## 18.4 Standard errors

As we have seen,
each sample is likely to be different,
so *any* statistic estimated from the sample
is likely to be different for each sample.
This is called *sampling variation*.

**Definition 18.2**Sampling variation refers to how much a sample estimate (a statistic) is likely to vary from sample to sample, because each sample is different.

The value of the sample statistic can vary for every possible sample that we could select, so the actual value of the sample statistic that we observe depends on which sample we have.

That is, all the possible values of sample statistics that we could observe have a distribution (a *sampling distribution*).
Perhaps surprisingly, under certain conditions,
the sampling distribution is a *normal distribution*.

If the sampling distribution is a normal distribution,
it is reasonable to ask
what the value of the *standard deviation*
of that normal distribution is.

Figs. 18.1 and
and 18.2
show that the standard deviation appears to get *smaller*
as the sample sizes get *larger*:
the sample statistics show less variation for larger \(n\).
This makes sense:
*larger* samples
generally produce more precise estimates.
(After all, that’s the advantage of using larger samples:
all else being equal,
larger samples are preferred
as they produce
more precise estimates.)

In other words,
the sample statistic varies *less* in larger samples:
the value of the standard deviation of the sampling distribution
is *smaller* for *larger* samples.
The standard error is
a measure of how precisely
the *sample* statistic
estimates the *population* parameter.

**Example 18.1 (Standard errors) **Suppose the sample proportion
of odd-spins
on the roulette wheel
(Sect. 18.2)
is estimated as \(\hat{p} = 0.51\).
If the standard error was 0.01,
this estimate is relatively precise:
the standard error is very small,
which means
the value of \(\hat{p}\)
is not likely to vary greatly from one sample to the next.
Any single *estimate* of
\(p\) is likely to be close to \(p\).

**Definition 18.3 (Standard error) **A *standard error* is
the standard deviation of the
sampling distribution of a statistic.

*Any*quantity estimated from a sample has a standard error.

To expand:
If every possible sample
(found a certain way, and of a given size)
was found,
and the statistic computed from each sample,
the standard deviation of these estimates
is the *standard error*.

Recall from
Sect. 18.1 that,
for many sample statistics,
*the variation from sample to sample
can be approximately described by a normal distribution*
(the *sampling distribution*)
if certain conditions are met.
Furthermore,
the
*standard deviation of this normal distribution is the standard error*.

Notice that the standard error is a special
type of *standard deviation*;
the variation in a sample estimate *from sample to sample*.

*error*or

*mistake*, or even

*standard*. (For example, there is no such thing as a ‘

*non*-standard error.’)