A Definitive Guide to Understand the Difference Between Covariance and Correlation

What Is Covariance?

Covariance is a statistical term that refers to a systematic relationship between two random variables in which a change in the other reflects a change in one variable.

The covariance value can range from -∞ to +∞, with a negative value indicating a negative relationship and a positive value indicating a positive relationship.

The greater this number, the more reliant the relationship. Positive covariance denotes a direct relationship and is represented by a positive number.

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A negative number, on the other hand, denotes negative covariance, which indicates an inverse relationship between the two variables. Covariance is great for defining the type of relationship, but it's terrible for interpreting the magnitude.

Let Σ(X) and Σ(Y) be the expected values of the variables, the covariance formula can be represented as:

Where,

• xi = data value of x
• yi = data value of y
• x̄ = mean of x
• ȳ = mean of y
• N = number of data values.

Application Of Covariance

The following are the most common applications of Covariance:

• Simulating systems with multiple correlated variables is done using Cholesky decomposition. A covariance matrix helps determine the Cholesky decomposition because it is positive semi-definite. The matrix is decomposed by the product of the lower matrix and its transpose. 
• To reduce the dimensions of large data sets, principal component analysis is used. To perform principal component analysis, an eigen decomposition is applied to the covariance matrix

What Is Correlation?

In statistics, correlation is a measure that determines the degree to which two or more random variables move in sequence. When an equivalent movement of another variable reciprocates the movement of one variable in some way or another during the study of two variables, the variables are said to be correlated. The formula for correlation is:

where,

var(X) = standard deviation of X

var(Y) = standard deviation of Y

Positive correlation occurs when two variables move in the same direction. When variables move in the opposite direction, they are said to be negatively correlated.

Correlation is of three types:

1. Simple Correlation: In simple correlation, a single number expresses the degree to which two variables are related.
2. Partial Correlation: When one variable's effects are removed, the correlation between two variables is revealed in partial correlation.
3. Multiple correlation: A statistical technique that uses two or more variables to predict the value of one variable.

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Methods of Calculating Correlation

There are a number of methods to calculate correlation coefficient. Here are some of the most common ones:

Coefficient of correlation

This is the most common method of determining the correlation coefficient of two variables. It is obtained by dividing the covariance of two variables with the product of their standard deviations.

Rank Correlation Coefficient 

A rank correlation coefficient measures the degree of similarity between two variables, and can be used to assess the significance of the relation between them. It measures the extent to which, as one variable increases, the other decreases. 

where,

ρ = coefficient of rank relation

D = difference between paired ranks

N = number of items ranked

Coefficient of Concurrent Deviations

Coefficient of concurrent deviations is used when you want to study the correlation in a very casual manner and there is not much need to attain precision.

where,

rc = coefficient of concurrent deviations

n = number of pairs of deviations

We will continue our learning of the covariance vs correlation differences with these applications of the correlation matrix.

Applications of Correlation 

A correlation matrix is computed for three main reasons:

• The goal when dealing with large amounts of data is to find patterns. As a result, a correlation matrix is used to look for a pattern in the data and determine whether the variables are highly correlated.
• For use in other analyses. When excluding missing values pairwise, correlation matrices are commonly used as inputs for exploratory factor analysis, confirmatory factor analysis, structural equation models, and linear regression.
• When checking other analyses, as a diagnostic. When it comes to linear regression, for example, a large number of correlations indicate that the linear regression estimates will be unreliable.

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Correlation Vs Covariance

Now you will see the differences between Covariance and Correlation.

Similarities: Covariance vs Correlation

Correlation and Covariance both measure only the linear relationships between two variables. This means that when the correlation coefficient is zero, the covariance is also zero. Both correlation and covariance measures are also unaffected by the change in location.

However, when it comes to making a choice between covariance vs correlation to measure relationship between variables, correlation is preferred over covariance because it does not get affected by the change in scale. 

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Example in Python

Now, calculate and understand the covariance and correlation in Python. Here you will take two variables X and Y. 

The matrix here is 2X2. Let’s calculate the covariance for cov(a,b).

Now, calculate the correlation between (a,b)

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