the population mean of 3 is computed. You should notice that the former formula will always produce a smaller value than the latter formula unless the sample mean is the same as the populaton mean. In this case, the two computations lead to the same result.

Notice the text fields to the right of the formulas. They are used to store the results of the simulation. The values of the variances are stored, and the mean of all the values is displayed at the bottom. After only one sample, the mean equals the single value.

Click the Draw 4 numbers" button again. Another sample will be taken and the computations will be done as before. Each text field will have two variances in it. Look to see which formula is giving the more accurate estimates of the population variance of 2.0.

With only two samples, it is hard to be sure which formula is more accurate. Continue sampling until you have taken about 20 samples. For each sample, note which formula gives you an answer closer to 2.0. You will probably find that formula 2 usually, but not always comes closer.

Look at the means for the two formulas. The mean for the upper formula will be lower than the mean for the lower formula. Look to see which is closer to the population variance of 2.0. You should find that the mean of the values for the upper formula is too low, probably somewhere around 1.50. The mean for the lower formula should be closer to 2.0. If fomula 1 had divided by N -1 (which is 3) rather than N (which is 4), it s values would have been larger. Specifically, they would have been 4/3 times larger. Multiply the mean from forumula 1 by 4/3 and see if it comes closer to the populaton variance of 2.

This makes sense because you would expect to be better able to estimate the variance if you knew the population mean (as you do in formula 2) than if you had to estimate it (as you do in formula 1).

The critical point is that when you have to estimate the population mean, you get values that are, on average, too low. This does not mean that every value will be too low. Look through the variances based on formula 1. Even though the mean is lower than 2.0, you will find that some of the values are above 2.0. This means that even though this formula tends to give you values that are too low, there are instances when it gives you values that are too high.