1. Standard errors and the Central Limit Theorem
The Gaussian distribution (also known as the normal distribution) plays an important role in statistics.
You may have noticed that its distinctive bell-shaped curve keeps cropping up in this course.
2. Sampling distribution of mean cup points
Here is the approximate sampling distribution of mean cup points from the coffee dataset. Each histogram shows five thousand replicates, with different sample sizes in each case.
Look at the x-axis labels. You already saw how increasing the sample size results in greater accuracy of estimating the population parameter, so the width of the distribution shrinks as the sample size increases. When the sample size is five, the x-axis ranges from seventy six to eighty six, whereas for a sample size of three hundred and twenty, the range is from eighty one point eight to eighty two point six.
Now look at the shape of each distribution. As the sample size increases, you can see that the shape of the curve gets closer and closer to being a normal distribution. At sample size five, the curve is only a very loose approximation since it isn't very symmetric. By sample size eighty, it is a very reasonable approximation.
3. Consequences of the central limit theorem
What you just saw on the previous slide is, in essence, what the central limit theorem tells us.
The means of independent samples have normal distributions.
Then as the sample size increases, you see two things. The distribution of these averages gets closer to being normal, and the width of this sampling distribution gets narrower.
4. Population & sampling distribution means
Recall the population parameter of the mean cup points. You've seen this calculation before, and its value is eighty two point one five.
We can also calculate summary statistics on our sampling distributions to see how they compare. For each of our four sampling distributions, if we take the mean of our sample means, you can see that we get values that are pretty close to the population parameter that the sampling distributions are trying to estimate.
5. Population & sampling distribution standard deviations
Now let's consider the standard deviation of the population cup points. It's about two point seven.
By comparison, if we take the standard deviation of the sample means from each of the sampling distributions, we get much smaller numbers, and they decrease as the sample size increases. What are these new values?
6. Population mean over square root sample size
One other consequence of the central limit theorem is that if you divide the population mean by the square root of the sample size, you get an estimate of the standard deviation of the sampling distribution.
It isn't exact because of the randomness involved in the sampling process, but it's very close.
7. Let's practice!
Let's explore some sampling distributions.