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Distribution of Sample Means

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Hi, this tutorial covers the distribution of sample means. So let's start by defining that term. So the distribution of sample means is the distribution of the means of all possible samples for a certain size n. All right, so let's construct a distribution of sample means for a simple population, 1, 2, 3, 4, so where these are the only four elements possible in the population, where n equals 2, the order of sampling does not matter, and the sampling is done without replacement.

OK, so let me construct it here. Let's, first of all, think about all of the possible samples of size n equals 2. So we know that for sure, 1, 2, is part of sample is one of the possible samples, 1, 3, 1, 4, 1-- excuse me. 2, 3, 2, 4, and 3, 4.

Since the order doesn't matter, and we have sampling without replacement, 1, 2 will be the same as 2, 1. 1, 1 won't be possible because we're sampling without replacement. So now that we have all of the possible samples of size 2, let's go ahead now and calculate x bar for each of those samples.

So if we're thinking about the sample mean of 1 and 2, that's going to end up being 1.5. OK, 1 and 3 is going to have a sample mean of 2. 1 and 4 we'll have a sample mean of 2.5. 2 and 3 will have a sample mean also of 2.5. 2 and 4 we'll have a sample mean of 3. And 3 and 4 will have a sample mean of 3.5.

OK, so now if we want the distribution of sample means, what we do-- I think it's easiest to make a dot plot in this case. These are going to represent my x bar values. So let's start at 1, and then 2, 3, 4. And then I'm going to display each of these possible sample means.

So 1.5 would be here. 2 would be here. Now, 2.5, I have two possible ways of getting 2.5, so I have 2.5 here and here. 3 would be here. And a sample mean of 3.5 would be here.

So this represents a very simple distribution of sample means. This is kind of how-- the thinking behind how to create one of these sample-- these distributions of sample means. All right, so now that we kind of how to make it and how it's defined, let's think about some different features of the distribution of sample means.

Now, we know for any distribution, it's important to consider the distribution's center, spread, and shape. So we're going to address all three of those things for the distribution of sample means. All right, so let's start with the center.

The mean of the population is equal to the mean of the distribution of sample means. OK, so the mean of the distribution of sample means gets notated as mu sub x bar. So it's basically your average sample mean.

Now, that's equal to the mean of the population, which we know is just mu. So this is an important relationship. The mean of the distribution of sample means is equal to the mean of the population means. That's how the center is Defined.

OK, now, let's move to spread. A common way of measuring the spread of a distribution is to use the standard deviation. So we're going to-- let's start by defining the standard deviation of the distribution of sample means. And what it is, it's a measure calculated by taking the population standard deviation divided by the square root of n.

So if we kind of put that into symbolic form, this would be sigma sub x bar. This is the standard deviation of the distribution of sample means. So sigma sub x bar is the standard deviation of the sample means. And what that equals is the population standard deviation, which is defined as just sigma divided by the square root of n. So this is a way of measuring the spread of the distribution of sample means.

And then to define standard error, a standard error is the standard deviation of a sample statistic. So sigma sub x bar, it can also be known as a standard error. So the standard error for the distribution of x bar is sigma over the square root of n.

All right, so we've looked at center, we've looked at spread, the last thing we'll look at is shape. And we usually like to have a normal distribution. So the distribution of sample means is approximately normal as long as the conditions of the central limit theorem are met. And the central limit theorem, you need finite population, mean, and standard deviation, so they do need to be finite mean and standard deviation, and n needs to be sufficiently large.

So you'll notice that in that sampling distribution that I created, this one, we didn't really get a normal distribution here because n was only 2. So n was not sufficiently large for the central limit theorem to be met. But a lot of times, when you're dealing with a distribution of sample means, you're going to be taking a much larger sample. So as long as n is sufficiently large, usually n and is greater than or equal to 30, that constitutes a sufficiently large, and then you can use then the central limit theorem. So again, as long as the central limit theorem is met, the distribution of sample means is approximately normal.

So just to recap, the center of the distribution of sample means is the same, so sigma-- excuse me-- mu sub x bar is equal to mu sub x bar is equal to just mu. Sigma sub x bar, measuring the spread, is equal to sigma over the square root of n. And then the shape is going to be approximately normal if the central limit theorem-- the conditions of the central limit theorem are met. All right, so this has been your tutorial on the distribution of sample means. Thanks for watching.