Overall | Normality | Mean | Standard Deviation | Illustrations
Overall: In these illustrations, read in columns. Each column illustrates a different population.
At the top of each column is the distribution of values in a population. That same distribution is duplicated with the dots throughout all the rest of the figures in that same column.
The second figure in a column has several parts.
One of the curved lines shows the sampling distribution of the sample mean (for simple random samples) for samples of size 2. Under the uniform population (first column) the sampling distribution of the sample mean has a triangular shape. See if you can find that shape.
The other curved line is the normal distribution with the same mean and standard deviation as the sampling distribution of the sample mean. Seeing how close the two curved lines are to each other helps you see whether, for this particular population, the sample size of 2 is large enough that you can use the Central Limit Theorem to approximate the sampling distribution of the sample mean. Not that we'd expect a sample of size 2 to be large enough. But it's interesting to see how close it is.
And, of course, the dots remind you of the population of values that all the samples came from.
Now look down at the subsequent figures in the column. These show the sampling distribution of the sample mean for larger sample sizes. After a certain point, you can't tell the difference between the curve that shows the sampling distribution and the curve that shows a normal distribution with that same mean and standard deviation. At that point, you could definitely use the Central Limit Theorem to approximate the sampling distribution of the sample mean by a normal distribution.
Noticing the mean of the sampling distribution of the sample mean.
In the original population (top of the column) you can estimate where the mean is. Notice that each of the sampling distributions seems to be centered on that same value. That's not just a coincidence. One can prove mathematically that that must be the case. Generally that is done in a post-calculus statistics course -- not MATH 1342. Perhaps these figures will help you understand it without the proof.
Noticing the standard deviation of the sampling distribution of the sample mean.
In the original population (top of the column) you can make a very rough estimate of the standard deviation. Notice that each of the sampling distributions is narrower, so it has a smaller standard deviation than the standard deviation of the original population. Can you estimate the standard deviation for each of these sampling distributions?
Notice that these sampling distributions have smaller and smaller standard deviation as the sample size increases. There is a formula for the standard deviation of the sampling distribution of the sample mean -- sigma divided by the square root of n. It is stated in the conclusion of the Central Limit Theorem and can be proved mathematically. Generally that is done in a post-calculus statistics course -- not in MATH 1342.
Last updated February 25, 2003 . Mary Parker