Covariance vs. Correlation: What is the Difference

Whether you are a newbie or a budding professional in the field of Data Science, having a thorough grasp on your basics is a must. And one of them is the very fundamental concept of the difference between covariance and correlation.

With the enormous amount of information generated, consumed, and stored globally, companies have a gold mine of data at their disposal. However, all that data is practically useless if not analyzed and manipulated to derive actionable insights. Here’s where Data Science comes into the picture – with its invaluable arsenal of statistical methods, data analytics, scientific methods, and artificial intelligence algorithms, Data Science is the ultimate savior. Data Science enables business analysts to discover trends and insights from large datasets, which can be further used to shape business decisions.

With the inarguable importance Data Science has in moulding the course of technology, let’s dive into the fundamentals of covariance vs. correlation and upGrad courses that can help you learn them.

Covariance vs. Correlation: What do they mean?

Covariance and correlation are two prevalent terms that one comes across in statistics and probability theory. While both have very similar connotations and describe the dependency and linear relationship between variables, there are stark differences between the two. Covariance signifies the direction of the linear relationship between two variables, whereas correlation indicates both the direction and strength of the linear relationship between variables. 

Before we get into the detailed explanation of covariance vs. correlation, it is essential to understand two other fundamental terms – variance and standard deviation.

Variance

Variance is the measure of the spread between variables in a dataset. In simpler terms, variance measures how far each variable in the dataset is from the average value and thus from every other variable in the set. The larger the spread, the more the variance with respect to the mean (average). Variance is denoted by the symbol S2 (sample variance).

Mathematically, variance is depicted using the formula:

S2 = Σ(X – x̄)2 / n – 1

where,

S2 = sample variance

Σ = sum of

X = each value

x̄ = sample mean

n = number of data values 

Standard Deviation

Standard Deviation measures the amount of dispersion or variation of a dataset relative to its mean. While a high value of standard deviation indicates that the data points are spread out over a broader range, a low value of standard deviation would mean that the data points are close to the mean of the dataset. Standard deviation is denoted by the symbol ‘s’ (sample standard deviation) or σ (population standard deviation).

Mathematically, the standard deviation is depicted using the formula:

s = √Σ(X – x̄)2 / n – 1

where,

s = sample standard deviation

Σ = sum of

X = each value

x̄ = sample mean

n = number of data values 

Before getting into in-depth correlation and covariance difference, let’s first get familiar with covariance and correlation.

Covariance

Covariance is an extension of variance and determines the direction of the relationship between two variables. In other words, covariance indicates whether the two variables are directly proportional or inversely proportional to one other. Therefore, a change in the value of one variable will inevitably affect the other. However, it is pertinent to mention that covariance only measures the change of one variable with respect to another and not their inter-dependency.

• Covariance can take any value between -∞ and +∞. 
• A positive covariance value signifies a direct relationship between the variables. So, an increase in the value of one variable would lead to a corresponding increase in the other variable, with other conditions remaining constant. Thus, both the variables move together in the same direction as they change.
• In contrast, a negative covariance would mean an inverse relationship between the two variables. When the value of one variable increases, the other will decrease. Essentially, these variables are said to be inversely related and move in opposite directions.

Mathematically, the covariance between two variables x and y is represented as follows:

Cov(X,Y) = Σ(Xi – x̄)(Yi – ȳ) / n – 1

where,

Cov(X,Y) = covariance between x and y

Σ = sum of

Xi = data value of X

Yi = data value of Y

x̄ = mean of X

ȳ = mean of Y

n = number of data values 

Source

Correlation

In contrast to covariance that only measures the direction of the relationship between two variables, correlation also measures the relationship’s strength. Thus, correlation quantifies the relationship between the variables and signifies how strong or weak the relationship is. The primary outcome of correlation is the correlation coefficient ( r ).

It not just demonstrates the type of relationship but also indicates the power of the relationship. Therefore, we can understand that the correlation values have standardized representation, but the covariance values are not standardized and can’t be used to evaluate the weak or strong relationship. The reason is the magnitude doesn’t have direct significance.

