University of Minnesota, Twin Cities     School of Statistics     Stat 5101     Rweb

The Central Limit Theorem (Stat 5101, Geyer)

Contents

General Instructions

To do each example, just click the "Submit" button. You do not have to type in any R instructions (that's already done for you). You do not have to select a dataset (that's already done for you).

Normal Population Distribution

If we have a random sample of size n from a normally distributed population, we know the sampling distribution of the sample mean is exactly normal with

E(sample mean) = population mean
and
sd(sample mean) = (population standard deviation) / sqrt(n)

The simulation below makes a random sample of size n from a normal population and calculates the sample mean. It does this repeatedly nsim times, thus obtaining a random sample from the sampling distribution of the sample mean. The histogram of sample mean values is plotted with a superimposed normal density curve that is the theoretical sampling distribution of the sample mean.

Gamma Population Distribution

If we have a random sample of size n from a non-normally distributed population, we know the sampling distribution of the sample mean is not exactly normal, only approximately normal for large sample sizes, but we do know the mean and sd are exactly

E(sample mean) = population mean
and
sd(sample mean) = (population standard deviation) / sqrt(n)

The simulation below makes a random sample of size n from a gamma population and calculates the sample mean. It does this repeatedly nsim times, thus obtaining a random sample from the sampling distribution of the sample mean. The histogram of sample mean values is plotted with a superimposed non-normal density curve that is the theoretical sampling distribution of the sample mean and the normal density curve (red) that is the approximately sampling distribution for large sample sizes. For comparison, the population density is shown in another plot.

Bimodal Skewed Population Distribution

If we have a random sample of size n from a non-normally distributed population, we know the sampling distribution of the sample mean is not exactly normal, only approximately normal for large sample sizes, but we do know the mean and sd are exactly

E(sample mean) = population mean
and
sd(sample mean) = (population standard deviation) / sqrt(n)

The simulation below makes a random sample of size n from a bimodal skewed population and calculates the sample mean. It does this repeatedly nsim times, thus obtaining a random sample from the sampling distribution of the sample mean. The histogram of sample mean values is plotted with a superimposed non-normal density curve that is the theoretical sampling distribution of the sample mean and the normal density curve (red) that is the approximately sampling distribution for large sample sizes.