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Distribution of Sample Means

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This tutorial reviews the distribution of sample means.

Now, with the distribution of sample means, it's when we're taking all the means of all the possible samples of size n. And then when we have a really large distribution, it's really difficult to do this to get all of those combinations because there's too many possible samples.

However, when we do the main distribution-- sorry, when we do the distribution of sample means, the mean of that distribution is the same as the population mean. So the sample mean is the same as the population mean. And then, additionally, we can also apply the Central Limit Theorem to this distribution because our population mean and standard deviation are finite. And then if our sample size is greater than 30, then the distribution of the sample means is going to be approximately normal.

When we look at the standard deviation of the distribution of sample means, another name for this is the standard error. So if you hear the term "standard error," they're talking about the standard deviation of the distribution of sample means. Now, if that distribution is approximately normal-- so if it fits the conditions for the Central Limit Theorem-- then we know that the standard deviation of the distribution of sample means-- and this is the symbol for that, the x bar in the subscript is indicating that's for the sample means-- is the same as the population standard deviation divided by the square root of the sample size.

Now, what's important about this formula is that it shows us that as sample size increases-- so if n is increasing-- then the standard deviation of the sample means-- otherwise known as the sampling error-- is going to be decreasing. We'll see an example now.

So here's our example. In our example, we talk about an office that has 200 employees. And on average, employees take 5.3 sick days per year with standard deviation of 0.7 sick days. If we selected a simple random sample of 20 employees, what will our standard deviation of the distribution of sample means be if we select a simple random sample of 20 employees?

So first of all, the mean of our distribution of sample means is going to be the same as the population mean. However, the standard deviation of our distribution of sample means is going to be different. We're going to use this formula here. So we start off with the population standard deviation on top, 0.7, and divide by the square root of the sample size.

Now, when we do that, we will calculate to see what we get. So the square root of 20 is 4.47, and 0.7 divided by 4.47 is 0.157. Now, if we repeated this, but instead of doing a sample size of 20, we did a sample size of 10, let's see what we get then.

So for this problem here, we still start with the population standard deviation on top, and then we're dividing by the square root of 10 instead. So the square root of 10 is 3.16, so we're going to do 0.7 divided by 3.16. And this time, we had 0.222.

So we can see that if our sample size increases-- so when we increased from 10 to 20-- our standard deviation of the distribution of sample means decreases. So this has been your tutorial on the distribution of sample means. The mean of the distribution of sample means is the same as the population mean. And the standard deviation, we need to use this formula in order to calculate that to find the standard deviation of the distribution of sample means.