## Difference Between Covariance Vs Correlation

Covariance Vs Correlationgives us the differences between the two statistical concepts used to determine the relationship between two random variables and that are exactly opposite to each other. However, they both are used in statistics and regression analysis.Covarianceshows us how the two variables vary or differ from each other, whereasCorrelationshows us the relationship and how they are related, i.e., how modification of one variable impacts the other.

For example, let us express these concepts mathematically for two random variables, A and B, with mean values as **Ua** and **Ub** and standard deviation as Sa and Sb, respectively.

Effectively we can define the relationship between the two:

Both **Correlation**s and **Covariance** find application in statistical and financial analysis fields. Since **Correlation** standardizes the connection, it is helpful in the comparison of any two variables. In addition, it helps analysts develop strategies like pair trade and **hedging** for efficient returns on the portfolio and safeguarding these returns in terms of adverse movements in the stock market.

We will consider some real-time values to understand the differences or the relation between them with specific examples.

##### Table of contents

### Key Takeaways

**Covariance****and****Correlation**are very closely related to each other, and yet they differ a lot.**Covariance**defines the type of interaction, but**Correlation**represents the type and the strength of this relationship. Due to this reason,**Correlation**is often termed as the special case of**Covariance**.- Most analysts prefer
**Correlation**as it remains unaffected by the changes in dimensions, locations, and scale. Also, since it is limited to a range of -1 to +1, we can compare variables across domains. However, an important limitation is that these concepts measure only the linear relationshipLinear RelationshipA linear relationship describes the relation between two distinct variables - x and y - in the form of a straight line on a graph. When presenting a linear relationship through an equation, the value of y is derived through the value of x, reflecting their correlation.read more.

### What Is Covariance?

**Covariance** measures how the two variables move concerning each other and is an extension of the concept of variance (which tells about how a single variable varies). It can take any value from -∞ to +∞.

- The higher this value, the more dependent the relationship is. A positive number signifies positive
**Covariance,**that means that an increase or variation will correspondingly increase in the other variable, provided other conditions remain constant. - A negative number signifies negative
**Covariance**, which denotes an opposite relationship between the two variables. Though**Covariance**is perfect for defining the type of relationship, it is not good for interpreting its magnitude.

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Source: Covariance Vs Correlation (wallstreetmojo.com)

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**Covariance** **Formula And Example**

**Covariance**

Let us understand the covariance formula to calculate and assess the movements of variables. The formula is,

where,

**R**=Return of stock A in the_{Ai}**i**interval^{th}**R**_{Bi}_{ }=Return of stock B in the**i**interval^{th}**R**=Mean of the return of stock A_{A}**R**=Mean of the return of stock B_{B}**n**= Sample size or the number of intervals

The covariance calculation between stock A and stock B can also be obtained by multiplying the standard deviation of returns of stock A, the standard deviation of returns of stock B, and the correlation between stock A’s and stock B’s returns.

Mathematically, it represents as,

**Cov (R _{A}, R_{B}) = ρ(A, B) * ơ_{A }* ơ_{B}**

where,

**ρ(A, B)**= Correlation between returns of stock A and stock B

**ơ**= Standard deviation of returns of stock A_{A}**ơ**= Standard deviation of returns of stock B._{B}

Let us consider an example to understand the Covariance.

Consider the data given below of stock A and stock B with the following daily returns for three days.

We will determine the covariance between stock A and stock B.

Given, **R _{A1 }= 1.2%, R_{A2 }= 0.5%, R_{A3 }= 1.0%**

**R _{B1}= 1.7%, R_{B2 }= 0.6%, R_{B3 }= 1.3%**

Therefore, the calculation will be as follows,

Now, mean return of stock A, **R _{A}= (R_{A1 }+ R_{A2 }+ R_{A3 }) / n**

**R**= (1.2% + 0.5% + 1.0%) / 3_{A}**R**=_{A}**0.9%**

Next, Mean Return of Stock B, **R _{B}= (R_{B1 }+R_{B2}+ R_{B3 }) / n**

**R**= (1.7% + 0.6% + 1.3%) / 3_{B}**R**=_{B}**1.2%**

Therefore, one can calculate the covariance between stock A and stock B as:

**Covariance between Stock A and Stock B will be –**

**=((B2-C5)*(C2-C6)+(B3-C5)*(C3-C6)+(B4-C5)*(C4-C6))/(COUNT(A2:A4)-1)**

= **[(1.2 – 0.9) * (1.7 – 1.2) + (0.5 – 0.9) * (0.6 – 1.2) + (1.0 – 0.9) * (1.3 – 1.2)] / (3 -1)**

= **0.200**

- Therefore, the
**Cov(R**=_{A}, R_{B})**0.200**

Therefore, the covariance between stock A and stock B is **0.200**, as shown above, which is positive. As such, both returns move in the same direction, i.e., either with positive or negative returns.

