Question

In what way(s) does a larger sample size affect the standard deviation of a sampling distribution? Explain your answer from a mathematical standpoint (a formula), and also a "logical" standpoint (like an explanation in words).

Answer 1

The sample size n affects the standard error or standard deviation of a sampling distribution for that sample. The following points highlights the importance of large sample size.

1. The first reason to understand why a large sample size is beneficial is simple. Larger samples more closely approximate the population. Because the primary goal of inferential statistics is to generalize from a sample to a population, it is less of an inference if the sample size is large.

2. A second reason is kind of the opposite. Small samples are bad. Why? If we pick a small sample, we run a greater risk of the small sample being unusual just by chance. Choosing 5 people to represent the entire U.S., even if they are chosen completely at random, will often result if a sample that is very unrepresentative of the population. Imagine how easy it would be to, just by chance, select 5 Republicans and no Democrats for instance.

3. Another reason why bigger is better is that the value of the standard error is directly dependent on the sample size.To calculate the standard error, we divide the standard deviation by the sample size (actually there is a square root in there).

In this equation, is the standard error, s is the standard deviation, and n is the sample size. If we were to plug in different values for n (try some hypothetical numbers if you want!), using just one value for s, the standard error would be smaller for larger values of n, and the standard error would be larger for smaller values of n.

4. There is a rule that someone came up with that states that if sample sizes are large enough, a sampling distribution will be normally distributed. This is called the central limit theorem. If we know that the sampling distribution is normally distributed, we can make better inferences about the population from the sample. The sampling distribution will be normal, given sufficient sample size, regardless of the shape of the population distribution.