• Correlation can only take values between -1 and +1.
• A correlation of +1 signifies a direct and strong relationship between the variables. The increase in one variable leads to a corresponding rise in the other. On the other hand, a correlation of -1indicates a solid, inverse relationship. An increase in one variable will cause an equal and opposite decrease in the other. A correlation value of 0 means that the variables do not have any linear relationship.
• A correlation value closer to -1 or +1 would mean a close relationship between the variables.
• If you want to determine whether the covariance between the two variables is small or large, you must evaluate it with respect to the standard deviations between the two variables.  You can do this by normalizing the covariance after dividing it with the multiplication of the standard deviations of the two variables. Hence, it provides a correlation between the two variables. Moreover, correlation is an important method for examining relations between two variables before executing statistical modeling.

The mathematical expression of correlation is as follows:

r = Cov(x,y) / σX – σY

where,

Cov(x,y) = covariance between X and Y

σX = standard deviation of X

σY = standard deviation of Y

Source

If there is no relationship between two variables, the correlation coefficient will be 0. But if this value is 0, you can only conclude that there is no linear relationship. There may exist other functional relationships among the variables.

When the correlation coefficient’s value is positive, if one variable’s value increases, the others also increase. But when the correlation coefficient’s value is negative, the alterations in the two variables happen in opposite directions. If the correlation coefficient shows “0”, a decrease or increase in one variable doesn’t affect another.

You can better evaluate the correlation and covariance difference if you understand the types of correlation. Here are its types: 

1. Simple Correlation: A single number expresses the amount by which two variables are related.
2. Partial Correlation: When one variable’s effects are eliminated, the correlation between the two variables is discovered in partial correlation.

3. Multiple Correlation: It is a statistical technique that uses two or more variables to forecast the value of one variable.

Methods of calculating the correlation

• The scatter method
• The graphic method
• Co-relation Table
• Coefficient of Concurrent deviation
• Karl Pearson  Coefficient of Correlation
• Spearman’s rank correlation coefficient
• Kendall rank correlation

Now let’s understand the correlation matrix.

What is a correlation matrix?

The correlation coefficients are used to establish the relationship between two variables. For instance, to determine the number of hours a student should spend working to accomplish a project before the deadline. But if you want to assess the correlation between multiple pairs of variables, you can use a correlation matrix.

A correlation matrix is a table demonstrating the correlation coefficients for different variables. The columns and rows include the variables’ values. Each cell indicates the correlation coefficient.

How is the correlation matrix useful?

The correlation matrix is used to analyze various data-driven problems. Here are a few common use cases:

• To perform regression testing
• To determine the input for various analyses
• To easily encapsulate datasets

With enough details on these two terms, let’s now go through the difference between correlation and covariance.

Difference Between Covariance and Correlation

Now that we have covered the basic concepts related to covariance and correlation, it is time to delve into their differences. No doubt, the two statistical terms seem pretty similar at first glance. However, a more detailed study reveals that covariance and correlation are distinct in several aspects.

The following section discusses covariance and correlation difference from various perspectives to ensure a thorough analysis. So, let us look at the difference between covariance and correlation:

• Meaning

Covariance is a measure of the extent to which two variables change together. 

On the other hand, correlation is a measurement of the strength of the linear relationship between variables.

• Values

Covariance can take any value between -∞ and +∞. 

The correlation value can be anywhere between -1 and +1.

• What do they represent?

Covariance shows the direction of the linear relationship between the variables. While a positive value indicates a direct relationship, a negative covariance value means an inverse relationship.

In contrast, correlation indicates both the direction and strength of the linear relationship between the variables. The closer the value to +1 or -1,  the stronger the relationship.

• Scalability

Another important covariance vs correlation difference is based on scalability. The change of scale affects covariance. For instance, if the value of two variables is multiplied by the same or different constants, the calculated covariance of the two variables will change.

