# Knowing both sample mean and sample range improves estimate of the the variance?

It is known that sample variance and sample mean are independent for normally distributed variables, which means knowing sample mean does not say anything about to estimate the variance of the underlying distribution.

However, intuitively range of a sample (max of sample-min of sample) should say something about the variance of the underlying distribution. So by knowing just the sample range, we should able to reach (I am not sure but feel in that way) an estimate of the variance of the underlying distribution.

Let say we have an unbiased estimator of variance of the related distribution by using range: var1

I wonder both knowing range and sample mean at the same time; can we have more efficient unbiased estimator than var1 .

I feel "no" as answer of this question for normal distribution but "yes" for asymmetric distributions. However, I can not figure it out. However, I am not clear about this subject. I will be very glad for any help.

• Look into the Rao-blackwell theorem May 14 '17 at 19:10
• I can't figure out the relation with Rao-Blackwell theorem. Can you please explain a bit? . Sample mean is not a sufficient estimator for variance. May 14 '17 at 19:36
• i don't know much about estimation theory, but i know that sample min/max can be used to estimate variance, see for exampleHozo, Djulbegovic, Hozo; Estimating the mean and variance from the median, range, and the size of a sample bmcmedresmethodol.biomedcentral.com/articles/10.1186/… Sep 25 '20 at 18:16

In other cases, the sample range may indeed add information. For example, consider a uniform distribution on $(\mu-\sqrt{3}\,\sigma,\mu+\sqrt{3}\,\sigma)$ -- which has mean $\mu$ an standard deviation $\sigma$. Then the sample range will contain all the information about $\sigma$ but the sample variance will add nothing to that. [Note that this is symmetric, not skew]