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The Effect of Sample Size on the Distribution of the Sample Means

Soc 5183 Intermediate Statistics Spring 2002 Example:

This example has been put together to demonstrate for you the effect of different sample sizes on how closely the mean of all sample means approximates the population mean for a given sample size. Table 1 shows the population statistics (mean, standard deviation, median and mode). Each of the following tables shows the same information for samples of 5, 10 and 20. I have also included the minimum and maximum sample means in each of these distributions.

We randomly selected samples of size 5, 10 and 20 from the GSS data set provided with Adventures in Social Research. The Variable is Age. The following tables present the information for samples of size 5, of 10 and of 20. The first table presents the information for the 'population.'

In the following you will find tables showing the parameters for each of the distributions -- 'population,' samples of size 5, 10 and 20. I have also included figures to show the shape of each distribution. Please compare the Mean for the population (Table 1) with each sample mean (Tables 2, 3, and 4). Then compare the variability of each of the samples with one another and with the population. You should also compare each of the distributions with one another and with the population.

Table 1 The Mean Age of the Population

N (Size of Population) / 1497
Missing / 3
Mean / 44.54
Median / 42.00
Mode / 38
Std. Deviation / 16.69
Minimum / 18
Maximum / 89

Figure 1
Distribution of Respondents by Age
Population (GSS 1996)

As you examine the histogram in Figure 1, note the shape of the distribution. You will find imposed on this histogram the outline of the normal curve. How does the distribution of age in this population compare to a normal curve?

Table 2 Mean of Means: Samples of Size 5

N (number of samples) / 126
Missing / 0
Mean / 46.95302
Median / 46.40000
Mode / 48.800
Std. Deviation / 7.93214
Minimum Sample Mean / 25.80
Maximum Sample Mean / 69.60

Figure 2. Distribution of Means: Samples of Size 5 drawn from GSS 1996 (Mean age in Sample)

Notice the number of samples and the general shape of the distribution of the sample means. You can see the normal curve imposed on this distribution for reference. Although the sample size is only 5, the distribution of sample means is beginning to approach normality. Once we get more samples in here we will see the areas filled in and the distribution of sample means begin to come closer to the normal curve. Take note of the standard deviation of this distribution and also of the maximum and minimum values reported in the table.

Table 3 Mean of Means Samples of 10

N (number of samples) / 104
Missing / 0
Mean / 44.7497
Median / 43.6500
Mode / 42.00
Std. Deviation / 6.850
Minimum Mean of Means / 31.80
Maximum Mean of Means / 63.50

Figure 3.
Distribution of Means, Samples of Size 10drawn from GSS 1996 (Mean age in Sample)

Figure 3 shows the distribution of sample means for a sample size of 10. Notice that we have only 94 samples in this distribution. You can also see what the normal curve looks like imposed on the figure. Note the Mean of means and standard deviation for this collection of samples. How do they compare to the samples of size 5 and to the population values? What seems to be happening to the standard deviation of the samples (the standard error of a mean)?

Table 4 Mean of Means Sample of 20

N (Number of samples) / 80
Missing / 0
Mean / 46.225
Median / 46.025
Mode / 48.50
Std. Deviation / 3.9856
Minimum Mean of Means / 39.00
Maximum Mean of Means / 55.80

Figure 4. Distribution of Means, Samples of Size 20 drawn from GSS 1996 (Mean age in Sample)

Figure 4 shows the distributions of means for samples of size 20. There are only 70 samples in this distribution (students got tired when it came to doing this part of the assignment and made a lot of errors, so I am missing the samples). Notice that the sample appears to be bi-modal. How does the mean of these samples compare with the population mean? What do you notice about the variation in the samples -- has it gotten larger or smaller as we increased the sample size? What do you think is happening to create that pattern?

Now you have had a chance to study the effects of sample size on the mean of means. Make some notes about the difference of the mean of means for a given sample size from the population mean. Do the same for the standard deviations. Comment on these differences. What do the seem to be showing you? Do the differences of the mean of means get greater or less as the size of the sample on which they are based increases? What sort of pattern does the standard deviation seem to follow?

Class Home Page
Topic 2: Computers and computer applications / Topic 3: Samples and Sampling Distributions / Topic 4: Hypothesis Testing / Topical Outline of the course
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This page last updated on January 15, 2002. All contents copyright © 1998, 1999, 2000, 2001, 2002 by Richard H. Anderson, the Department of Sociology, University of Colorado at Denver, Denver, Colorado.