What is covariance

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What is covariance?

Covariance measures how many random (X, Y) variables in each population are distinct. When the population is greater or random, the matrix represents the relation between different dimensions.

The covariance matrix can be easier to understand by describing this relationship as the relationship between the two random variables in the entire dimension.

Covariance may be used to quantify variables not having the same measuring units. We can decide whether units increase or decrease by using covariance. It cannot be measured to what point the variables change together due to various measuring units for covariance.

What is population covariance?

Usually, the whole population data is not available to us. Only small amounts of samples are available most of the times. Such samples can provide an estimate of population covariance for random variables X and Y.

The below formula is used to calculate the population covariance:

Cov pop (X, Y) = sum (xi – x mean) (yi – y mean) / (n-1)

The value n-1 is slightly better than the sample covariance. As small samples are not actually the entire variance between entire populations, it is natural to infer that denominator N-1 is the best correction factors.

Relationship between sample and population covariance

The relationship between sample and population variance can be expressed as:

Cov pop (X, Y) = (n / n-1) × Cov sam (x, y)

Nevertheless, the difference between n and n-1 becomes less when the sample size increases. Therefore, the population covariance and sample covariance formula give comparable results for large samples.

Covariance vs Correlation

The relationship between covariance and correlation can be stated as:

  • Another way to express the difference between two random variables X and Y is called correlation, while,
  • Covariance computes the difference between X and Y, two random variables.

The relationship between both can be express in equation form as:

Corr (X, Y) = Cov (X, Y) / (σX × σY)


 σX represents the standard deviation of X, and

σY represents the standard deviation of Y.

The correlation is considered as a stable form of covariance. The correlation should be between 1 and -1 according to the above equation. That is why correlation is used more often than covariance, even if they perform the same function.


Can covariance be negative?

Yes. In a case, where two variables are inversely related, covariance can be negative.

What does a negative covariance mean?

Negative covariance depicts that the both variables and are inversely proportional to each other. If X goes down, Y goes up and if Y goes down, X increases.

If the covariance is negative, the variables are considered related in reverse.

For example:

If temperature goes up, use of heating instruments also goes up. It demonstrates that the relationship between both variables is inversely proportional.

What does a covariance of 0 mean?

The covariance is zero if the two random variables are not mutually dependent. However, a covariance at zero does not imply the variables are independent. A non-linear relationship can still occur, resulting in a zero covariance value.

What does covariance tell us?

Covariance shows us the extent to which two variables differ. It shows us how different a variable is from a related one. It calculates, how two variables relate monotonically to each other.

What is a high covariance?

A high covariance indicates a strong relation between the two variables. In the case of low covariance it is the other way around. The weaker relationship between two variables are represented by low covariance.

What does a covariance of 1 mean?

Covariance of 1 means that the both variables are directly proportional to each other. If one variable reduces other will increase, and in case, one variable gets a boost, other goes down.

What does analysis of covariance mean?

Covariance analysis (ANCOVA) is used to analyze differences in the mean values of the variable that are dependent on each other. These variables are related to the impact of the independent controlled variables thus taking into account the effects of the independent uncontrolled variables.