As the sample size increases, the standard deviation of the sampling distribution

This simulation simulates the sampling distributions from a variety of population distributions for a selection of statistics.

Basic Instructions:

1. Use the top-left menu to select the population distribution. The population mean and standard deviation are displayed.
2. Use the middle-left menu to select the sample size.
3. Use the bottom-left menu to select the sample statistic to be observed.
4. Click on the "1 Sample" button a few times to see what is happening one sample at a time. Then speed up the generation of the sampling distribution by clicking on the "100 Samples" button.

The three menus yield 210 different combinations. In order not to get lost, it is important to have some strategies. Here are a few suggestions of demonstrations to be done with this simulation.

Some Suggested Demonstrations:

1. Central Limit Theorem: Observe the sampling distribution of the mean for a number of different, very non-normal population distributions. The sampling distribution of the mean always approaches a normal distribution.
2. Sample Size: Use different sample sizes for a given population distribution to observe that the sampling distribution for the mean narrows as the sample size increases. The standard deviation of the sampling distribution of the mean will equal the population standard deviation divided by the square root of the sample size.
3. Sample Size n = 2: Compare the sampling distribution for n = 1 (which simply reproduces the population distribution) with the sampling distribution for n = 2. Even an average of only two observations appreciably improves the precision of the sample estimate of the population mean.
4. Mean vs. Median: For a given population distribution and sample size, compare the sampling distributions of the mean and the median. Do they both estimate the population mean? Which distribution is narrower and hence more accurate?
5. Standard Deviation dividing by n-1 vs. n: For a fixed population distribution and sample size (a moderate size works best), the sampling distribution using n as the divisor for the variance or standard deviation produces a biased estimate that slightly underestimates the true population standard deviation. The sampling distribution for the standard deviation when dividing by n-1 yields an unbiased estimate (i.e., the mean of the sampling distribution of the standard deviation using n-1 estimates the true population standard deviation).

Technical Notes:

The distributions are across the integers from 0 to 32. The depicted distributions are technically beta-binomial distributions with parameters n = 32, alpha, and beta. Beta-binomial distributions with alpha = beta are symmetric. Values of alpha = beta = 10 approximates a normal distribution. Values of alpha = beta = 1 yield a normal distribution. The extreme skewed distribution has alpha = 1 and beta = 5, while the less extreme "partial" skew distribution has alpha = 3 and beta = 10. Finally, values of alpha less than 1.0 produce bimodal distributions; in this case, alpha = beta = 0.5.

Distribution of Normal Means with Different Sample Sizes

Initializing live version

Samples of a given size were taken from a normal distribution with mean 52 and standard deviation 14. The distribution of sample means for samples of size 16 (in blue) does not change but acts as a reference to show how the other curve (in red) changes as you move the slider to change the sample size. Distributions of sample means from a normal distribution change with the sample size. This Demonstration lets you see how the distribution of the means changes as the sample size increases or decreases.

Snapshots

Details

The population mean of the distribution of sample means is the same as the population mean of the distribution being sampled from. Thus the mean of the distribution of the means never changes. The standard deviation of the sample means, however, is the population standard deviation from the original distribution divided by the square root of the sample size. Thus as the sample size increases, the standard deviation of the means decreases; and as the sample size decreases, the standard deviation of the sample means increases.

Reference:

Michael Sullivan, Fundamentals of Statistics, Upper Saddle River, NJ: Pearson Education, Inc., 2008 pp. 382–383.

The central limit theorem states that the sampling distribution of the mean approaches a normal distribution, as the sample size increases. This fact holds especially true for sample sizes over 30.

Therefore, as a sample size increases, the sample mean and standard deviation will be closer in value to the population mean μ and standard deviation σ .

The central limit theorem tells us that no matter what the distribution of the population is, the shape of the sampling distribution will approach normality as the sample size (N) increases.

This is useful, as the research never knows which mean in the sampling distribution is the same as the population mean, but by selecting many random samples from a population the sample means will cluster together, allowing the research to make a very good estimate of the population mean.

Thus, as the sample size (N) increases the sampling error will decrease.

• As the sample size increases, the distribution of frequencies approximates a bell-shaped curved (i.e. normal distribution curve).

• Sample size equal to or greater than 30 are required for the central limit theorem to hold true.

• A sufficiently large sample can predict the parameters of a population such as the mean and standard deviation.

How to reference this article:

How to reference this article:

McLeod, S. A. (2019, Nov 25). What is central limit theorem in statistics? Simply psychology: https://www.simplypsychology.org/central-limit-theorem.html

How to reference this article:

How to reference this article:

McLeod, S. A. (2019, November 25). What is central limit theorem in statistics? Simply Psychology. www.simplypsychology.org/central-limit-theorem.html

What happens to the standard deviation as the sample size increases?

What happens to the sampling distribution when the sample size increases?

How does sample size affect the standard deviation of a sampling distribution?

What increases when sample size increases?