What Is Conditional And Marginal Distribution?

6 Mins Read Red Hat Media 30 Nov 2021
Marginal Distribution

In statistics, we often examine two discrete variables simultaneously to seek correlations in a dataset. For example, we might be interested in a population’s dietary habits. Then, we can speculate their commute time or other observable variables like age or money that may impact their dietary behaviors. Marginal distribution and conditional distribution are two approaches that may help you analyze specific combinations of bivariate data like this.

You must evaluate every potential combination of variables while collecting data for a statistical problem with multiple variables. However, when analyzing the data, you may be required to focus on data distribution for one of the variables while completely ignoring the others. Finding the marginal distribution of a multi-variable data collection entails doing just that. So, let’s find out how marginal vs conditional probability works. 

What Is Marginal Distribution?

The marginal distribution uses the formula Probability of event X=A given variable Y. That’s probably all you need to know how to find marginal distribution if you are good at math. However, if that formula gives you a headache, you can always find marginal distribution using a frequency distribution table.

A frequency distribution table gives you a snapshot of the data that allows you to find patterns. Of course, it isn’t relatively that straightforward. You can’t merely assert the last column or row of any frequency distribution table as marginal distribution. You need to know a few rules if you want to know how to calculate marginal distribution. 

The marginal probability is the chance that a single event will occur. We ignore any secondary variable calculations while calculating marginal probabilities. For example, we may calculate two marginal probabilities in our hypothetical example or look at individual variables. We are, in essence, estimating the probability of a single independent variable.

What Is Conditional Distribution?

A probability distribution for a sub-population is known as a conditional distribution. In other words, it shows the likelihood that a randomly selected object in a sub-population possesses a trait you care about. 

For example, if you’re researching eye colors in a specific population, you might want to discover how many people have blue eyes. It means you have to research the sub-population. Just like marginal distribution, with the help of a frequency distribution table, conditional distributions are easy to find.

A conditional probability is the likelihood of an event occurring, given the occurrence of another specific event. So, for example, if we are given a population’s commuting times, we may determine their probability of some eating behavior. That means, in conditional distribution, we must put a condition on the more extensive data distribution, where that one variable’s calculation is dependent on another variable.

Marginal Vs Conditional Distribution-Examples

You can use the same table to calculate both marginal and conditional distributions. The totals of the probabilities are called marginal distributions. They are located on the edges of things; thus, the name marginal probability is given to them. For example, the likelihood for genderwise commute preference is shown in the table below. 


The marginal distributions are the overall probabilities in the margins. So, you have to count the total population first, which is 22 in this table. Now, you have to count the number of people preferring each type of commute and turn that ratio into a probability. 

  • People preferring bus = 8 / 22 = .36.
  • People preferring car = 7 / 22 = .32.
  • People preferring train = 7 / 22 = .32.

If you want to check if you did the marginal distribution calculation right, just make sure all the probabilities sum up to 1. That means .36 +.32 +.32 = 1.

On the other hand, a sub-population in this table would be a conditional distribution. The varied pet preferences would be the subpopulations in this scenario. For example, if you want to count women’s commute preferences, you have to calculate the total number of women in this table, 12. Now, count the number of women who prefer each type of commute and turn that ratio into a probability. 

  • Women preferring bus = 2 / 10 = .3
  • Women preferring car = 5 / 10 = .2
  • Women preferring train = 3 / 10 = .5

If you want to check if you did the marginal distribution calculation right, just make sure all the probabilities sum up to 1. That means .3 +.2 +.5 = 1.

Marginal Distribution Calculation

There are two key points to remember about marginal probability. Firstly, you can express your marginal distribution as counts or percentages. Secondly, if the ratio is defined as a percentage, the total of all marginal values must equal 100 percent, or if the percentages are expressed as decimals, they must equal 1. 

If you’re describing the data as counts, the total number of trials or data items in your collection should equal the total number of counts. You may find out more about it in the example. For example, suppose a classroom has 18 students, therefore 1 + 4 + 5 + 6 + 2 = 18.

Divide the count for each category by the total number of data points to get your marginal distribution values as percentages. The marginal value stated as a percentage can alternatively be called the marginal probability in data sets to extend the probability. This is also an important parameter if you want to calculate marginal revenue.

Conditional Distribution Calculation

The conditional distribution of Y given X is the probability distribution of Y when X is known to be a definite value if X and Y are two jointly distributed random variables. So, the conditional distribution formula is  P(X | Y)=P(X∩Y)P(Y). 

A conditional distribution is used to determine the likelihood that a person thing based on their gender, age, financial status, etc. Here, the value of one random variable is known to you, while the value of the second random variable is unknown.

You have to use the earlier formula to use the conditional distribution ratio of the unknown variable based on the known variable dataset. Thus, you will find the conditional distribution of male/ female sports, commute preference, or pet preference by looking at the numbers in the male/female row of the table.


1: What Are Marginal Density Functions?

The marginal Density Function is the one that denotes the marginal probability of a continuous variable. It means we will determine the likelihood of a specific event taking place without knowing the occurrence probability of other related variables. So, a marginal probability density function indicates the possibility of a single occurring variable.

2: How To Find Marginal Distribution From Joint Distribution?

If you take X and Y as two discrete random variables and f (x,y) is the value of their joint probability distribution at (x,y), the functions given by the marginal distributions of X and Y will be g(x) = Σy f (x,y) and h(y) = Σx f (x,y). This is how you derive marginal distribution from joint distribution.

3: What Is Marginal Distribution In Stats?

The marginal probability of a subset of a whole collection of random variables is the probability distribution of those variables included in the subset. It is the definition of marginal distribution in statistical theory. You get the probabilities of various values of the variables in the subset without referring to the values of other variables.


We have elaborated on what marginal distribution is, and we have also elaborated on what conditional distribution is. This article has also elaborated on marginal vs conditional probabilities. The formulas that are used to calculate these statistical details are also attached here. So, if you have any more questions, you can ask us in the comment section.

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