In contrast, correlation is immune to the change in scale. Hence, multiplication by constants does not change the initial correlation value. 

• Units

The unit of covariance is the product of the units of the two variables.

On the other hand, correlation is dimensionless. Therefore, it is a unit-free measure of the relationship between the variables that makes the comparison of calculated correlation values easier across variables.

• Utility

Covariance can be computed for only two variables.

On the other hand, correlation can be calculated for multiple sets of variables, a quality that makes it a more convenient choice for data analysts. 

• Applications

Covariance mostly finds its use as an input to other analyses. Typical use cases are in stochastic modelling and principal component analysis.

Common applications of correlation include summarizing large amounts of data, input into other analyses, and as a diagnostic for further analyses.

Applications of covariance:

• One prominent correlation vs covariance difference is that covariance is used in Molecular and Genetics Biology to measure certain DNAs.
• It is used in the forecast of the amount of investment on various assets in the financial markets.
• It is extensively used to gather data acquired from oceanographic /astronomical studies and derive conclusions.
• It is used in Statistics to analyze a data set with logical allegations of a principal component through a covariance matrix. The Principal Component Analysis is implemented to decrease the dimensions of huge data sets. An Eigen decomposition is implemented to the covariance matrix to carry out a principal component analysis.
• It is used to study signals collected in different forms.
• Cholesky decomposition is used to simulate systems having multiple correlated variables. A covariance matrix helps define the Cholesky decomposition since it is a positive semi-definite

Applications of correlation

• Correlation determines time vs money spent by a customer on different online e-commerce websites.
• It compares previous records of weather forecasts with that of the current year.
• It is commonly used in pattern recognition when dealing with huge amounts of data. It checks whether the variables are highly correlated or not.
• It analyzes the increase in temperature during summer vs. water consumption in a family.
• It determines the relationship between poverty and population.
• When eliminating missing values pairwise, the correlation matrices are used as inputs for confirmatory factor analysis, exploratory factor analysis, linear regression, and structural equation models.

How are covariance and correlation relevant to data analytics?

Statistics lays the foundation of several data analysis techniques. To have a detailed correlation vs covariance comparison, you must know how they are relevant to data analytics. Although there exist covariance and correlation difference, both of them are uniquely useful to data analytics. Certain common use cases of correlation and covariance in the field of data analytics are:

They are used to compare samples among two or more diverse populations. This helps in analyzing the common patterns and trends in different samples.

They are useful in data-driven industries to identify the multivariate data that eventually helps in data processing and performing analytical operations.

PCA (principal component analysis) is employed using correlation and covariance to reduce the dimensions of huge datasets and thus improve interpretability. Commonly, data scientists use PCA to perform exploratory data analysis and predictive analysis.

Analytical processes like multivariate analysis and feature selection are fulfilled by implementing correlation and covariance methods.

Which one to choose?

Having gone through the detailed discussion on covariance vs correlation, it’s now imperative to decide which one to choose. Though there is a difference between correlation and covariance, they are closely related to each other. Covariance indicates the type of interaction, whereas correlation indicates the strength and the type of this relationship. This is why correlation is usually designated as the special case of covariance. But when it comes to a choice between the two, most analysts choose correlation as it stays unaffected by the alteration in locations, dimensions, and scale. Moreover, since its range is limited to -1 to +1, it helps to derive comparisons between variables across the domains. But, the key limitation of correlation and covariance is that they evaluate only the linear relationship.

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To Wrap It Up

Both covariance and correlation measure the linear relationship between variables. Nonetheless, given a choice between the two, correlation is favoured over covariance for two primary reasons. First, the correlation coefficient remains unaffected by the change in scale, and second, it is a unitless measure that simplifies comparisons. 

A strong foundation of mathematical and statistical concepts is crucial to a promising career in Data Science and Artificial Intelligence. However, with the cut-throat competition and the constant need for professional upskilling, the best way to future-proof your resume is by choosing the right program – a step that you can take with upGrad.