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Sample Size Simulation

Module by: David Lane

General Instructions

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

Depicted on the top graph is the population distribution. By default it is a uniform distribution (all values are equally likely). The sampling distributions for two different sample sizes are shown in the lower two graphs. The starting values are 2 and 10. By default, the statistic to be computed is the mean, although you can also specify to compute the median.

For both the population distribution and the sampling distribution, the mean and the standard deviation are depicted graphically on the frequency distribution itself. The blue-colored vertical bar below the X-axis indicates the mean value. 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 this simulation, you specify two sample sizes (the defaults are set at N = 2 and N = 10), and then sample a sufficiently large number of samples until the sampling distributions stabilize. Compare the mean and standard deviaiton of the two sampling distributions. Repeat the process a couple times and watch the results. Do you observe a general rule regarding the effect of sample size on the mean and the standard deviation of the sampling distribution?

You may also test the effect of sample size with a normal population or with a different sample statistic (the median).

When you have discovered the rule, go back and answer the questions again.

Step by Step Instructions

  1. With the default setting (uniform population, sample statistic set to mean, sample sizes set at 2 and 10, respectively), click the button "5 Samples" a couple times. Notice how the sample means accumulate at the bottom two graphs. Then click the button "10,000 Samples" multiple times until the two sampling distributions stablize (don't change much in shape with the addition of new samples). Compare the two sampling distributions (mean and standard deviation). How do the means compare? (Don't pay attention to very small differences since they can occur by sampling error.) How do the standard deviations compare?
  2. Find the vertical bar at the right-hand end of the X-axis on the middle graph (N = 2). Click and drag the bar to a position x = 10. When you release the mouse, the area falling to the left of the bar is displayed on top of the graph. (The location of the bar is rounded to the nearest integer.) Determine the probability of getting a sample mean less than 10 for a sample of size 2 and for a sample of size 10? What is the probability of a sample mean being greater than 22 for each sample size? (Hint the probability of being less than 22 is shown. You will have to subtract from 1.0)
  3. Set sample sizes to be 10 and 25, respectively. Sample 50,000 times for each sample size. For each of the resulting distributions, calculate the probability of a sample mean falling within an interval that encloses the population mean 16. For example, use the interval betwen x = 12 and x = 20. To find the probability of a sample mean being in the interval, find the proportion of means below the low end of the interval (12) and subtract this from the proportion of means below the high end of the interval (16).Which sample mean is more likely to be close to the population mean, the one with the smaller sample size or the one with the larger sample size?
  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 mean and standard deviation associated with each sample size on a piece of paper. Answer the following question: Does sample size significantly affect the mean of the distribution of sample means? Does sample size significantly affect the standard deviation of the distribution of sample means??
  5. Select "Median" as the sample statistic and repeat the above steps. Does sample size affect the sampling distribution of the median (mean and standard deviation)?

Summary

The mean and standard deviation of the distribution of sample means are systematically related to those of the population. The mean of the sampling distribution of the mean is the population mean. The mean of the sampling distribution of the median is the population median. As sample size increases the standard deviation of the sampling distribution of the mean (also called the standard error of the mean) decreases. The same is true for the standard error of the median.

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