Why does the variance of sampling distribution of sample mean become smaller as the sample size increase?

Question

In my statistics book, it says that the variance of sampling distribution of sample mean become smaller as the sample size increases. But it doesn't say why it tends to be smaller than the population distribution. Are there any theories or conditions for the tendency?

Answer 1

Two basic properties of variances:

• The variance of the sum of independent random variables is the sum of their individual variances

• The variance of a constant $k$ times a random variable is $k^2$ times the variance of a random variable

So if your sample values are the random variables $X_1,X_2,\ldots,X_n$, independent each with variance $\sigma^2$ then

• $\text{var}\left(\sum X_i\right) =\sum\text{var}\left( X_i\right)= \sum \sigma^2 = n\sigma^2$
• $\text{var}\left(\overline{X}\right) =\text{var}\left(\frac1n\sum X_i\right) =\frac{1}{n^2}\text{var}\left(\sum X_i\right) =\frac{1}{n^2}(n \sigma^2)= \frac1 n\sigma^2$