Why is the median better than the mean for measuring “typical” values? The best way to understand what is wrong with the mean is to look at how both behave in answering a simple question: how well have Stephon Marbury’s Lincoln High School basketball teammates done in their careers in the last 10 years?
I started thinking again about the difference between mean and median while listening to an NPR story about Stephon Marbury. For those who don’t know, he is a 29 year-old from Coney Island, New York, who plays professional basketball for the New York Knicks.
Imagine Stephon Marbury went back last year to meet with 9 teammates on the 10th anniversary of their winning a high school basketball championship. You might ask, “Where are they now? Have they done well? Did they turn what they learned playing hoops into the start of a successful career?”
I haven’t actually researched where Stephon Marbury’s teammates are now, so let’s just make up some wages that would be typical for people in their late twenties in New York. Remember, 50% of high school seniors do not go to college, so we could expect a pretty wide range of pay: a janitor ($13/hour), a delivery truck driver ($14/hour), a retail store assistant manager ($18/hour), automobile mechanic ($20/hour), fire fighter ($24/hour), a nurse ($25/hour), a department store buyer ($28/hour), an automobile salesman ($32/hour with commissions), and an IT project manager ($41/hour).
What is the mean (average) wage of these nine? It is easy to calculate: add up the wages and divide by 9:
13+14+18+20+24+25+28+32+41 = 215
mean wage = 215 / 9 = $23.90/hour
The median is $24/hour. The fire fighter earns the middle wage: 1/2 are lower, and 1/2 are higher. For this particular sample, the mean and median give a similar answer for what a “typical” person on that team now earns.
Now let’s look at Stephon’s hourly wage. Stephon played an average of 36.5 minutes over the course of 60 games last year. For this, Stephon earned about $20 million/year, or about $550,000/hour.
If we include Stephon’s wage, his high school team’s mean wage is $55,000/hour!
By contrast, the median wage is $24.50/hour, now half-way between the fire fighter and the nurse. The median is not significantly changed by this one “outlier,” while the mean becomes a wage that no one earns: it is 2000X too high for 9 of the teammates, and 10X too low Stephon.
Like standard deviation, mean is very sensitive to the most abnormal of values, particularly very high values. Why would one use a measure for what people “typically” earn, that is so strongly affected by atypical salaries?
The answer is historical, and the subject of a future post.