Covariance Formula

Covariance Formula

Mathematics is an essential component of the student curriculum. To flourish in many phases of life, a student must be well acquainted with numbers from the start. As a result, a full grasp of Mathematics from a young age would provide a solid foundation for the future. Understanding fundamental ideas, calculating strategies, and simple formulae come in helpful for whatever test students encounter in their life, whether it’s their class exams or JEE, or the banking and management entrance examinations later in life. Although Mathematics is a fascinating topic to learn, students frequently become bored and disinterested. It is mostly due to how the topic is taught. The Extramarks have highly qualified educators who educate using audiovisual components in a real-time online setting.

Mathematics formulae are expressions developed through decades of research that aid in the rapid resolution of problems. Straightforward arithmetic computations, such as addition and subtraction, are simple to accomplish. However, when it comes to algebraic expressions, geometry, and other areas, Mathematics formulae are required to simplify the procedure and save time. At Extramarks, students will not only get formulas for every topic, but they will also learn how that equation is created. As a result, they won’t have to remember formulae since they grasp the logic behind them. When they use these formulas to solve problems creatively, they will notice an improvement in their Mathematics abilities.

The Covariance Formula is used in statistics to evaluate the connection between two variables. It is simply a variance measure between two variables. The Covariance Formula is measured in units and determined by multiplying the two variables’ units. The variance might be either positive or negative. The interpreted values are as follows:

A positive Covariance Formula occurs when two variables move in the same direction.

In contrast to the preceding statement, when two variables move in opposite directions, the consequence is a negative covariance.

Statistics is an important part of Mathematics that is applied in a variety of fields. As a result, students must gain a grasp of statistics and covariance since it will not only help them score higher on the final test but will also aid them in the future. In Mathematics terms, what is covariance? Let’s look at the definition of covariance to find out. The calculation of the connection between any two random variables from their mean is what covariance means. It calculates the total variance of any two random variables from their predicted values.

Covariance types include:

Covariance Formula may be classified into two forms based on the movement of the two variables. There are two kinds of covariance:

  • Positive Covariance Formula:

When both variables move in the same direction, the covariance is said to be positive. When both variables rise or decrease, this is referred to as positive covariance.

  • Negative Covariance Formula

When two variables move in opposing directions, the covariance is said to be negative, indicating that they are inversely connected. If the X variable grows, the Y variable decreases, and vice versa.

Formulas for Covariance (Population and Sample)

Population Covariance Formula  = Cov(x, y)=(xi-x̄)(yi-ȳ)N

Sample Covariance Formula = Cov(x, y)=(xi-x̄)(yi-ȳ)N-1

Methods for Reciting Mathematics Formulas:

Hundreds of Mathematics formulae must be learned by students during their school careers. As a result, pupils must have a few pointers to assist them to memorise arithmetic formulae correctly. A few tips are provided below:

Understanding: A student must understand the notion underlying a Mathematics calculation. If a student r understands the derivation and has a clear understanding of the purpose of the arithmetic formula, he may not even need to remember it. He will intuitively understand how and when to use the appropriate Mathematics formula.

Practice: It will be impossible for them to measure the utility of a formula unless they have solved an issue that demands it. They begin remembering a formula through muscle memory by answering several questions. As a result, They won’t have to make an active effort to learn formulas because they already know how to use them.

Make revision sheets: Once they have been introduced to a topic and begin answering questions about it, make a sheet with all of the arithmetic formulae that they are using. Continue to update it as their learning cycle progresses. This document can be reviewed when students need to have a fast glance at these formulae or if they wish to memorise them. One may also use the Extramarks to aid them with the Mathematics formulae provided above.

Real-World Mathematics Formula Applications:

Students benefit from knowing the real-life applications of the corresponding arithmetic formulae when learning a topic since it makes the subject more accessible. The following are some of the applications of Mathematics formulas:


Algebraic formulae are commonly utilised in computer science to accomplish a variety of analytical operations. They are also employed in cryptography to safeguard financial information. Furthermore, they utilise Mathematics daily, both consciously and unconsciously, to arrange their calendar and complete their activities.


Calculus integration and differentiation formulae are commonly used in engineering. It is also utilised in rocket trajectory research, materials science, shock wave physics code creation, modelling airflow over aerodynamic bodies, heat transfer, seismic wave propagation for earthquake analysis, signal processing, and many more applications.


Geometry formulae are commonly used in building and design to build various structures. They are also utilised in landscape modelling, mechanical part design, cryptography, and airflow patterns.

