What is a two way table?
Let’s consider an example before defining two way tables.
Example:
A group of 80 students composed of 50 boys and 30 girls, were part of a survey to check whether they were part of a local music concert. Out of the 50 boys, 28 boys went to the concert. Of the 30 girls, 18 of them went to the concert.
Let’s construct a two way table for this data.
Boys | Girls | Total | |
Yes | 28 | 18 | 46 |
No | 22 | 12 | 34 |
From this table we can clearly observe the frequency of two data categories, boys and girls, and the number of boys and girls who went to the concert. This two way table is a summary of the data that was collected during the survey.
Hence, this shows us that a two way table is a summary of the data collected so that it is easier to collect as well as to analyze data in a tabular form.
There are also three types of relative frequencies that can be calculated from the two way table. They are:
- Joint relative frequency
- Marginal relative frequency
- Conditional relative frequency
In this article we will look at how to calculate the marginal relative frequency from the data provided in a two way table.
What is Marginal Relative Frequency?
The marginal relative frequency of a data set is calculated by dividing the sum or total of a row or the sum or total of a column by the total number of observations in a dataset. The dataset in consideration here is represented in the form of a two way table. The marginal relative frequency is expressed as both a decimal and a percentage value, but the percentage value is usually preferred. Let’s consider the same example of the students attending the concert. We already have the two way table:
Boys | Girls | Total | |
Yes | 28 | 18 | 46 |
No | 22 | 12 | 34 |
Total | 50 | 30 | 80 |
This is the method we apply to calculate the marginal relative frequency for the data:
The marginal relative frequency of boys who attended the concert \(=28/80 = 0.35 = \frac{35}{100}\text{ }=\text{ }35%\)
The marginal relative frequency of boys who did not attend the concert \(=22/80 = 0.275 = \frac{27.5}{100}\text{ }=\text{ }27.5%\)
The marginal relative frequency of girls who attended the concert \(= 18/80 = 0.225 = \frac{22.5}{100} = 22.5%\)
The marginal relative frequency of girls who did not attend the concert \(= 12/80 = 0.15 = \frac{15}{100} = 15%\)
The marginal relative frequency of students who attended the concert \(= 46/80 = 0.575 = \frac{57.5}{100} = 57.5%\)
The marginal relative frequency of students who did not attend the \(= 34/80 = 0.425 = \frac{42.5}{100} = 42.5%\)
Observe that we can quickly calculate the percentage values of different frequencies that are part of the two way table.
Rapid Recall
Solved Examples
Example 1:
A two way table for the observations noted after 75 people were asked about whether they like salted peanuts or plain peanuts. Among the men 20 of them liked salted peanuts, 10 liked plain peanuts and 10 of them didn’t eat peanuts. Among the women, 15 liked salted peanuts, 15 liked plain peanuts and the remaining 5 women didn’t eat peanuts. Calculate the marginal relative frequency for the following:
- Men who didn’t eat peanuts
- Women who preferred salted peanuts
- Men who preferred plain peanuts
- People who liked salted peanuts
Men | Women | Total | |
Salted | 20 | 15 | 35 |
Plain | 10 | 15 | 25 |
Don’t Eat Peanuts | 10 | 5 | 15 |
Total | 40 | 35 | 75 |
Solution:
Formula for marginal relative frequency =
\(Marginal Relative Frequency = \frac{Row \text{ }Total (or) \text{ }Column\text{ } Total}{Total \text{ }of \text{ }all\text{ } Observations}=Decimal Frequency = Percentage Frequency\)
1. Men who didn’t eat peanuts\(= \frac{10}{75}=0.13=\frac{13}{100}=13%\)
2. Women who preferred salted peanuts \(= \frac{15}{75}=0.20=\frac{20}{100}=20%\)
3. Men who preferred plain peanuts \(= \frac{10}{75}=0.13=\frac{13}{100}=13%\)
4. People who liked salted peanuts \(= \frac{35}{75}=0.46=\frac{46}{100}=46%\)
Hence,
- Men who didn’t eat peanuts – 13%
- Women who preferred salted peanuts – 20%
- Men who preferred plain peanuts – 13%
- People who liked salted peanuts – 46%
Example 2:
There are 200 people who are interviewed about how they like to drink coffee. Among the youngsters of the group 35 preferred cold coffee, 40 liked dark coffee and 35 of them preferred orange juice instead of coffee. Amongst the middle aged people 15 of them liked cold coffee, 60 preferred dark coffee and 15 of them drank orange juice. Draw a two way frequency table and calculate the marginal relative frequency for:
- Youngsters who prefer cold coffee
- Youngsters who drink orange juice
- People who like cold coffee
- Middle aged people who drink dark coffee
Solution:
a. Draw the two way table.
Youngsters | Middle Aged | Total | |
Cold Coffee | 35 | 15 | 50 |
Dark Coffee | 40 | 60 | 100 |
Orange Juice | 35 | 15 | 50 |
Total | 110 | 90 | 200 |
b. Calculate marginal relative frequency.
Formula for marginal relative frequency
\(= Marginal Relative Frequency = \frac{Row Total (or) Column Total}{Total of all Observations}=Decimal Frequency = Percentage Frequency\)
- Youngsters who prefer cold coffee \(= \frac{35}{200}=0.17=\frac{17}{100}=17%\)
- Youngsters who drink orange juice \(= \frac{35}{200}=0.17=\frac{17}{100}=17%\)
- People who like cold coffee \(= \frac{50}{200}=0.25=\frac{25}{100}=25%\)
- Middle aged people who drink dark coffee \(= \frac{60}{200}=0.3=\frac{30}{100}=30%\)
Hence,
- Youngsters who prefer cold coffee = 17%
- Youngsters who drink orange juice = 17%
- People who like cold coffee = 25%
- Middle aged people who drink dark coffee = 30%
A two way table is a way to represent frequencies of two unique data categories. The rows display one category and the columns of the table represent the second category.
In statistics, the term frequency represents the number of times that an observation repeats itself in a specific data set.
In statistics, the central tendency is represented by the mean,the mode and the median of a given set of observations.
We can calculate the joint relative frequency, marginal relative frequency and conditional relative frequency from a two way table.