# Sampling Distribution of the Mean

*… a core concept used in Quantitative Methods and Atlas104*

### Concept description

Wikipedia (reference below) defines a sampling distribution as “the probability distribution of a given statistic based on a random sample.”

OnlineStatBook (reference below) notes that:

“The concept of a sampling distribution is perhaps the most basic concept in inferential statistics. It is also a difficult concept because a sampling distribution is a theoretical distribution rather than an empirical distribution.”

“… sampling distributions are important for inferential statistics. In the examples given so far, a population was specified and the sampling distribution of the mean and the range were determined. In practice, the process proceeds the other way: you collect sample data and from these data you estimate parameters of the sampling distribution. This knowledge of the sampling distribution can be very useful. For example, knowing the degree to which means from different samples would differ from each other and from the population mean would give you a sense of how close your particular sample mean is likely to be to the population mean. Fortunately, this information is directly available from a sampling distribution. The most common measure of how much sample means differ from each other is the standard deviation of the sampling distribution of the mean. This standard deviation is called the **standard error of the mean**. If all the sample means were very close to the population mean, then the standard error of the mean would be small. On the other hand, if the sample means varied considerably, then the standard error of the mean would be large.

“To be specific, assume your sample mean were 125 and you estimated that the standard error of the mean were 5 … If you had a normal distribution, then it would be likely that your sample mean would be within 10 units of the population mean since most of a normal distribution is within two standard deviations of the mean. … Keep in mind that all statistics have sampling distributions, not just the mean.”

###### Central limit theorem

OnlineStatBook describes the concepts involved in the central limit theorem as follows:

“The **mean of the sampling distribution of the mean** is the mean of the population from which the scores were sampled. Therefore, if a population has a mean μ, then the mean of the sampling distribution of the mean is also μ. The symbol μ_{M} is used to refer to the mean of the sampling distribution of the mean. Therefore, the formula for the mean of the sampling distribution of the mean can be written as:

μ_{M} = μ

“The **variance of the sampling distribution of the mean** is computed as follows:

“That is, the variance of the sampling distribution of the mean is the population variance divided by N, the sample size (the number of scores used to compute a mean). Thus, the larger the sample size, the smaller the variance of the sampling distribution of the mean.

“The **standard error of the mean** is the standard deviation of the sampling distribution of the mean. It is therefore the square root of the variance of the sampling distribution of the mean and can be written as:

“The standard error is represented by a σ because it is a standard deviation. The subscript (M) indicates that the standard error in question is the standard error of the mean.

“The **central limit theorem** states that:

Given a population with a finite mean μ and a finite non-zero variance σ^{2}, the sampling distribution of the mean approaches a normal distribution with a mean of μ and a variance of σ^{2}/N as N, the sample size, increases.

“The expressions for the mean and variance of the sampling distribution of the mean are not new or remarkable. What is remarkable is that regardless of the shape of the parent population, the sampling distribution of the mean approaches a normal distribution as N increases.”

**Atlas topic, subject, and course**

Sampling Distributions (core topic) in Quantitative Methods and Atlas104 Quantitative Methods.

###### Sources

Wikipedia, Sampling distribution, at https://en.wikipedia.org/wiki/Sampling_distribution, accessed 11 June 2017.

David Lane, OnlineStatBook, at http://onlinestatbook.com/2/sampling_distributions/sampling_distributions.html, http://onlinestatbook.com/2/sampling_distributions/intro_samp_dist.html, and http://onlinestatbook.com/2/sampling_distributions/samp_dist_mean.html, accessed 11 June 2017.

**Page created by:** Ian Clark, last modified 11 June 2017.

**Image:** OnlineStatBook, at http://onlinestatbook.com/2/sampling_distributions/samp_dist_meanM.html, accessed 11 June 2017.