Covariance from Variance Calculator

Calculate the statistical covariance between two variables using their individual variances and the variance of their sum.

Covariance (Cov(X, Y))

What is Calculating Covariance using Variance?

Calculating covariance using variance is a specific method to determine the joint variability of two random variables, X and Y. While covariance is often calculated from raw data points, it can also be derived if you already know the individual variances of the variables (Var(X) and Var(Y)) and the variance of their sum (Var(X+Y)). This method is based on a fundamental property of variances. Covariance itself is a measure that indicates the directional relationship between two variables. A positive covariance means the variables tend to move in the same direction, while a negative covariance means they move in opposite directions. For more advanced analysis, you might use a Correlation Coefficient Calculator.

Covariance Formula and Explanation

The ability to find covariance from variance comes from the identity for the variance of the sum of two variables. The formula is as follows:

Cov(X, Y) = (Var(X + Y) - Var(X) - Var(Y)) / 2

This formula rearranges the more standard identity: Var(X + Y) = Var(X) + Var(Y) + 2 * Cov(X, Y). By isolating the covariance term, we can solve for it using the three known variance values.

Description of variables used in the formula. Values are typically unitless or in squared units of the original data.

Variable	Meaning	Unit	Typical Range
Cov(X, Y)	The covariance between variables X and Y.	Squared units of the source data	-∞ to +∞
Var(X)	The variance of variable X.	Squared units of the source data	≥ 0
Var(Y)	The variance of variable Y.	Squared units of the source data	≥ 0
Var(X+Y)	The variance of the sum of variables X and Y.	Squared units of the source data	≥ 0

Visual representation of input variances.

Practical Examples

Example 1: Financial Portfolio Analysis

An analyst is studying two stocks, A and B. They don’t have the raw daily returns but have calculated the following variances:

• Variance of Stock A’s returns (Var(A)) = 0.04
• Variance of Stock B’s returns (Var(B)) = 0.09
• Variance of the combined portfolio (A+B)’s returns (Var(A+B)) = 0.10

Using the formula for calculating covariance using variance:

Cov(A, B) = (0.10 - 0.04 - 0.09) / 2 = -0.03 / 2 = -0.015

The negative covariance suggests that the returns of Stock A and Stock B tend to move in opposite directions, which is a desirable property for portfolio diversification. A similar analysis can be performed with a Standard Deviation Calculator by first squaring the standard deviations to get variance.

Example 2: Agricultural Science

A researcher is examining the relationship between hours of sunlight (X) and plant growth in centimeters (Y). From their experiment, they have the variance data:

• Variance of Sunlight (Var(X)) = 10 (hours²)
• Variance of Growth (Var(Y)) = 25 (cm²)
• Variance of the sum of the two measures (Var(X+Y)) = 65

Cov(X, Y) = (65 - 10 - 25) / 2 = 30 / 2 = 15

The positive covariance of 15 indicates that more hours of sunlight are associated with greater plant growth.

How to Use This Covariance Calculator

1. Enter Var(X): Input the variance of your first variable into the first field.
2. Enter Var(Y): Input the variance of your second variable into the second field.
3. Enter Var(X+Y): Input the variance of the sum of the two variables. This is a crucial piece of information for this calculation.
4. Review the Results: The calculator will automatically display the calculated covariance. The intermediate steps are also shown to provide transparency on how the result for calculating covariance using variance was achieved.

Key Factors That Affect Covariance

• Direction of Relationship: If variables move together, covariance is positive. If they move in opposite directions, it’s negative.
• Magnitude of Variances: The scale of the variables heavily influences the magnitude of the covariance. Larger variances will generally lead to larger covariance values, which is why correlation is often preferred for comparing relationship strength.
• Strength of the Linear Relationship: A stronger linear relationship (either positive or negative) will result in a larger absolute covariance value, all else being equal.
• Outliers: Extreme data points can significantly skew the variance calculations, which in turn will dramatically affect the calculated covariance.
• Units of Measurement: The units of covariance are the product of the units of the two variables (e.g., cm * hours). Changing the units of the underlying data will change the value of the covariance.
• Independence: If two variables are independent, their covariance is zero. However, a covariance of zero does not necessarily mean the variables are independent. To learn more, see our article on variance explained.

FAQ

1. What does a negative covariance mean?

A negative covariance indicates an inverse relationship between two variables. When one variable’s value increases, the other variable’s value tends to decrease.

2. Is covariance the same as correlation?

No. Covariance measures the directional relationship. Correlation, which is derived from covariance, measures both the direction and the strength of the linear relationship, and is scaled to a range between -1 and +1.

3. What are the units of covariance?

The units are the product of the units of the two variables being compared. If you are comparing height (cm) and weight (kg), the covariance unit is cm-kg. This is a primary reason correlation is often used, as it is unitless.

4. Can variance be negative?

No, variance cannot be negative because it is calculated from squared differences, which are always non-negative.

5. Why would I use this method for calculating covariance using variance?

This method is useful in situations where you have access to variance data from a study or a dataset, but not the raw data points themselves. It’s a valid statistical identity.

6. What does a covariance of zero imply?

A covariance of zero implies that there is no linear relationship between the two variables. It does not rule out the possibility of a non-linear relationship.

7. Does the order of Var(X) and Var(Y) matter?

No, because addition is commutative. Cov(X,Y) is the same as Cov(Y,X).

8. Can I use standard deviation instead of variance in this calculator?

No, this calculator requires variance. However, you can easily calculate variance by squaring the standard deviation (Variance = Standard Deviation²). Our variance calculator can help with this.