Concept of a Sampling Distribution

Statistical inference involves using sample data to draw conclusions about a population. For example, if we want to say something about the average income of all St. John's residents (our population), it would be difficult and time-consuming to ask every resident what their income is. Instead we could take a random sample of 100 (for example) residents, and use the average income of this group to estimate the population average income. If we let the population mean (or average) be denoted by $\mu$ (mu), and the sample mean be denoted by $\bar{x}$, then $\bar{x}$ would be used as our estimate of $\mu$. But to use $\bar{x}$ to make some inferential statement about $\mu$, we need some idea of how $\bar{x}$ behaves, in general, as an estimate. One way to do this is to consider the sampling distribution of the estimate. As we will briefly illustrate here, the concept of a sampling distribution is a long-run, or repeated sampling one.

Although we will describe the sampling distribution of $\bar{x}$, the procedure applies to any statistic, or quantity calculated from sample data: the sample median, sample standard deviation, etc.

Suppose we have a huge pot of numbers that represents that population distribution of interest. This may be a normal distribution, or something with a very non-normal distribution.

1.

Draw a random sample from the pot, and find the sample mean. Call it $\bar{x}_{1}$.

2.

Toss the values back in the pot. Pull out another sample, find its mean. Call it $\bar{x}_{2}$.

3.

Repeat these steps 1000's of times, get 1000's of $\bar{x}$'s.

4.

Draw a histogram (barchart) of these $\bar{x}$ values.

The histogram you get in step 4 is the sampling distribution of $\bar{x}$. It is this distribution that is used to derive confidence intervals and test statistics used hypothesis testing.

A very important result in statistics comes from the idea of sampling distributions. The result is the Central Limit Theorem:

Central Limit Theorem: If x is a random variable that comes from a distribution with mean $\mu$ and standard deviation $\sigma$, then for a ``large enough'' sample size n, $\bar{x}$ is approximately normally distributed with mean $\mu$ and standard deviation $\sigma/\sqrt{n}$.

This tells us that, even if we're dealing with a complicated population distribution, the sample mean still behaves as if comes from a normal distribution.

So, how large does the sample size have to be? The rule of thumb is that we need $n \geq 30$ in order for the Central Limit Theorem to hold.