16 Distribution of Sample Means
Jenna Lehmann
Up until this point, as far as distributions go, it’s been about being able to find individual scores on a distribution. Moving into hypothesis testing, we’re going to switch from working with very concrete distributions with scores to hypothetical distributions of sample means. In other words, we’re still working with normal distributions, but the points that make up the distribution will no longer be individual scores, but all possible sample means which can be drawn from a population with a given 
or number of scores in them.
We use these kinds of distributions because with inferential statistics we’re going to want to find the probability of acquiring a certain sample mean to see if it’s common or very rare and therefore perhaps significantly different from another mean.
There are some concepts you will have to keep in mind for this shift including sampling error, the central limit theorem, and standard error.
Sampling Error
Sampling error is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter. So each sample is different because you’re likely drawing separate samples from the same population; you hope to get a diverse group, but you don’t really get to pick what you get, and you’re not likely to get the exact same group twice. Samples can’t be entirely representative of a population and always have less variability than the population. So even though we take a sample in order to run statistics that can be generalized back to the population, there is always going to be some error.
Central Limit Theorem
The central limit theorem is a set of rules that dictate how a distribution of sample means will look given certain criteria. For any population with mean 
and standard deviation 
, the distribution of sample means for sample size 
will have a mean of and a standard error of 
(which we will talk about more in a minute) and will approach a normal distribution as n approaches infinity. So this practically means that the distribution of sample means is almost perfectly normal in either of two conditions: the population from which the samples are selected is a normal distribution or the number of scores in each sample (also known as sample size) is relatively large (around 30 or more). The central limit theorem also mentions that as n increases, variability decreases. In other words, the greater the sample n, the pointer your distribution.
These videos will help your understanding:
Standard Error
The standard error provides a measure of how much distance is expected on average between a sample mean 
and the population mean 
. Essentially, it’s the standard deviation of sample means from the mean of sample means. It specifies precisely how well a sample mean estimates its population mean. The magnitude of the standard error is determined by two factors: the size of the sample and the standard deviation of the population from which the sample is selected. We can see by the equation 
that the greater is, the greater its square root and the more that the standard deviation will have to be divided by, making the standard error smaller. But if the population standard deviation is already small, that will make the standard error small too.
Inferential Statistics
Inferential statistics are methods that use sample data as a basis for drawing general conclusions about populations, and as mentioned before, are the reason why we’re learning about distributions of sample means. It’s important to know how much a sample differs from the population because we can’t draw many conclusions about the population from a sample that is very different. The error is important to keep in mind too when creating a control group. If a study with treated and untreated patients is to be generalized to the general population, you don’t just want to know if there was a significant difference between the two groups, but you want to make sure that the untreated group represents the general population.
This chapter was originally posted to the Math Support Center blog at the University of Baltimore on June 6, 2019.
