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Center and Variation of a Sampling Distribution

Center and Variation of a Sampling Distribution

Author: Katherine Williams
Description:

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

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Tutorial

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This tutorial covers center and variation of a sampling distribution. First off, we have the mean of the sampling distribution of the sample mean. Now I know it looks like it's a lot of words, but it's really not so bad.

First of all, we have a set of sample means. We create a sampling distribution of them. And then we find the mean of that.

Now the mean of the sampling distribution of the sample mean is the same as the mean of the underlying distribution. Sometimes it's called the grand mean. So here, the mean of our sampling distribution is the same as the mean of our underlying distribution, our population of x's.

Now, we also can talk about the standard deviation of the sampling distribution of the sample mean. And again, we have the set of sample means, we take the sampling distribution and create one for those sample means, and then we can find the standard deviation of that.

Now here, we have a little bit of a formula to use. The standard deviation of the underlying distribution, when we divide that by the square root of the sample size that gives us the standard deviation of the sampling distribution of the sample mean. We'll work through an example now.

Here is our example. We'll let x stand for the number of pounds the lawn chair can support before breaking. We sample 10 chairs at a time. And we find that the underlying mean is 250 and the standard deviation is 33.

So now in order to find the mean of the sampling distribution of the sample means, it's actually quite easy. It's just the same. So our mean of the sampling distribution of the sample mean is also going to be 250 pounds.

Now for the standard deviation of the sampling distribution of the sample mean, again, we've got a little bit of a formula to do. We need to take the underlying distributions standard deviation, 33, and we divide by the square root of the sample size divided by the square root of 10. When we do that, we get 10.44.

So that's all we have to do in order to find the mean of the sampling distribution of the sample mean and the standard deviation of the sampling distribution of the sample mean. This has been your tutorial on the center and variation of a sampling distribution.

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