What does “Minimizing the variance” mean in Statistics?

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?

• A statistical estimator is a random variable, hence is endowed with a variance of its own. – 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.