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A conditional distribution is a probability distribution that describes the likelihood of a given event occurring, given that another event has already occurred. It is determined by combining the joint probability distribution of two random variables with the marginal probability distribution of one of the variables. This allows us to determine the probability of one variable given the probability of the other variable.
A two-way frequency table is a table that displays the frequencies (or “counts”) for two categorical variables.
For example, the following two-way table shows the results of a survey that asked 100 people which sport they liked best: baseball, basketball, or football.
The rows display the gender of the respondent and the columns show which sport they chose:
In this example, there are two variables: Sports and Gender.
A marginal distribution is simply the distribution of each of these individual variables. In a two-way table, the marginal distributions are shown in the margins of the table:
For example, we would say that the marginal distribution of sports is:
- Baseball: 36
- Basketball: 31
- Football: 33
We could also write the marginal distribution of sports in percentage terms (i.e. out of the total of 100 respondents):
- Baseball: 36 / 100 = 36%
- Basketball: 31 / 100 = 31%
- Football: 33 / 100 = 33%
And we would say that the marginal distribution of gender is:
- Male: 48 (or 48%)
- Female: 52 (or 52%)
Note: Marginal distributions always add up to 100%.
Why Use Marginal Distributions?
Marginal distributions are useful because although we often collect data for two variables (like Sports and Gender), sometimes we have specific questions about just one variable.
For example, we might want to know the distribution of survey respondents based purely on gender.
Use the following examples as practice for gaining a better understanding of marginal distributions.
Example 1: Marginal Distributions for Percentages
The following two-way table shows the results of a survey that asked 238 people which movie genre they liked best:
Question: What is the marginal distribution for movie genre (in percentages)?
Answer: The marginal distribution for movie genre is:
- Fantasy: 47/238 = 19.7%
- Drama: 88/238 = 37%
- Action: 103/238 = 43.3%
Question: What is the marginal distribution for gender (in percentages)?
Answer: The marginal distribution for gender is:
- Male: 122/238 = 51.3%
- Female: 116/238 = 48.7%
Example 2: Marginal Distributions for Counts
The following two-way table shows the exam scores of 64 students in a class based on how many hours they spent studying:
Question: What is the marginal distribution for exam scores (in counts)?
Answer: The marginal distribution for exam scores is:
- 71-80: 22
- 81-90: 22
- 91-100: 20
Question: What is the marginal distribution for hours studied (in counts)?
Answer: The marginal distribution for hours studied is:
- 1 Hour: 14
- 2 Hours: 14
- 3 Hours: 16
- 4 Hours: 20
Notice that the total of the marginal distribution adds up to the table total of 64 students.