4.2.1 Two way Tables

A two-way table, also known as a contingency table, is a way of organizing and displaying data that relates to two different categorical variables.

• It consists of rows and columns, with each row representing one category of the first variable, and each column representing one category of the second variable.

• The cells of the table contain the frequency or count of each combination of the two variables.

• For example, let’s say we are conducting a survey on the preferred modes of transportation among people of different age groups. We can create a two-way table to show the results, with one variable being age group (e.g., \( 18 \)-\( 24 \), \( 25 \)-\( 34 \), \( 35 \)-\( 44 \), etc.) and the other variable being mode of transportation (e.g., car, public transportation, bike, etc.).

  • The resulting table would show the frequency or count of each combination of age group and mode of transportation.

• Here is an example of what a two-way table might look like:

• This table shows the number of respondents in each age group who prefer each mode of transportation.

• For example, we can see that \( 50 \) respondents in the \( 18 \)-\( 24 \) age group prefer to travel by car, while \( 30 \) prefer public transportation and \( 20 \) prefer a bike.

  • This information can be useful for analyzing and understanding the relationship between the two variables.

• To calculate probability from a two-way table

  • First determine the total number of observations (or sample size) in the table.

  • Use the formula:

\( P(A \ and \ B)=\frac{Number \ of \ observations \ in \ both \ A \ and \ B}{Total \ number \ of \ observations} \)

where \( P (A \ and \ B) \) represents the probability of both events \( A \) and \( B \) occurring together.

• For example, let’s say we have the following two-way table that shows the distribution of students by gender and major:

• If we want to calculate the probability of a randomly chosen student being male and majoring in engineering, we can use the formula above:

\( P (Male \ and \ Engineering) =\frac{50}{80+60}=0.45 \)

So, the probability of a randomly chosen student being male and majoring in engineering is \( 0.45 \), or \( 45 \) %.

Similarly, we can calculate the probability of other events by using the formula and the information provided in the two-way table.