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What Is a Sampling Distribution?
When we have a random variable
and take simple random samples1 from
that random variable, we often introduce additional random variables
that are related to the original random variable. The values of the
related random variable depend on the particular sample. The
distribution of such a related random variable is called a sampling distribution.
Example: Suppose we start with
a random variable Y which has a normal distribution. If we take a
simple random sample y1, y2, ...,
yn of size n from Y, we can form the sample mean y-bar of this sample:
y-bar is just the usual arithmetic mean:
y-bar = (1/n)(y1+ y2,+...,+ yn)
Now imagine all possible random samples of
size n from Y. For each one, we have a (typically different) value of
y-bar. In other words, we have a new random variable. In keeping with
the usual notation in this website, we will use capital letters to
distinguish a random variable from a value of that random variable.
Thus we will call this new random variable Y-bar. The
distribution of Y-bar is a sampling distribution.
Note that in this example, we had a fixed
value of the sample size n. So it would be better to use the notation
(Y-bar)n to make this clear.
1. We can also talk about
sampling distributions when we use a probability samping scheme other
than simple random samples. The idea is similar, so this page restricts
to simple random samples for simplicitiy.