4.1.3 - Impact of Sample Size

There is an inverse relationship between sample size and standard error.  In other words, as the sample size increases, the variability of sampling distribution decreases.

Also, as the sample size increases the shape of the sampling distribution becomes more similar to a normal distribution regardless of the shape of the population.

Example: Mean NFL Salary

The built-in dataset "NFL Contracts (2015 in millions)" was used to construct the two sampling distributions below. In the first, a sample size of 10 was used. In the second, a sample size of 100 was used.

Sample size of 10:

200 Show Data Table Generate 100 Samples Choose samples of size n = samples = 5000 mean = 2.195 std. error = 0.936 Generate 10 Samples Generate 1 Sample Left Tail Two - Tail Right Tail NFL Contracts (2015 in millions) Sampling Dotplot of Mean Edit Data Upload File Change Colu Reset Plot 10 150 100 50 0 1 2 2.195 3 4 5 6 7 8 9 Generate 1000 Samples

 

Sample size of 100:

120 Show Data Table Generate 100 Samples Choose samples of size n = samples = 5000 mean = 2.236 std. error = 0.296 Generate 10 Samples Generate 1 Sample Left Tail Two - Tail Right Tail NFL Contracts (2015 in millions) Sampling Dotplot of Mean Edit Data Upload File Change Colu Reset Plot 100 100 80 60 40 20 0 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 Generate 1000 Samples 2.236

With a sample size of 10, the standard error of the mean was 0.936. With a sample size of 100 the standard error of the mean was 0.296. When the sample size increased the standard error decreased. 

Also know that the population was strongly skewed to the right. With the smaller sample size, the sampling distribution was also skewed to the right, though not as strongly skewed as the population. With the larger sample size, the sampling distribution was approximately normal.

Example: Proportion of College Graduates Section

The built-in dataset "College Graduates" was used to construct the two sampling distributions below. In the first, a sample size of 10 was used. In the second, a sample size of 100 was used.

Sample size of 10:

1200 1400 Edit Proportion Generate 100 Samples Choose samples of size n = samples = 5000 mean = 0.273 std. error = 0.143 Generate 10 Samples Generate 1 Sample Left Tail Two - Tail Right Tail College Graduates Sampling Dotplot of Proportion Edit Data Reset Plot 10 1000 800 600 400 200 0 0.1 0.0 0.2 0.3 0.4 0.5 0.7 0.6 0.8 0.9 0.273 Generate 1000 Samples

 

 

Sample size of 100:

Edit Proportion Generate 100 Samples Choose samples of size n = samples = 5000 mean = 0.275 std. error = 0.044 Generate 10 Samples Generate 1 Sample Left Tail Two - Tail Right Tail College Graduates Sampling Dotplot of Proportion Edit Data Reset Plot 100 400 300 200 100 0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.275 Generate 1000 Samples

With a sample size of 10, the standard error of the mean was 0.143. With a sample size of 100 the standard error of the mean was 0.044. As the sample size increased the standard error decreased.

Also note how the shape of the sampling distribution changed. With the smaller sample size there were large gaps between each possible sample proportion. When the sample size increased, the gaps between the possible sampling proportions decreased. With the larger sampling size the sampling distribution approximates a normal distribution.