DA8. CLT and Large Numbers

Imagine that you had to explain the law of large numbers and the central limit theorem to a person who has never heard of these things. Write an explanation to help this person understand these concepts. If you use a technical term, be sure to explain it.

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” (Yakir, 2011).

Simply put, The sample size can not ever exceed the population size. So the more the sample size expands, the more this sample approaches the actual population, which means that this sample will include more and more of the population observations so that its distribution gets closer and closer to the population’s distribution until they match.

With a sample size large enough to the point where one can ignore the difference between its size and the population size, we can approximately say that the two distributions are the same (since the sample and the population almost represent the same set of observations or values).

The Law of Large Numbers states that “the distribution of the sample average tends to be more concentrated as the sample size increases” (Yakir, 2011).

With a sample size large enough to be almost close to the population size, we are sampling the same set (almost the entire population) n times. So the sampling distribution would be the distribution of the same set repeated multiple times (exactly n times where n is the number of the samples that constitute this distribution).

With a distribution of n observations, where each observation almost represents the same set, the average of each set becomes more concentrated around the average of this set.

The variance means how the elements of the set spread around its expectation. The variance of such distribution logically should go smaller and smaller where the means of the sample get closer to each other until we finally conclude that the variance of the sampling distribution equals the population variance divided by the sample size.

References

• Yakir, B. (2011). Introduction to statistical thinking (with R, without Calculus). The Hebrew University of Jerusalem, Department of Statistics.