## 25.3 Sampling distribution: Comparing odds

From the numerical summary table
(Table 25.2),
the odds of a student eating most meals *off-campus* is:

- \(26\) for students
*living with their parents*. - \(4.375\) for students
*not living with their parents*.

So the OR of eating most meals *off-campus*,
comparing
students living with parents to students *not* living with parents,
is \(26 \div 4.375 = 5.943\).
The odds are different in each group,
and hence the OR is not one.
The OR means that the odds of eating most meals off-campus
is 5.943 times larger
for students living *with* their parents.

Of course,
every sample of students is likey to be different,
so the OR *varies* from sample to sample,
so there is *sampling variation*.
This means that the odds ratio has a *sampling distribution*
and a *standard error*.

Unfortunately,
the sampling distribution of the sample OR
is not a normal distribution.^{8}
Fortunately,
a simple transformation to the sample OR has a normal distribution.
For this reason,
we will use software output for finding the CI for the odds ratio,
and not discuss the sampling distribution directly.
In other words,
we will rely on software to find CIs for odds ratios.

For those who want to know (this is

*optional*): The OR is only defined for*non-negative*values so a normal distribution is inappropriate. However, the*logarithm*of the OR has an approximate normal distribution under certain conditions.↩︎