I am wondering what "Minimizing the variance" in Statistics mean. As far as I understand that the variance is one of the parameters of a probability distribution, for example, a normal distribution and it is a measure of how spread the data points are from the mean. However, what is there between "Minimizing the variance" and the spread of data points?

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    $\begingroup$ A statistical estimator is a random variable, hence is endowed with a variance of its own. $\endgroup$ – Xi'an Apr 18 at 12:41

A common task in statistics is to perform parameter estimation.

The parameter estimator is a function of the sample.

For example, consider the following two parameter estimator for the mean of normal distribution:

  • $\hat{\mu}_1=\frac{X_1+X_2}{2}$

  • $\hat{\mu}_2=X_1$

We have $Var(\hat{\mu}_1)=\frac{\sigma^2}{2}$ but $Var(\hat{\mu}_2)=\sigma^2$. Even though both are unbiased estimator of the mean, we would prefer the first estimators.

Given a few unbiased estimators, we would prefer the one with the smallest variance.


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