A group of 20 people takes a memory span test. You are told that the mean memory
span is 7.5 and are asked to predict each person's score. Since you know nothing
about the individuals except that they are memebers of a group with a mean of 7.5,
the best you can do is predict that each person will have a memory span of 7.5. Therefore,
for each person in the group, the predicted score will be 7.5 and the error of prediction
will be their score minus 7.5.
The errors of prediction will always have a mean of 0. One way to measure the accuracy
of prediction is to compute the mean squared error of prediction. This measure of
accuracy will be smaller when the mean is used as the prediction then it would be
if any other measure (such as the median) were used as the predicted score. (Click
here for a proof of this).
Each individual score can be thought of as consisting of the sum of two parts: the
predicted score and the error of prediction. For example, if the mean were 4.0 and
a particular score were 5.0 then the predicted score would be 4.0 and the error of
prediction would be = 1.0.
How do the mean and median compare? Although the average squared error of prediction
will be higher for predictions based on the median than on the mean, the average
absolute value of the difference will be lower for the median than for any other
number. (Click here for a proof of this).
Change the distribution in the applet and compare the error of prediction and squared
errors of prediction for the mean and median.