I’ve noticed when the olympic gymnasts perform, their scores are recorded with a number like 9.762.
There is no way an individual judge can make a score determination with that kind of accuracy. (What is the difference between a 9.762 and a 9.763 in a judges eye/mind?)
From my high school physics days, if you start out with 2 digits of accuracy (like 9.72) and average several scores…you can not claim 3 digits in your final average…am I wrong in my understanding of this?
I think the most they would go is two digits, and the averaging produces the third.
Even a two scores of 9.72 and 9.73 would result in a 3-digit average.
The reason significant figures do not count here is because it’s a subjective rating. If you have judges that score 9.9 9.8 9.7 9.6 9.6 9.5 and 9.7, you come up with an average of 9.6857… Now, for another competitor, the scores are the same, except the 9.5 judge rated a 9.6 instead. Now you get an exact average of 9.7. Obviously, the second competitor scored higher, since all the other judges thought they were equal, with one thinking higher…thus that person wins. Sticking with Sig Figs, you’d end up with 9.7 for both, and a tie. Since it’s not a measurement, you can easily ignore sig figs, and get a true, correct, average. When measuring things, you can’t get MORE accuracy than your least accurate measurement. See the difference?
Jman
thanks Jman…you just fought a little bit of ignorance today 
beagledave, you’re mixing physics and statistics, here, a very dangerous combination!
An easy example to consider with which most of us are familiar is a GPA.
People get 4’s (A’s), 3’s (B’s), 2’s, well, you get the picture.
In my high school class, though 2 students were co-valedictorians with 4.0’s. 2 other girls each had 1 B for 1 semester grade in their high school career. 1 girl had taken driver’s ed in the summer because that’s when she was first old enough. She was the salutatorian on the basis of having something like a 3.9796 GPA where the other girl had a 3.9791 GPA.
I’ll leave it to you to decide whether that is a good or bad use of our ability to calculate fractions out to absurdity, though.
This is a bad use of our ability to determine what place someone came in. The salutatorian is supposed to be the person in the class with the second highest GPA. Ms 3.9796 had the third highest GPA in the class and Ms 3.9791 had the fourth highest. Neither one should have been salutatorian.
I see this mistake made a lot when people talk about sports. A friend told me that her softball team was in second place with a 2-2 record, but there were three teams tied for first at 3-1. I tried to explain to her that her team was in fourth place because there were three teams in front of her, but I did not succeed.
This response, of course, has nothing to do with the OP, but that was asked and answered, so I thought I’d throw some useless crap out there.
There was some debate in baseball circles about whether or not Todd Helton would have been considered to have been a .400 hitter, if he finished with a batting average of something like .3996
Presently, baseball just uses three digits and rounds up (I’m not sure what they do in case the terminal digit is a five, but I doubt that happens frequently [a .XXX5])
A few times, they have had to use the fourth digit to figure out the batting champion. You could say it’s not relevant since no one gets a 1/10000 of a hit. However, nobody gets a 1/10 of a hit either.
Perhaps batting averages should be expressed as either 0 or 1. 
I don’t understand any debate on Helton. The issue came up in 1941, where .3996 was considered a .400 average. Of course, Ted played in the final doubleheader and raised his average to .406.
There’s a joke in there somewhere.
[Bart]Damn TV! Ruined my imagination! [/Bart]
I’m not sure what they would do either, but for all of you out there that are tearing their hair out wondering what to do if you run into this situation while using sig figs, I have the answer. You’re supposed to round to the even number by convention. i.e. .3455 goes to .346 and .3445 goes to .344. You’re welcome.
That’s not necessarily true. By averaging many measurements of the same thing, it’s often possible to get more resolution than any individual measurement. Note that resolution is not the same thing as accuracy, though, but it can be a very usefull technique. In essence you average out any noise in the measurement process.
Arjuna34