Univariate vs Bivariate Data

Univariate and Bivariate distribution

“If it is proved true that in a large number of instances two variables tend always to fluctuate in the same or in opposite directions, we consider that the fact is established and that a relationship exists. This relationship is called correlation.”

Univariate distribution: These are the distributions in which there is only one variable such as the heights of the students of a class.
Bivariate distribution: Distribution involving two discrete variable is called a bivariate distribution. For example, the heights and the weights of the students of a class in a school.
Bivariate frequency distribution: Let x and y be two variables. Suppose x takes the values x₁, x₂, …., x_n and y takes the values y₁, y₂, ….., y_n then we record our observations in the form of ordered pairs (x₁, y₁), where 1 ≤ i ≤ n, 1 ≤ j ≤ n. If a certain pair occurs f_ij times, we say that its frequency is f_ij.

The function which assigns the frequencies f_ij’s to the pairs (x_i, y_j) is known as a bivariate frequency distribution.

Difference between Univariate and Bivariate Data

Univariate data means “one variable” (one type of data).
Bivariate data means “two variables” (two types of data).

Univariate Data	Bivariate Data
involving a single variable	involving two variables
does not deal with causes or relationships	deals with causes or relationships
the major purpose of univariate analysis is to describe	the major purpose of bivariate analysis is to explain
central tendency – mean, mode, median dispersion – range, variance, max, min, quartiles, standard deviation. frequency distributions bar graph, histogram, pie chart, line graph, box-and-whisker plot	analysis of two variables simultaneously correlations comparisons, relationships, causes, explanations tables where one variable is contingent on the values of the other variable. independent and dependent variables
Sample question: How many of the students in the freshman class are female?	Sample question: Is there a relationship between the number of females in Computer Programming and their scores in Mathematics?

Covariance

Let (x_i, y_j); i = 1, 2, …., n be a bivariate distribution, where x₁, x₂, …., x_n are the values of variable x and y₁, y₂, ….., y_n those of y. Then the covariance Cov (x, y) between x and y is given by
Univariate vs Bivariate Data 1
Covariance is not affected by the change of origin, but it is affected by the change of scale.