Expert: Paul Klarreich - 5/23/2008

Question
Eldon received 72 on his first calculus test. The marks were normally distributed with a mean of 60 and a standard deviation of 8. He received 80 on the next test, which was normally distributed with a mean of 75 and a standard deviation of 6. Which was the better mark, relatively speaking?

Answer
Questioner:   Joel
Category:  Advanced Math
Private:  No
 
Subject:  Standard Deviation
Question:  Eldon received 72 on his first calculus test. The marks were normally distributed with a mean of 60 and a standard deviation of 8. He received 80 on the next test, which was normally distributed with a mean of 75 and a standard deviation of 6. Which was the better mark, relatively speaking?
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Hi, Joel,

In our declining years, my wife and I have been attending piano competitions. (No, we don't go to Russia for the Tchaikovsky.)  In the old days, the jury (a group of famous piano teachers, usually) would get together in a room and decide on the winner -- leading to some bad feelings, to put it mildly.

Nowadays, the judges listen to a performance and immediately assign a score (from 1 to 10, I assume) to the performer, and a computer totals up the scores and spits out the winner.

So it came to pass that Performers A,B,C played and got these scores:

Judge 1:  A: 9.3 -- he stunk.
         B: 9.4 -- he was pretty good.
         C: 9.6 -- he was great.
Judge 2:  A: 10.0 -- fantastic!
         B: 4.0  -- wonderful!
         C: 1.0  -- excellent!

Who won?  You will be amazed to hear that the contest ended in a 3-way tie for first place!  The computer (said the contest brochure) adjusts for the mean and standard deviation of the scores of each judge.  

So what happened with your Eldon?
 
72 on a mean of 60, sd = 8, is  1.5 s.d.'s above the mean.
80 on a mean of 75, sd = 6, is 5/6 s.d.s above the mean.

You decide.