統計迷思 : Average of averages

今天老闆寄來的。
唉！看了就頭痛。
工作難為啊。


Average of averages
You're living dangerously.
If the averages don't have equal weight, your average of averages will yield skewed results. Here's why.
Let's say you're trying to calculate your overall average grade for all the tests you've taken in school, and let's say you are taking only math, English, and science. Let's say your average in math is 80, English is 85, and science is 90. The average of those averages is, of course, 85. But is that your true average?
Not if you took ten math tests, five English tests, and only two science tests!! If all those tests have equal value, then your average is closer to 80, because the ten math tests dominate the true average. In other words, your true average is the WEIGHTED average of the three averages. To weight them properly, you must think of the math average as not one data point but ten data points, the English average as not one data point but five, and the science average as not one but two data points.
Here's how the weighted average math looks:
10 x 80 = 800
5 x 85 = 425
2 x 90 = 180
__ ___
17 1405
Now, divide 1405 by 17. So, 1405/17 = 82.7. (The 17 is the sum of the weighting factors, which, as you can see, is equal to the total number of tests you took.)
So, you can see that taking the average of averages may give erroneous results -- unless you find the weighted average.