Tuesday, October 29, 2013
The three averages—means, medians, and modes—are basic mathematical concepts. Nevertheless, they seem to have generated an inordinate amount of confusion among some critics of the Economic Value of a Law Degree (see here, here, and here).
Some of the critics have emphasized modes and medians while downplaying the importance of means. Steven Harper, for example, has claimed that the mean is a “meaningless” statistic and we should instead focus on the medians and modes while ignoring the mean.
To understand his error, imagine two lecture halls, each with 100 seats. Underneath each of those seats is a suitcase full of cash. The individuals sitting in those seats will get to keep whatever cash they find when they open the suitcase.
In Lecture Hall A, every suitcase contains $600,000. $600,000 is the mean, median, and mode value.
In Lecture Hall B, 60 of the suitcases contain $600,000, but the remaining 40 suitcases each contain $1,600,000. The median and mode is exactly the same as in Lecture Hall A--$600,000. But the mean is much higher in Lecture Hall B—it is $1,000,000 instead of $600,000.
If you didn’t know how much money would be in your suitcase, but you could choose between sitting in Lecture Hall A and Lecture Hall B, which room would you choose?
You would be wise to choose Lecture Hall B. But the only reason to choose Lecture Hall B is because the mean (average) is higher in Lecture Hall B. The median and mode in both lecture halls is identical.
Now imagine a slightly different fact pattern. In Lecture Hall A there are three suitcases each containing $1.6 million, while the remaining 97 suitcases contain amounts that are close to $600,000 (i.e., a range from $599,950 to $600,050), with none of these 97 suitcases containing the exact same dollar value. The mode value in Lecture Hall A is $1.6 million, while the median is $600,000 and the mean is $630,000.
Lecture Hall B is the same as in the previous fact pattern—60 suitcases contain $600,000 while 40 suitcases contain $1.6 million. The mode value in Lecture Hall B is $600,000, while the mean is $1,000,000.
In other words, the mode is higher in Lecture Hall A, but the mean is higher in Lecture Hall B. The medians are identical.
Which room would you choose to sit in?
Once again, you would be wise to choose Lecture Hall B. This suggests that you believe that means (averages) are more important than modes.
The money at the top (or the bottom) matters. Means provide useful information that is not available from medians alone, and that is not reflected in modes. That’s why we provide both means and medians, as well as 75th percentile and 25th percentile values in Economic Value of a Law Degree.
(Posted by Michael Simkovic)