5.2 The Sampling Distribution of the Sample Mean (Central Limit Theorem)
5.2 The Sampling Distribution of the Sample Mean (Central Limit Theorem)
Let’s start our foray into inference by focusing on the sample mean. Why are we so concerned with means? Two reasons: they give us a middle ground for comparison, and they are easy to calculate. In this section, we will see what we can deduce about the sampling distribution of the sample mean.
The Central Limit Theorem for a Sample Mean
The central limit theorem (CLT) is one of the most powerful and useful ideas in all of statistics. There are two alternative forms of the theorem, and both forms are concerned with drawing finite samples sizes, n, from a population with a known mean, μ, and a known standard deviation, σ. One of the forms says that, if we collect samples of size n with a “large enough” n, then the resulting distribution can be approximated by the normal distribution.
Applying the law of large numbers here, we could say that taking larger and larger samples from a population brings the mean, , of the sample closer and closer to μ. From the central limit theorem, we know that the sample means increasingly follow a normal distribution as n gets larger and larger. The larger n gets, the smaller the standard deviation gets. (Remember that the standard deviation for is .) This means that the sample mean, , must be close to the population mean μ. We can say that μ is the value that the sample means approach as n gets larger. The central limit theorem illustrates the law of large numbers.
The size of the sample, n, that is considered “large enough” depends on the original population from which the samples are drawn (the sample size should be at least 30, or the data should come from a normal distribution). If the original population is far from normal, then more observations are needed for the sample means or sums to be normal. Sampling is done with replacement.
The following images look at sampling distributions of the sample mean built from taking 1,000 samples of different sample sizes from a normal population. What pattern do you notice?
The following images look at sampling distributions of the sample mean built from taking 1,000 samples of different sample sizes from a non-normal population (in this case, it happens to be exponential). What pattern do you notice?
What differences do you notice when sampling from normal and non-normal populations?
Example
Suppose:
- eight students roll one fair die ten times
- seven roll two fair dice ten times
- nine roll five fair dice ten times
- 11 roll ten fair dice ten times
Each time a person rolls more than one die, he or she calculates the sample mean of the faces showing. For example, one person might roll five fair dice once and get 2, 2, 3, 4, 6.
The mean is = 3.4. The 3.4 is one mean when five fair dice are rolled. Suppose this person then rolls the five dice nine more times and calculates nine more means (for a total of ten means).
As the number of dice rolled increases from one to two to five to ten, the following