Question

Several years ago a well-known public official left Illinois and moved to Texas.  A local regional reporter revealed both his regional chauvinism and his feelings about the official when he remarked that “on this occasion he raised the mean IQ in both states”.

Such a statistical change is possible. Explain how.

Considering this, and since the mean IQ of the US is the average of the means of all the states, it appears that by a mere reshuffling of populations between states, one could increase the mean IQ of the U.S! Is this so? Explain.

Answer 1

When the public official moved from Illinois to Texas, it is possible that he raised the mean IQ in both states if the IQ of the public official is less than the mean IQ of Illinois but higher than the mean IQ of Texas. This also implies that the mean IQ of Texas is lower than that of Illinois.

For example let us say there are only 3 people in Illinois and their IQs are 120,130,140. Thus the mean IQ of Illinois is 130.

Let us say that there are only 3 people in Texas and their IQs are 100, 105, 110 and the average IQ of Texas is 105.

Now let us say the public official currently in Illinois (one of the 3 residents), has an IQ of 120. This official with an IQ of 120 moves from Illinois to Texas. Illinois now has 2 residents having IQs 130 and 140 and the new mean IQ of Illinois is 135.

Texas now has 4 residents, with IQs 100,105,110,120 and the new mean IQ of Texas is 108.75.

Thus, the mean IQ of both Illinois and Texas are raised.

The mean IQ of the US is not a simple average of the means of all the states, but a weighted average of the means of all the states. Due to the movement of the public official, the weight of Illinois has reduced and the weight of Texas has increased. But the mean of Illinois is higher than the mean of Texas and hence they balance out, thus keeping the mean IQ of the US unchanged.

In the example above the population of Illinois has fallen from 3 to 2 and Texas has increased from 3 to 4. That is the weight of Illinois which has a higher mean has reduced and the weight of Texas which has a lower mean than Illinois has increased. They (the weights) balance each other exactly in this case and the mean IQ of the US remains the same