Statistics and Probability:

Real-life examples may be used to apply probability and statistics calculations, as well as skills learnt in school. They are utilised in domains such as Monte Carlo simulation, signal processing, reliability analysis, risk analysis, stock market prediction models, computer network designs, insurance pricing, and so on.

Mathematics Formulas Are Important for Students

Mathematics formulae are designed for a reason by some of the most clever individuals. They assist children in answering questions quickly and accurately. It also makes the process of finding a solution to a sum much easier than starting from scratch. The following are the advantages of Mathematics formulas:

A student must adhere to the time-sensitive curriculum established by the school. Children’s knowledge is checked regularly through various tests such as units, half-yearly, and finals. Mathematics formulae are so required to ensure that pupils have prepared the subject matter on time and with a buffer for review.

A student is unlikely to solve numerous problems using a pen and paper while reviewing. Thus, to receive a rapid overview of sums and how to solve them, students must be familiar with formulae, which are the keys to obtaining the right solutions.

During exams, students do not have the luxury of deriving a full formula to answer a question, suggesting that they cannot begin at step 1. They must memorise and recall formulae to finish their question paper in the allotted time, which aids them with time management and scheduling.

Students taking competitive examinations must not only understand formulae, but also the many tips and tactics linked with them. Because these tests are typically in the form of MCQs, students must have a strong understanding of arithmetic formulae.

Notations in Covariance Formulas

 xi = data value of x

yi = data value of y

x̄ = mean of x

ȳ = mean of y

N = number of data values.

Relation Between Correlation Coefficient and Covariance Formulas

The correlation coefficient formula is as follows 

Correlation = Cov(x,y)σx×σy


Cov (x, y) is denoted by the covariance between x and y  

The standard deviations of x and y are denoted by σx and σy, respectively.

Using the aforementioned method, the correlation coefficient formula may be determined using the Covariance Formula, and vice versa. The Covariance Formula is measured in units, which may be calculated by multiplying the units of the two variables in question. The variance values are interpreted as follows:

Positive covariance indicates that the two variables tend to move in the same direction.

When two variables have negative covariance, they tend to move in opposing directions.

This is not the case with correlation.

What Is Covariance Formula?

In statistics, the Covariance Formula is a measure of the connection between two random variables. The Covariance Formula illustrates the relationship between the two variables and helps determine if they fluctuate together. The Covariance Formula between two random variables X and Y are indicated as Cov in the  Covariance Formula (X, Y).

Applications of Covariance Formula

The Covariance Formula is used in finance, most notably in portfolio theory. Thus, assets that do not have a strong positive correlation with each other can be picked, hence reducing unsystematic risk.

In statistics, covariance has several uses. Many biological, genetic, financial, and meteorological concepts use covariance. Its applications include micrometeorology and feature extraction.

The Covariance Formula and correlation equations are the foundation for measuring the Genetic connection matrix in genetic and microbiological research. The Covariance Formula equation also serves as a foundation for understanding essential concepts like evolution, gene transmission, and natural selection.

The Covariance Formula is important in finance since it aids in the identification of risks and opportunities. It enables investors to determine if the instruments are moving in positive or negative covariance. It aids them in avoiding losses. The equation is utilised in investigations of data embedding for dimensionality reduction. Its features are used to carry out a variety of procedures and duties.

Canonical correlation analysis is another prominent use of the algorithm (CCA). This explains how and where the sets of variables are migrating.

The features are utilised to facilitate multi-view learning.

Examples Using Covariance Formula

  1. Find covariance for following data set x = {2,5,6,8,9}, y = {4,3,7,5,6}


Given data sets x = {2,5,6,8,9}, y = {4,3,7,5,6} and N = 5

Mean(x) = (2 + 5 + 6 + 8 + 9) / 5 

= 30 / 5

= 6

Mean(y) = (4 + 3 +7 + 5 + 6) / 5

= 25 / 5

= 5

Sample  Covariance Formula = Cov(x,y)=(xi-x̄)(yi-ȳ)N-1

= [(2 – 6)(4 – 5) + (5 – 6)(3 – 5) + (6 – 6)(7 – 5) + (8 – 6)(5 – 5) + (9 – 6)(6 – 5)] / 5 – 1

= 4 + 2 + 0 + 0 + 3 / 4

= 9 / 4

= 2.25

Population  Covariance Formula  = Cov(x,y)=(xi-x̄)(yi-ȳ)N

= [(2 – 6)(4 – 5) + (5 – 6)(3 – 5) + (6 – 6)(7 – 5) + (8 – 6)(5 – 5) + (9 – 6)(6 – 5)] / 5

= 4 + 2 + 0 + 0 + 3 / 

= 9 / 5

= 1.8

The population covariance is 1.8, but the sample covariance is 2.25.

