Let's start with the default case of a normal distribution for

the population with mean zero and standard deviation 20.

Let's take samples of, let's say, size 45 from this population and

what we can see here is that each one of these dot plots show us one

sample of 45 observations from the normal population.

We can see that the centers of each one of these samples is close to 0,

though not exactly 0.

And we can also see that the sample mean varies from one sample to another.

Since these are random samples from the population, each time we reach out to

the population and grab 45 observations We may not be getting the same sample,

in fact we will not be getting the same sample and

therefore the for each samples are slightly different.

The standard deviation of each one of these samples should be roughly equal to

the population standard deviation because after all each one of these samples

are simply a subset of our population We have illustrated 8 of the first

samples here, but we are actually taking 200 samples from the population.

We can make this a very large number, say 1,000 samples from the population.

And what we have at the very bottom is basically our sampling distribution.

Each one of the sample means, once calculated, get dropped to the lower plot.

And what we're seeing here is a distribution of sample means.

Since we saw that the sample means had some variability among them, the sampling

distribution basically illustrates for us what this variability look like.

The sampling distribution, as we expected, is.

Looking just like the population distribution.

So nearly normal.

And the center of the sampling distribution so that is the mean of

the means is close to the true population mean of 0.

However one big difference between our population distribution up top And

our sampling distribution at the bottom is the spread of these distributions.

The sampling distribution at the bottom is much skinnier

than the population distribution up top.

And if you think about it, while the standard deviation

of the population distribution is 20, the standard error.

So the standard deviation of the sample means, is only 2.93.

The reason for this is that while individual observations can be very

variable, it is unlikely that sample means are going to be very variable.

So if we want to decrease the variability of the sample means,

what that means is you're taking samples that have more consistent means.

In order to do that we would want to increase our sample size.

Let's say that we increase our sample size all the way to 500.

All right, so what we have here is again our same population distribution.