What Is a Sampling Distribution?

Suppose you randomly sampled 10 women between the ages of 21 and 35 years from the population of women in Houston, Texas, and then computed the mean height of your sample. You would not expect your sample mean to be equal to the mean of all women in Houston. It might be somewhat lower or higher, but it would not equal the population mean exactly. Similarly, if you took a second sample of 10 women from the same population, you would not expect the mean of this second sample to equal the mean of the first sample.

Houston Skyline

Suppose you randomly sampled 10 people from the population of women in Houston, Texas between the ages of 21 and 35 years and computed the mean height of your sample. You would not expect your sample mean to be equal to the mean of all women in Houston.

Inferential statistics involves generalizing from a sample to a population. A critical part of inferential statistics involves determining how far sample statistics are likely to vary from each other and from the population parameter. These determinations are based on sampling distributions. The sampling distribution of a statistic is the distribution of that statistic, considered as a random variable, when derived from a random sample of size $n$. It may be considered as the distribution of the statistic for all possible samples from the same population of a given size. Sampling distributions allow analytical considerations to be based on the sampling distribution of a statistic rather than on the joint probability distribution of all the individual sample values.

The sampling distribution depends on: the underlying distribution of the population, the statistic being considered, the sampling procedure employed, and the sample size used. For example, consider a normal population with mean $\mu$ and variance $\sigma$. Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean for each sample. This statistic is then called the sample mean. Each sample has its own average value, and the distribution of these averages is called the "sampling distribution of the sample mean. " This distribution is normal since the underlying population is normal, although sampling distributions may also often be close to normal even when the population distribution is not.

An alternative to the sample mean is the sample median. When calculated from the same population, it has a different sampling distribution to that of the mean and is generally not normal (but it may be close for large sample sizes).