  1. Using the Covariance Formula, find the covariance for the following data set x = {5,6,8,11,4,6}, y = {1,4,3,7,9,12}.

Given data sets x = {5,6,8,11,4,6}, y = {1,4,3,7,9,12} and N = 6

Mean(x) = (5 + 6 + 8 + 11 + 4 + 6) / 6 

= 40 / 6

= 6.67

Mean(y) = (1 + 4 + 3 + 7 + 9 + 12) / 6

= 36 / 6

= 6

Sample  Covariance Formula = Cov(x, y)=(xi-x̄)(yi-ȳ)N-1

= – 0.4

Population  Covariance Formula = Cov(x, y)=(xi-x̄)(yi-ȳ)N-1

= – 0.33

The sample covariance is -0.4 and the population covariance is -0.33.

  1. Find covariance for the following data set x = {13,15,17,18,19}, y = {10,11, 12,14,16} using the Covariance Formula.


Given data sets x = {13,15,17,18,19}, y = {10,11,12,14,16} and N = 5

Mean(x) = (13 + 15 + 17 + 18 + 19) / 5 

= 82 / 5

= 16.4

Mean(y) = (10 + 11 +12 + 14 + 16) / 5

= 63 / 5

= 12.6

Sample  Covariance Formula = Cov(x, y)=(xi-x̄)(yi-ȳ)N-1

= [(16.4 – 13)(12.6 – 10) + (16.4 – 15)(12.6 – 11) + (16.4 – 17)(12.6 – 12) + (16.4 – 18)(12.6 – 14) + (16.4 – 19)(12.6 – 16)] / 5 – 1

= (8.84 + 2.24 – 0.36 + 2.24 + 8.84) / 4

= 21.8 / 4

= 5.45

Population  Covariance Formula = Cov(x,y)=(xi-x̄)(yi-ȳ)N-1

= [(16.4 – 13)(12.6 – 10) + (16.4 – 15)(12.6 – 11) + (16.4 – 17)(12.6 – 12) + (16.4 – 18)(12.6 – 14) + (16.4 – 19)(12.6 – 16)] / 5

= (8.84 + 2.24 – 0.36 + 2.24 + 8.84) / 5

= 21.8 / 5

= 4.36

The sample covariance is 5.45 and the population covariance is 4.36.

Maths Related Formulas
Rectangle Formula Gaussian Distribution Formula
Slope Formula Geometric Distribution Formula
Area Formula For Quadrilaterals Parallel Line Formula
Arithmetic Mean Formula Pearson Correlation Formula
Geometry Formulas Population Mean Formula
Interest Formula Sum Of Arithmetic Sequence Formula
Selling Price Formula Cos Inverse Formula
Circumference Formula Direct Variation Formula
Cone Formula Direction Of A Vector Formula
Correlation Coefficient Formula Fibonacci Formula
FAQs (Frequently Asked Questions)

1. What is the relation between the formulas for covariance and correlation coefficient?

The correlation coefficient, denoted as r, is the covariance of variables divided by the product of their standard deviations.

2. What does the concept of Covariance mean?

The Covariance Formula is a statistic that defines the direction of any two random variables’ connection. The Covariance Formula determines whether the link between the two variables is positive or negative. It aids in comprehending the overall deviation of two variables from their predicted values. The connection can be gauged if the variables move in parallel or have an inverse relationship. In any instance, one cannot deduce the strength of the link or the interdependence of the variables. The covariance values span from positive (+) infinity to negative (+) infinity. The Covariance Formula has dimensions or units.

3. What is the correlation?

Correlation measures the strength or degree of the link between two variables. In contrast to the covariance values, the correlation coefficient values vary from -1 to +1. Correlation has no dimensions. Several +1 indicates a strong positive association, whereas a value of -1 suggests a strong negative relationship. The Covariance Formula and correlation formula clearly explain the link between the Covariance Formula and correlation. Correlations and the Covariance formula are both used to evaluate the relationship between variables. Correlation aids in determining the binding strength of the variables’ connection. It is the scaled covariance measure. It lacks dimension. To put it another way, the Covariance Formula is always true.

4. What is the significance of Mathematics formulas?

Mathematics formulae let students solve tough problems quickly and easily. All students have to do is figure out the proper formula to solve the problems, and they’ll be OK. When we examine the case of a timed test, it becomes vital to employ these Mathematics formulae so that the paper may be completed well within the allotted limit.