Two-Way Frequency Table

Welcome back to Statistics! Recall that a regular frequency table looked at how often one set of data occurs. We know that compound frequency tables compounds the frequency down the table. We have histograms and frequency distributions to graphically show frequency data.

That wasn't enough frequency nor table for math, so scientists devised the two-way frequency table. More table, more frequency, more math homework for you.

A two-way frequency table is just a frequency table for two sets of data. One set will be separated by columns and the other set will be separated by rows.

$$\begin{array}{|c|c|c|} \hline &\text{Option A}&\text{Option B}\\ \hline \text{Option }1\\ \hline \text{Option }2\\ \hline \end{array}$$

An Example

Suppose you want to see how often you visit a specific coffeeshop. At the same time you want to see how often you order certain drinks. Just whip out your handy two-way frequency table. $$\begin{array}{|c|c|c|c|} \hline &\text{Crazy Mocha}&\text{Dunkin}&\text{Starbucks}\\ \hline \text{Coffee}&3&5&2\\ \hline \text{Cold Brew}&1&9&2\\ \hline \text{Latte}&7&1&3\\ \hline \end{array}$$

If you want to see how many times you ordered a cold brew at Dunkin, just go to the cold brew row and move to the Dunkin column: $9$.

$$\begin{array}{|c|c|c|c|} \hline &\text{Crazy Mocha}&\text{Dunkin}&\text{Starbucks}\\ \hline \text{Coffee}&3&5&2\\ \hline \text{Cold Brew}&1&\color{#8925AE}{9}&2\\ \hline \text{Latte}&7&1&3\\ \hline \end{array}$$

Totals

Another addition we can add to the table is totals. Let's look at our previous example with a total column and row added in.

$$\begin{array}{|c|c|c|c|c|} \hline &\text{Crazy Mocha}&\text{Dunkin}&\text{Starbucks}&\text{Total}\\ \hline \text{Coffee}&3&5&2\\ \hline \text{Cold Brew}&1&9&2\\ \hline \text{Latte}&7&1&3\\ \hline \text{Total}\\ \hline \end{array}$$

To find the total number of times you ordered coffee, you just add up the values in the coffee row.

$$3+5+2=10$$ $$\begin{array}{|c|c|c|c|c|} \hline &\text{Crazy Mocha}&\text{Dunkin}&\text{Starbucks}&\text{Total}\\ \hline \text{Coffee}&3&5&2&\color{#8925AE}{10}\\ \hline \text{Cold Brew}&1&9&2\\ \hline \text{Latte}&7&1&3\\ \hline \text{Total}\\ \hline \end{array}$$

Similarly, if you want to the number of times you went to Crazy Mocha, you add up the values in the Crazy Mocha column, putting the result in the total row.

$$3+1+7=11$$ $$\begin{array}{|c|c|c|c|c|} \hline &\text{Crazy Mocha}&\text{Dunkin}&\text{Starbucks}&\text{Total}\\ \hline \text{Coffee}&3&5&2&10\\ \hline \text{Cold Brew}&1&9&2\\ \hline \text{Latte}&7&1&3\\ \hline \text{Total}&\color{#8925AE}{11}\\ \hline \end{array}$$

Filling in the other totals, we get:

$$\begin{array}{|c|c|c|c|c|} \hline &\text{Crazy Mocha}&\text{Dunkin}&\text{Starbucks}&\text{Total}\\ \hline \text{Coffee}&3&5&2&10\\ \hline \text{Cold Brew}&1&9&2&12\\ \hline \text{Latte}&7&1&3&11\\ \hline \text{Total}&11&15&7\\ \hline \end{array}$$

There's one box left. What's in the box!? Well, this is the grand total or the total of the totals. This will turn out to be the number of times you visited a coffee shop in general. We can find this by summing either the total column or the total row; they'll give you the same answer. Do not sum both!

$$\text{Row: }11+15+7=33\\ \text{Column: }10+12+11=33$$ $$\begin{array}{|c|c|c|c|c|} \hline &\text{Crazy Mocha}&\text{Dunkin}&\text{Starbucks}&\text{Total}\\ \hline \text{Coffee}&3&5&2&10\\ \hline \text{Cold Brew}&1&9&2&12\\ \hline \text{Latte}&7&1&3&11\\ \hline \text{Total}&11&15&7&\color{#8925AE}{33}\\ \hline \end{array}$$

Two-Way Tables

Two-way frequency tables let you look at $2$ variables at once. Think of it like the Venn diagram of frequency tables.

Let's look at an example of a two-way table.

Some two-way tables include the total frequencies shown in each column and row. If you've got a two-way table that doesn't show totals, it's a good idea to add them in yourself.

The totals are an example of cumulative frequencies.

(Video) Two Way Frequency Tables

by Milanese Math

This video by Milanese math covers two-way frequency tables.

$\large{\text{Summary}}$

A two-way frequency table is the same thing as a frequency table, but it allows us to keep track of two things at the same time, rather than one.

He asked his students to toss a paper ball into a trashcan. He then recorded how many made the shot and how many missed. He also recorded how many were girls and how many were boys. The two-way frequency table he made was:

$$\begin{array}{|c|c|c|c|} \hline &\text{Boys}&\text{Girls}&\text{Total}\\ \hline \text{Made}&7&6&\\ \hline \text{Missed}&11&9&\\ \hline \text{Total}\\ \hline \end{array}$$

The first cell of data, $7$, tells us that there were $7$ boys that made a basket. Moving over, we have $6$ girls that made a basket.

Similarly, $11$ boys missed the basket, and $9$ girls missed the basket.

You might also notice there is a $\text{Total}$ row and column. We use these to find the total for each category. For example, since we have $7$ boys that made a basket and $11$ that missed a basket, there are $7+11=18$ boys.

$$\begin{array}{|c|c|c|c|} \hline &\text{Boys}&\text{Girls}&\text{Total}\\ \hline \text{Made}&7&6&\\ \hline \text{Missed}&11&9&\\ \hline \text{Total}&7+11=18\\ \hline \end{array}$$

For the girls, we see there are $6+9=15$ of them.

$$\begin{array}{|c|c|c|c|} \hline &\text{Boys}&\text{Girls}&\text{Total}\\ \hline \text{Made}&7&6&\\ \hline \text{Missed}&11&9&\\ \hline \text{Total}&18&15\\ \hline \end{array}$$

For the rows, we find the total number that made the basket and the total number that missed the basket.

$$\begin{array}{|c|c|c|c|} \hline &\text{Boys}&\text{Girls}&\text{Total}\\ \hline \text{Made}&7&6&7+6=13\\ \hline \text{Missed}&11&9&11+9=20\\ \hline \text{Total}&18&15\\ \hline \end{array}$$

What about that last cell? This is the grand total, the total of the total, i.e. the total number of people in the class. To find this we can either add up the total row or the total column. Do not add both!

$$\text{Row: }\quad18+15=33\\ \text{Column:}\quad13+20=33$$ $$\begin{array}{|c|c|c|c|} \hline &\text{Boys}&\text{Girls}&\text{Total}\\ \hline \text{Made}&7&6&13\\ \hline \text{Missed}&11&9&20\\ \hline \text{Total}&18&15&\color{#8925AE}{33}\\ \hline \end{array}$$