# Covariance vs. Correlation: What's the Difference?

Edited by Aimie Carlson || By Janet White || Published on November 20, 2023
Covariance measures the directional relationship between two variables, while Correlation quantifies the strength and direction of that relationship.

## Key Differences

Covariance is a measure indicating the extent to which two variables change in tandem. On the other hand, Correlation quantifies the strength and direction of the linear relationship between those two variables.
In statistical terms, Covariance can take on any value between negative infinity and positive infinity, which can make interpretations challenging. Conversely, Correlation values range between -1 and 1, making it easier to discern the strength and direction of the relationship.
While Covariance might indicate that two variables increase or decrease together, it does not give a clear picture of the relationship's consistency. Correlation, however, directly signifies whether the relationship is positive, negative, or nonexistent.
If Covariance is positive, it suggests that the variables move in the same direction, and if negative, they move in opposite directions. Correlation, meanwhile, not only indicates direction but also how strong the relationship is, with values close to 1 or -1 indicating stronger relationships.
A key distinction between Covariance and Correlation is their units. Covariance has units that are a product of the two variables' units. In contrast, Correlation is dimensionless, as it's a standardized measure, providing more clarity in comparative scenarios.

## Comparison Chart

### Value Range

Negative infinity to positive infinity
-1 to 1

### Interpretability

Can be challenging due to range and units
Easier due to standardized range

### Measurement Focus

Directional relationship
Strength and direction of relationship

### Units

Product of the two variables' units
Dimensionless

### Indication

Movement of variables in tandem
Linear relationship and its consistency

## Covariance and Correlation Definitions

#### Covariance

Indicates whether an increase in one variable would result in an increase or decrease in another variable.
Based on the positive Covariance, sales seem to rise with advertising spends.

#### Correlation

A statistical measure expressing the extent of interdependence between two variables.
The Correlation between smoking and lung disease was found to be strong and positive.

#### Covariance

A measure used to gauge the linear direction of two sets of data.
We calculated the Covariance to understand the dynamic between age and spending.

#### Correlation

Represents the consistency and directionality of a relationship between two datasets.
The zero Correlation suggested no linear relationship between the datasets.

#### Covariance

A statistical measure indicating how two variables move in relation to one another.
The Covariance between stock A and stock B was positive, indicating they move together.

#### Correlation

An indicator of how closely two sets of data are related in a linear fashion.
The Correlation between hours studied and test scores was evident in the data.

#### Covariance

A value representing the degree to which two variables change together.
The Covariance between height and weight in the sampled data was noticeably high.

#### Correlation

Quantifies the strength and direction of a linear relationship between two variables.
The Correlation coefficient of 0.9 suggested a high positive relationship.

#### Covariance

Describes the joint variability of two random variables.
The Covariance between variables X and Y gave insights into their concurrent movements.

#### Correlation

A dimensionless value between -1 and 1 indicating the nature of the linear relationship between variables.
A Correlation of -0.8 between variables indicates a strong inverse relationship.

#### Covariance

(Statistics) A statistical measure of the tendency of two random variables to vary in the same direction (called positive covariance) or in an opposite direction (called negative covariance) over many observations. Covariance is equal to the summed products of the deviations of corresponding values of the two variables from their respective means.

#### Correlation

A relationship or connection between two things based on co-occurrence or pattern of change
A correlation between drug abuse and crime.

#### Covariance

(Physics) The principle that the laws of physics have the same form regardless of the system of coordinates in which they are expressed.

#### Correlation

(Statistics) The tendency for two values or variables to change together, in either the same or opposite way
As cigarette smoking increases, so does the incidence of lung cancer, indicating a positive correlation.

#### Covariance

(statistics) A statistical measure defined as $\scriptstyle\operatorname\left\{Cov\right\}\left(X, Y\right) = \operatorname\left\{E\right\}\left(\left(X - \mu\right) \left(Y - \nu\right)\right)$ given two real-valued random variables X and Y, with expected values $\scriptstyle E\left(X\right)\,=\,\mu$ and $\scriptstyle E\left(Y\right)\,=\,\nu$.

