## 14.2 Odds

Consider again the **small** kidney stone data
(Table 14.1).

For *Method A*,
the sample contains 81 successes and 6 failures.
Apart from proportions and percentages,
another way to numerically summarise this information is to see that
there are \(81\div 6 = 13.5\) *times* as many successes than failures in the sample.

In other words,
for small kidney stones,
the **odds** of success for Method A is 13.5 (in the sample).
The sample odds is a *statistic*,
and the population odds is a *parameter*.

**Definition 14.3 (Odds) **The **odds** are the proportion (or percentage, or number) of times that an event *happens*,
divided by the proportion (or percentage, or number) of times that the event does *not happen*:

\[ \text{Odds} = \frac{\text{Proportion of times that something happens}} {\text{Proportion of times that something doesn't happen}}, \] or (equivalently)

\[ \text{Odds} = \frac{\text{Number of times that something happens}}{\text{Number of times that something doesn't happen}}. \] The*odds*show how many

*times*an event

*happens*compared to

*not happening*. Alternatively, it is how many times the event

*happens*for every 100 times that it does

*not*happen.

Notice that, when computing odds,
we divide the relevant number by the *remaining number*,
which is different than how percentages are computed.

percentages and proportions,
we divide the relevant number by the *total number* relevant to the context.

Software usually works with odds rather than percentages
(for good reasons that we will not delve into).
However,
understanding *how* software computes the odds is important.

Software usually computes odds as comparing either

- Row 1 to Row 2; or
- Column 1 to Column 2.

**Example 14.1 (Interpreting odds) **For the *small* kidney stone data,
the odds of a success for Method A is
\(81\div6 = 13.5\) (in the sample).
This can be interpreted as:

- There are \(13.5\)
*times*as many successes as failures (in the sample); - There are \(13.5\times 100 = 1350\) successes for every 100 failures (in the sample).

*far*more common than failures, for small kidney stones using Method A.

**Think 14.3 (Odds)**What are the odds of finding a

*failure*for Method A? How is this value interpreted?

**Example 14.2 (Odds) **Suppose that about 67% of students at a particular university were female.
The *population* odds of finding a female is about
\(67 / (100 - 67) = 2.03\):
about twice as many females are students as non-females.

*sample*odds of finding a female in this class is \(18/5 = 3.60\). Another classes had 16 females and 9 non-females. The

*sample*odds of finding a female in this class is \(16/9 = 1.79\).

**Example 14.3 (Computing odds) **Consider again the **small** kidney stone data
(Table 14.1).
The odds of a success using *Method B* can also be found
(Table 14.1):

\[\begin{align*} &\text{Odds}(\text{Success with Method B})\\ = &\frac{\text{Number of successes for Method B}}{\text{Number of failures for Method B}} =\frac{234}{36} = 6.52. \end{align*}\] Working with the proportions (or percentages) (Table 14.2) rather than the numbers, the same value results:

\[\begin{align*} & \text{Odds}(\text{Success with Method B})\\ = &\frac{\text{Percentage of successes for Method B}}{\text{Percentage of failures for Method B}} =\frac{86.7}{13.3} = 6.52. \end{align*}\]When interpreting odds:

- When the odds are
*greater*than one: the event is*more*likely to happen than to not happen. - When the odds are
*equal to*one: the event is just as likely to happen as it is to not happen. - When the odds are
*less*than one: the event is*less*likely to happen than to not happen.