### What Is The Correlation?

**Correlation** is a step ahead of **Covariance** as it quantifies the relationship between two random variables. In simple terms, it is a unit measure of how these variables change concerning each other (normalized **Covariance** value).

- The
**Correlation**has an upper and lower cap on a range, unlike**Covariance**. It can only take values between +1 and -1. A**Correlation**of +1 indicates that random variables have a direct and strong relationship. - On the other hand, the
**Correlation**of -1 indicates a strong inverse relationship, and an increase in one variable will lead to an equal and opposite decrease in the other variable. 0 means that the two numbers are independent.

**Correlation** Formula And Example

**Correlation**Formula And Example

If we have two variables, x, and y, then the **correlation coefficient** between 2 variables can be found as:

**Correlation Coefficient = ∑(x(i)- mean(x))*(y(i)-mean(y)) / √ (∑(x(i)-mean(x)) ^{2} * ∑(y(i)-mean(y))^{2})**

where,

**x(i)=**value of x in the sample**Mean(x) =**mean of all values of x**y(i) =**value of y in the sample**Mean(y) =**mean of all values of y

Let us consider an example to understand the Correlation.

Below are the values of x and y, with all the necessary details.

The calculation is as follows.

Basis Excel formula = **CORREL (array(x), array(y))**

Therefore, the **Coefficient = +0.95**

Since this coefficient is near +1, x and y are highly positively correlated.

### Correlation Vs Covariance Infographics

Let us see the top differences between **Correlation and Covariance**.

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Source: Covariance Vs Correlation (wallstreetmojo.com)

### Correlation Vs Covariance Key Differences

**Covariance**is an indicator of how two random variables change concerning each other.**Correlation**, on the other hand, measures the strength of this relationship. The correlationCorrelationCorrelation is a statistical measure between two variables that is defined as a change in one variable corresponding to a change in the other. It is calculated as (x(i)-mean(x))*(y(i)-mean(y)) / ((x(i)-mean(x))2 * (y(i)-mean(y))2.read more value is bound to the upper by +1 and the lower by -1. Thus, it is a definite range. However, the range of**Covariance**is indefinite. It can take any positive or negative value (theoretically, the range is -∞ to +∞). You can rest assured that a**Correlation**of .5 is greater than .3, and the first set of numbers (with a**Correlation**of .5) are more dependent on each other than the second set (with a**Correlation**of .3). Interpreting such a result would be tough from**Covariance**calculations.- Change of scale affects
**Covariance**. For example, if the value of two variables is multiplied by similar or different constants, this affects these two numbers’ calculated**Covariance**. However, applying the same mechanism for**Correlation**, multiplication by constants does not change the previous result. That is because a change in measurements does not affect**Correlation**. **Correlation**defines the inter-dependence of two variables that helps us to compare any two variables irrespective of their units and dimensions.- We can calculate CovarianceCovarianceCovariance is a statistical measure used to find the relationship between two assets and is calculated as the standard deviation of the return of the two assets multiplied by its correlation. If it gives a positive number then the assets are said to have positive covariance i.e. when the returns of one asset goes up, the return of second assets also goes up and vice versa for negative covariance.read more for only two variables. On the other hand,
**Correlation**can be calculated for multiple sets of numbers. Another factor that makes the**Correlation**desirable to analysts compared to**Covariance**.

### Covariance Vs Correlation Comparative Table

Basis | Covariance | Correlation |
---|---|---|

Meaning | Covariance indicates how two random variables are dependent on each other. A higher number denotes higher dependency. | Correlation indicates how strongly these two variables are related, provided other conditions are constant. The maximum value is +1, representing a perfect dependent relationship. |

Relationship | We can deduct Correlation from a Covariance. | Correlation provides a measure of Covariance on a standard scale. It is deduced by dividing the calculated Covariance by the standard deviation. |

Values | The value of Covariance lies in the range of -∞ and +∞. | Correlation is limited to values between the range -1 and +1. |

Scalability | Covariance is affected. | Correlation is not affected by a change in scales or multiplication by a constant. |

Units | Covariance has a definite unit as deduced by the multiplication of two numbers and their units. | Correlation is a unitless absolute number between -1 and +1, including decimal values. |

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Mike Geddes says

Great article where information is presented in a simple, accurate, and concise manner.