# Finding the mean and standard deviation of an unknown distribution from a sample

I am interested in finding the mean and standard deviation of the whole distribution by looking only at a random sample. I don't know anything else about the distribution (for example I don't know if the distribution is normal or not). Is what I'm asking even possible?

• You cannot know the parameters for sure. You can estimate them (as estimates) from the (random & representative) sample, in point+error (interval) manner. Either (1) you assume that you know the type of the distribution in the population and then you estimate by formulas, or (2) you don't assume that but instead assume that the distribution shape there is exactly like your sample's, and do bootstrapping estimation (of intervals). – ttnphns Aug 26 '17 at 11:30

Sure, your best guess of the population mean is your sample mean, and the same is true for the standard deviation (assuming you use $\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}$) and not divide by just $n$).

There are distributions where the mean or standard deviation are less meaningful or even undefined, but that is up to you to decide. Also inferences may be more difficult with some distributions.

• Perhaps this comment is not so useful to the original poster, but may be helpful to future readers: there may not be a single "best guess", but there can be more than one estimator for a population parameter. Which is "best" depends on which estimator has the properties that are important for you. Note that the standard deviation estimator with $n-1$ in the denominator is not an unbiased estimator of the true standard deviation, but its square is an unbiased estimator of the true variance. – Silverfish Aug 26 '17 at 12:07

No. It isn't possible to know this exactly. The whole point of inferential statistics is that you infer things about the population from a sample. However, your knowledge of the population is not exact, Thus we have things such as standard errors, confidence intervals and so on.

Now, if your sample is randomly drawn from the population then it is possible to get some idea about the whole population. But if you don't have a random sample and you don't know in what way it is nonrandom, you can do much less.

• I forgot to mention my sample is random. What kind of things can I know about the distribution in that case? – Aleksandar Aug 26 '17 at 11:29
• Even though our answers are opposite I don't think we disagree in principle. My take on the differences in our answers is that i just assumed a random sample and an acceptance of uncertainty in an estimate, while you took the question literally. – Maarten Buis Aug 26 '17 at 11:30
• @MaartenBuis I agree. – Peter Flom Aug 26 '17 at 11:31