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Center and Variation of a Sampling Distribution

Center and Variation of a Sampling Distribution

Author: Ryan Backman
Description:

Calculate the mean and standard deviation of a population or a sampling distribution of sample means.

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Hi. This tutorial covers the center and variation of a sampling distribution. All right, suppose the population of some random variable x-- such as the height of all adult females, the number of siblings of US high school students, et cetera-- is distributed with a mean of mu sub x and a standard deviation of sigma sub x.

So sometimes you'll see this with a subscript of x. Other times, you'll just see these just as mu and sigma. Really, the x is just whatever variable-- it's the mean of this variable x. And these are both population parameters. This is a population mean, population standard deviation. OK, now suppose we took samples of size n from this population and calculated the mean x bar for each sample.

So we're putting together a lot of symbols here, so if we're thinking about sampling from this population and calculating a sample mean from each sample. Now, the collection of the sample means from our various samples is called the sampling distribution of sample means. So if we took all of those sample means and graphed them as a distribution, it would called the sampling distribution. And since it's a distribution of means, it's a sampling distribution of sample means.

So like any distribution, it's helpful to know about the center, the variation-- or the spread-- and the shape of the sampling distribution of sample means. So today we're going to focus on center and variation. All right, so usually the main way we use to measure the center is using the mean. So the mean of the sampling distribution of sample means is notated as mu sub x bar.

So instead of mu sub x-- so the average x value-- now we're looking for what's the average x bar value. So it's the average. So it's the average sample mean. The standard deviation of a sampling distribution of a sample means is notated as sigma sub x bar, and it's often referred to as the standard error.

So we have the mean and the standard deviation of the sampling distribution of sample means. So the mean of the sampling distribution is equal to the mean of the population distribution. So these two means are always going to be the same. Now, the standard error-- the standard deviation of the sampling distribution is going to equal the standard deviation of the population divided by the square root of n.

So notice that your mean doesn't really change, but your standard deviation will change when you go from a population to a sampling distribution. OK, so let's see if we can make some sense of that now by looking at an example. So we're going to let x denote the time it takes a sixth grader to complete a math problem.

Suppose the mean and standard deviation are mu equals 2.5 minutes, and sigma equals 0.7 minutes, respectively. A sample of 35 students is selected. So we're dealing with a sample of size 35. So what are the mean and standard deviation of the sampling distribution of sample means?

So what we're looking for is mu sub x bar and sigma sub x bar. So what mu sub x bar, in this case, represents is, on average, what's going to be the average time that a group of 35 students will do the math problem? So this 2.5 minutes represents what we would expect one student-- the time it would take one student to do the math problem.

What this is is it's what we would expect an average of 35 students to be. And remember, from the previous page, we said that mu sub x bar is just equal to mu sub x, or mu. So we would expect a group of 35 students to average 2 and 1/2 minutes. Now, the standard deviation of the sampling distributions is going to be a little different. Remember that that formula was sigma sub x over the square root of n.

So for our example, it's going to end up being 0.7 divided by the square root of 35. And if we do that on the calculator-- 0.7 divided by the square root of 35-- that's going to give me about 0.118. So really, the variation went down when we're dealing with a greater group of students. And I think that should make sense.

If we're dealing with a sample mean of 35 students, we would expect it to be the same as the mean for one student. But since we're dealing with 35 students, the x bar value should be a lot closer to 2.5 than if we were just looking at one student. So there's going to be more variability or variation in the scores of just single students versus the scores of-- the average scores of groups of 35 students. So that's why this number is smaller than the population standard deviation.

So really, these two formulas are the important things to get out of this tutorial-- mu sub x bar equals mu sub x and sigma sub x bar equals sigma sub x over the square root of n. So that's been the tutorial on mean and variation of a sampling distribution. Thanks.

Terms to Know
Mean of a Distribution of sample means

The average of all possible means from all possible samples of a given size. It will be equal to the mean of the original population.

Standard Deviation of a Distribution of sample means

The standard deviation of all possible means from all possible samples of a given size. It will be equal to the standard deviation of the original population, divided by the square root of the sample size.

Formulas to Know
Mean of a Distribution of sample means

mu subscript x with bar on top end subscript space equals space mu subscript o r i g i n a l end subscript

Standard Deviation of a Distribution of sample means

sigma subscript x with bar on top end subscript space equals space fraction numerator sigma over denominator square root of n end fraction