Covariance vs Correlation: The Most Comprehensive Guide

What are Covariance and Correlation?

In layman’s terms, both covariance and correlation are used to gauge the relationship and the dependency between two variables, usually a variable and the actual value it corresponds to.

Covariance

A coherent association between two random variables where a change in one variable reflects a change in the other is referred to as covariance. The direction of the linear relationship between the two variables is indicated by covariance. By direction, we mean whether the variables are proportional to each other directly or inversely.

The covariance values could be any real number lying between the positive and the negative infinities. Therefore, covariance values can be positive, negative, or even zero. A positive value represents positive covariance, which indicates a direct dependency, i.e, increasing the value of one variable will result in a positive change for the other variable and vice versa. On the other hand, a negative value signifies negative covariance, which indicates that the two variables have an inverse dependency, i.e., increasing the value of one variable will result in a negative change for the other variable and vice versa.

It is also worth noting that covariance simply gauges how two variables change together, not whether one variable is dependent on another. Covariance is useful for determining the relationship; however, it is ineffective for determining the magnitude.

Correlation

A correlation analysis is a statistical approach for assessing the intensity of a relationship between two numerically measured continuous variables. Correlation is a statistical metric that measures how closely two or more random variables move in time. The variables are considered correlated when an analogous movement of another variable imitates the direction of one variable in some way throughout the examination of the two variables.

It reveals not only the nature of the relationship but also its strength. As a result, we may argue that correlation values are standardized. Still, covariance values are not and, therefore, cannot be used to measure how strong or weak a relationship is since the magnitude has no direct meaning.

The value of the correlation coefficient ranges from -1 to +1. A correlation of -1 indicates that the two variables are negatively correlated, meaning that when one rises, the other falls. The maximum correlation value is +1, which indicates that the two variables are entirely positively connected, meaning that if one increases, the further increases. The two variables are unrelated if the correlation is 0.

There are three different types of correlation:

• Simple Correlation: A single number represents the degree to which two variables are associated in simple correlation.
• Partial Correlation: When one variable's effects are eliminated, partial correlation reveals the relationship between two variables.
• Multiple Correlation: A statistical strategy that predicts the value of one variable using two or more variables.

Covariance vs Correlation: Mathematically

Covariance Formula

The sum of the product of the differences from the means of the variables is used to calculate the value of covariance between two variables:

For Population:

cov(x,y) = i=1n(xi - x’)(yi - y’)n

For Sample:

cov(x,y) = i=1n(xi - x’)(yi - y’)n - 1

Here,
x’ and y’ = mean of the provided sample set
n = total number of sample
n - 1 = degree of freedom
xi and yi = individual samples of the set

What is the degree of freedom?

The number of independent data points used to calculate the estimate is called degrees of freedom.

Example:

Let us take s as a sample set of three integers.

The calculated mean of these three integers is 5, and two of the three variables are 3 and 7. As a result, the third variable has just one possible value: 5.
There is only one value for each two given values in any group of three integers with the same mean, such as 4, 6, and 5 or 2, 8, and 5.
You may adjust the first two numbers, and the third value will automatically correct itself.

Therefore, the degree of freedom of this sample set s is 2 (which is n - 1, if n = 3).

The variances of the variables involved determine the covariance's upper and lower bounds. However, these variances, in turn, might change depending on how the variables are scaled. Even a change in measuring units might affect the covariance. As a result, covariance is only helpful in determining the direction, not the size or the magnitude, of a relationship between any two variables.

Correlation Formula

To calculate correlation, we must first evaluate the covariance of the two variables in relation to their standard deviations. To do so, we need to divide the covariance by the product of the two variables' standard deviations, resulting in a correlation between the two variables.

The final product of a correlation is called the correlation coefficient, denoted by, corr(x,y).

corr(x,y) = i=1n(xi - x’)(yi - y’)i=1n(xi - x’)2i=1n(yi - y’)2
= i=1n(xi - x’)(yi - y’)ni=1n(xi - x’)2i=1n(yi - y’)2n2 (dividing both sides by n)

corr(x,y) = cov(x,y)xy

Note:
cov(x,y) = i=1n(xi - x’)(yi - y’)n
x = i=1n(xi - x’)2n and y = i=1n(yi - y’)2n

Here,
x’ and y’ = mean of the provided sample set
n = total number of sample
x = standard deviation of x
y = standard deviation of y
xi and yi = individual samples of the set

Covariance vs. Correlation: Similarities and Differences

Similarities

In probability theory and statistics, the concepts of covariance and correlation are pretty similar as they are used only to measure the linear relationships between two variables. Both concepts refer to how much a random variable or a group of random variables might depart from its anticipated value. This indicates that if the correlation coefficient is zero, so is the covariance. The change in location does not affect correlation and covariance measurements.

However, when choosing between covariance vs correlation to assess the relationship between variables, correlation is selected from over covariance since it is unaffected by scale changes.

Differences

Both the covariance and correlation measurements look at two variables throughout the entire domain, and not just one. For easy reference, the distinctions between them are summarized in a table. Let's look at covariance vs correlation and how different they are from each other.

Covariance vs Correlation: Applications

Application of covariance

• Cholesky decomposition is used to simulate systems with numerous interrelated variables. Due to its positive and semi-definite in nature, a covariance matrix aids in determining the Cholesky decomposition. The lower matrix's product and its transpose are used to deconstruct the matrix.
• Principal component analysis minimizes the dimensionality of huge data sets. An eigendecomposition is performed on the covariance matrix to perform principal component analysis.

Application of correlation

• When working with enormous volumes of data, the objective is to uncover patterns. Therefore, a correlation matrix is employed to search for patterns in the data and assess if the variables are highly connected.
• A correlation matrix is often used as input for exploratory component analysis, confirmatory factor analysis, structural equation models, and linear regression when missing values are excluded pairwise.
• Correlation matrix is also used as a diagnostic while verifying other analyses. For example, many correlations in linear regression imply that the linear regression estimates would be incorrect.

Wrapping Up

Here, we conclude the most comprehensive guide on covariance vs correlation. A journey that included some core components of mathematics and statistics, additionally, also understanding and establishing a paradoxical dependency between covariance and correlation. So to answer the question, covariance vs correlation, which is better?
Considering all the information in the above guide, a correlation has more use cases than covariance. However, this does not prove that correlation is better than covariance. To compute correlation, you need to calculate covariance as well. Correlation is a scaled version of covariance. Therefore it is impossible to answer which concept is better than the other when they are equally important.

Covariance vs Correlation, which is better? The correct answer is that both are equally important. It is essential to understand the areas where they excel and their limitations. Covariance vs Correlation, a complex dependency between two dependent concepts.