#### Correlation

An act of correlating or the condition of being correlated.

#### Covariance

(object-oriented programming) The conversion of data types from wider to narrower in certain situations.

#### Correlation

A reciprocal, parallel or complementary relationship between two or more comparable objects.

#### Covariance

A statistical measure of the relationship of two variables, formed by multiplying the difference of each variable from its mean, both variables being measured at the same time, and averaging all such products.

#### Correlation

(statistics) One of the several measures of the linear statistical relationship between two random variables, indicating both the strength and direction of the relationship.

#### Covariance

Statistical measure of the variance of two random variables measured in the same mean time period

#### Correlation

(algebra) An isomorphism from a projective space to the dual of a projective space, often to the dual of itself.

#### Correlation

Reciprocal relation; corresponding similarity or parallelism of relation or law; capacity of being converted into, or of giving place to, one another, under certain conditions; as, the correlation of forces, or of zymotic diseases.

#### Correlation

A reciprocal relation between two or more things

#### Correlation

A statistic representing how closely two variables co-vary; it can vary from -1 (perfect negative correlation) through 0 (no correlation) to +1 (perfect positive correlation);
What is the correlation between those two variables?

#### Correlation

A statistical relation between two or more variables such that systematic changes in the value of one variable are accompanied by systematic changes in the other

## FAQs

#### Is Correlation affected by changes in scale or units?

No, Correlation is dimensionless and not influenced by units, unlike Covariance.

#### Is a high Covariance always indicative of a strong relationship?

Not necessarily. A high Covariance indicates a relationship, but its strength and consistency are better gauged using Correlation.

#### Can Covariance be negative?

Yes, a negative Covariance indicates that as one variable increases, the other decreases.

#### Is it possible to have a strong Correlation but low Covariance?

It's rare because Correlation is a standardized version of Covariance. However, the magnitude of Covariance is influenced by the units.

#### What does a Correlation of 1 mean?

A Correlation of 1 indicates a perfect positive linear relationship between two variables.

#### Can Correlation determine causality?

No, Correlation can only indicate a relationship, not causation.

#### How does Covariance differ from Correlation?

Covariance measures the directional relationship between two variables, whereas Correlation quantifies the strength and direction of that relationship.

#### What does a Correlation of -0.5 indicate?

A Correlation of -0.5 suggests a moderate inverse linear relationship.

#### Is Covariance affected by changes in the center or spread of data?

Yes, Covariance is influenced by shifts in data scale and center.

#### Can two unrelated variables have a high Correlation by chance?

Yes, it's possible due to random variations or the presence of confounding variables.

#### Why might Covariance be used over Correlation in some cases?

Covariance can be useful when the scale of the original variables is of interest.

#### Why might one choose to use Correlation over Covariance?

Correlation offers clearer interpretability due to its standardized range and is not influenced by the variables' units, unlike Covariance.

#### What does a zero Covariance indicate?

A zero Covariance suggests no linear relationship between the variables.

#### What's the formula for Covariance?

Covariance is calculated as the sum of the product of deviations of two variables from their means, divided by the number of data points minus one.

#### How can Covariance be converted to Correlation?

By dividing Covariance by the product of the standard deviations of the two variables.

#### What does a Correlation of 0 signify?

A Correlation of 0 means there's no linear relationship between the variables.

#### Is the square of Correlation meaningful?

Yes, the square of the Correlation coefficient represents the proportion of variance shared between the two variables.

#### Can Correlation be greater than 1?

No, Correlation ranges between -1 and 1.

#### How is the direction of the relationship determined using Covariance?

Positive Covariance suggests variables move in the same direction, while negative indicates they move oppositely.

#### Why is Correlation a preferred measure in finance?

Correlation gives clearer insights into the strength and direction of relationships, crucial for portfolio diversification.