Constraints for variance given sample mean for positive-only variables

Question

Suppose we measure sample mean $\mu$ of a positive-only variable (e.g. number of cigarettes smoked by a person today) in a given sample (100 different people). Are there any constraints on the sample standard deviation $\sigma$, given a certain value of $\mu$? For example, if $\mu=1$ is it possible $\sigma=10$?

Answer 1

The bound is affected by the sample size.

In your example, the maximum variance case would be $1$ person smoking $100$ cigarettes and $99$ people smoking $0$ cigarettes.

Using the $\dfrac{\sum(x_i-\overline{x})^2}{n-1}$ definition of sample variance, this would give a variance calculation of $$\frac{1\times (100-1)^2+99\times (0-1)^2}{99}=100$$ and so a sample standard deviation of $10$.

So if your random variable takes non-negative values, then in general the extreme case is one observation of $n\mu$ and $n-1$ observations of $0$, giving a maximum sample variance of $$\frac{1\times (n\mu-\mu)^2+(n-1) \times (0-\mu)^2}{n-1}= n\mu^2$$ and a corresponding maximum sample standard deviation of $\sqrt{n}\mu$.

If instead you used the $\dfrac{\sum(x_i-\mu)^2}{n}$ definition of variance, the maximum would be $(n-1)\mu^2$ and a maximum standard deviation of $\sqrt{n-1}\mu$.