Rules of thumb for reporting precision for sample statistics I'm curious about your thoughts on reporting precision for sample statistics.  Are there good rules of thumb for this?
For example, I'm reporting on $X$ and $Y$ which have $n$ samples. Statistics of interest include the mean, standard deviation, skewness, kurtosis, and $\rm{Cov}(X,Y)$.
For some $(X,Y)$, the sample size $n$ is very large.  For some $(W,Z)$ the sample sizes are smaller.  
How should I determine the reporting precision (significant figures)? Very interested in your thoughts.   
 A: What you are getting at is how should the reporting precision (number of significant digits written out) depend on your certainty in your result. You have more trust in the accuracy of an empirical mean for a larger sample than for a smaller one.
The central limit theorem is the rigorous result that comes to my mind, at least for the difference between the empirical mean and the mean of the underlying distribution. You should compute the standard deviation of the limiting normal distribution in the CLT (estimate it from the sample variance) and divide it by square root of sample size; this gives a good measure of spread for the mean.
In experimental scientific papers people often report measurements like this: $$\textrm{measured mean}\ \pm\ \textrm{st. dev.}$$
I would consider the standard deviation to 2 or 3 significant values as relevant, and then I'd report the mean to include the first two digits that are covered by the order of magnitude of the st. dev. E.g. if your st. dev. is $0.3456$, then reporting the mean is meaningful to at most two decimal places. If your st. dev. is $65000$, then there is no point in writing out hundreds and lower in your mean.
A: Think about your audience and the real entities that the numbers represent. For example, if you're writing a research article in psychology and you're describing seventh-graders' mean self-reported number of friends, a tenths digit is expected, a hundredths digit might be appreciated, and a thousandths digit would probably be excessive. 8.1 and 8.4 friends sound meaningfully different to me, whereas I wouldn't make much of a distinction between 8.11 and 8.14 friends.
A: I'll add to @Kodiologist that you wouldn't want the numbers to be more than say 5 or 6 places total. So if you have things with an average of 3000, do not have a decimal. If it's in millions, report it in 000's or 000,000's (3.125 million, 3,125 in 000's). There's probably a good rule of thumb to be established that you don't need to add another decimal place if the precision will be less than x percent. In @Kodiologist example, .1 is a change of 1.25% in the mean of 8 which should probably be reported, but .01 (change of 0.125%) is probably not important.
