The Law of Large Numbers states that, as the sample size becomes larger, the
sampling distribution of the sample average becomes more and more concentrated
about the expectation.
the variances decrease with the increase of the sample sizes. The
decrease is according to the formula Var( ¯X ) = Var(X)/n.
The variance is a measure of the spread of the distribution about the expectation.
The smaller the variance the more concentrated is the distribution around
the expectation. Consequently, in agreement with the Law of Large Numbers,
the larger the sample size the more concentrated is the sampling distribution of
the sample average about the expectation.
The Law of Large Numbers states that the distribution of the sample average
tends to be more concentrated as the sample size increases. The Central Limit
Theorem (CLT in short) provides an approximation of this distribution.