6.13: Sampling Distribution

Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean.

This text is adapted from Openstax, Introductory Statistics, Section 2.7 Measures of Spread of Data

Consider spinning ten picker wheels and finding the mean of outcomes. This process is repeated, say 20,000 times.

The sample mean obtained for each repetition of the process is plotted, which looks similar to a normal distribution graph.

If the sample size is large, the distribution gets closer to the normal distribution, and the mean of the sample means gets closer to the population mean.

Such distribution of values of a statistic such as mean, variance, or a sample proportion is known as the sampling distribution.

Just like the mean, one can obtain the variance for each sample and plot the frequency distribution, which appears skewed to the right.

Even in this case, if the sample size is large, the mean of the sample variances is close to the population variance.

If one considers the proportion of odd numbers in each sample and plots the graph, the distribution follows approximately a normal distribution pattern.

Similar to mean and variance, if the sample size is large, the mean of the sample proportions is close to the population proportion.