>Rules of Probability>What Are Contingency Tables Used For?

What Are Contingency Tables Used For?

You can use a contingency table to organize information. A contingency table gives you an overview of how much you’ve got of at least two different things at the same time. Anything you can organize in a contingency table can also be presented in a Venn diagram, and you can often choose which one to use. Take a look at this table:

Contingency table of boys and girls and whether they like snickers

By reading off the information in the table, you learn this:

You’ve asked boys and girls whether or not they like the chocolate “Snickers”. They have either answered that they like Snickers or that they don’t.

43 boys like Snickers, while 8 don’t. 30 girls like Snickers, while 19 don’t.

In total, 73 boys and girls like Snickers, and 27 don’t.

You’ve asked 100 people in total, 51 boys and 49 girls. The number of girls and boys that have been asked is roughly equal.

In the table, the answers are divided between boys’ answers and girls’ answers. From this you can see if there’s a difference between how many girls like Snickers and how many boys do. You couldn’t have done that if you didn’t divide the answers between boys’ answers and girls’ answers.

That means you’ve gotten more information by making this partition and setting up a contingency table. From the table it seems that boys tend to like Snickers more than girls do.

Think About This

Can you be absolutely certain that boys like Snickers more than girls do, from this table?

No, you can’t. The reason is that you have to ask a massive number of people before you can say anything for certain about these kinds of matters. You should at least ask $1000$ people to get a decent impression of who likes “Snickers” more.

You also have to make sure you ask similar amounts of boys and girls. If you ask $900$ boys and $100$ girls, that’s not good enough to be sure.

A contingency table is a table that shows how elements are distributed between different groups. This kind of table is easy to read and easy to make when you know how.

Theory

Contingency Table

A contingency table presents a good overview of probabilities and data from complex experiments. The table is easy to read and extract information from, for further calculations. In addition to that, contingency tables are useful when you’ve been given some values, but not all of them. The missing ones can then be obtained by using the values you’ve already got in the contingency table.

The contingency table below shows you how many boys and girls go skiing and how many don’t. The row marked with “sum” shows the total number of boys (70) and girls (95) who participated in the survey.

The position in the bottom right corner shows us the total number of participants ( $70 + 95 = 165$ ). We add the numbers up horizontally. The column with “sum” at the top shows how many go skiing (146) and how many don’t (19). This also adds up to 165 when we add the numbers in the column together vertically.

In the column marked with “boy”, you can see that 55 go skiing and 15 don’t. This adds up to $55 + 15 = 70$ boys. In the column marked with “girl”, you can see that 91 girls go skiing and 4 don’t. This adds up to $91 + 4 = 95$ girls.

Contingency table of boys and girls and whether they go skiing

Example 1

Contingency Table

When giving birth, the probability of the baby being a girl is $0.486$ , and the probability of it being a boy is $0.514$ . The probability of a baby being born colorblind is $0.044$ . Make a contingency table that shows the probabilities of having a boy or girl that is colorblind or not.

From the text you can see that the events are sex and colorblindness. That gives you the following possibilities:

The event regarding the sex of the baby has only two possible outcomes, Boy or girl. Call the outcome that a baby is a boy $Y$ , and the event that the baby is a girl $\overset{}{Y}$
The event regarding whether the baby is colorblind or not also has only two possible outcomes, Colorblind or not colorblind. Call the outcome that a baby is colorblind $C$ , and the event that the baby is not colorblind $\overset{}{C}$

You already know the probabilities of both boy and girl, but you still only have the probability of the baby being colorblind. That means you have to find the probability of the baby not being colorblind. Colorblind and not colorblind are complementary events, which gives you

P (\overset{}{C}) + P (C) = 1

We find the probability:

\begin{array}{l} P (\overset{}{C}) + 0.044 & = 1 \\ P (\overset{}{C}) & = 1 - 0.044 \\ = 0.956 \end{array}

So far you can fill in the contingency table in this way:

	$Y$	$\overset{}{Y}$	Sum

$\overset{}{C}$	?	?	$0.956$
$C$	?	?	$0.044$

Sum	$0.514$	$0.486$	1

Now you just have to find the values to put in place of the question marks. As we have two events occurring together, this is a case of multiple trials, and we have to use the chain rule.

Here are the calculations you have to do:

\begin{array}{l} P (\overset{}{C} ∣ Y) & = 0.514 \cdot 0.956 \\ = 0.491 \\ P (\overset{}{C} ∣ \overset{}{Y}) & = 0.486 \cdot 0.956 \\ = 0.465 \\ P (C ∣ Y) & = 0.514 \cdot 0.044 \\ = 0.023 \\ P (C ∣ \overset{}{Y}) & = 0.486 \cdot 0.044 \\ = 0.021 \end{array}

These numbers are the ones you can switch the question marks for in the contingency table. Then it’ll look like this:

	Boy	Girl	Sum

Not colorblind	$0.491$	$0.465$	$0.956$
Colorblind	$0.023$	$0.021$	$0.044$

Sum	$0.514$	$0.486$	1

Now you can read the different probabilities of boy and girl, colorblind and not colorblind off the contingency table.

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