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# CLT Simulation

Module by: David Lane. E-mail the author

## General Instructions

This simulation demonstrates the effect of sample size on the shape of the sampling distribution of the mean.

Depicted on the top graph is the population which is sometimes referred to as the parent distributoin. Two sampling distributions of the mean, associated with their respective sample size will be created on the second and third graphs.

For both the population distribution and the sampling distributions, their mean and the standard deviation are depicted graphically on the frequency distribution itself. The blue-colored vertical bar below the X-axis indicates where the mean value falls. The red line starts from this mean value and extends one standard deviation in length in both directions. The values of both the mean and the standard deviation are also given to the left of the graph. Notice that the numeric form of a property matches its graphical form in color. In additon, the skew and the kurtosis of each distribution are also provided to the left. These two variables are determined by the shape of distribution. The skew and kurtosis for a normal distribution are both 0.

In this simulation, you need to first specify a population (the default is uniform distribution). Take note of the skew and kurtosis of the population. Then pick two different sample sizes (the defaults are N=2N2 and N=10N10), and sample a sufficiently large number of samples until the sampling distributions change relatively little with additional samples (about 50,000 samples.) Observe the overall shape of the two sampling distributions, and further compare their means, standard deviations, skew and kurtosis. Change the sample sizes and repeat the process a few times. Do you observe a general rule regarding the effect of sample size on the shape of the sampling distribution?

You may also test the effect of sample size with populations of other shape (uniform, skewed or customed ones).

## Step by Step Instructions

1. With the default setting (uniform population, sample sizes set at 2 and 5, respectively), click the button "5 Samples" a couple of times. Notice how the sample means accumulate at the bottom two graphs. Then click the button "5000 Samples" multiple times until the total number of samples exceeds 50,000. Observe the shape of the two distributions, and compare their variance, skew and kurtosis. Write these numbers down on a piece of paper for future reference. (Square the standard deviation to get the variance).

2. Set the sample sizes to be 10 and 15, respectively. Sample 50,000 times for each sample size. Observe the shape of the two distributions, and compare their variance, skew and kurtosis. Write them down for future reference. Repeat for samples size 25.

3. Review the data you have written down. Answer the following question: How does sample size affect the shape of the sampling distribution of the mean? What is the effect of sample size on the variance. What is the effect on the variance of doubling the sample size (Compare N=5N5 to N=10N10). What is the effect of tripling the sample size? How does sample size affect skew and kurtosis?

4. Set the population to be "Normal", set the sample size to be 2, 5, 10, 15, 25, respectively. Sample 50,000 times in each case. Write down the variance associated with each sample size on a piece of paper. Does the rule you found with the uniform population hold here?

5. Set the population to be "Skewed" and repeat steps 1-3.

6. Set the population to be "Custom", click and drag mouse in the top graph to construct a distribution of your own., then repeat steps 1-3.

## Summary

Skew and kurtosis are statistics that reflect the shape of a distribution. The shape of a sampling distribution of the mean is affected by the sample size. As sample size increases, the sampling distribution of the mean approaches a normal distribution. This is an important part of the "Central Limit Theorem".

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