Last modified: September 21, 2024

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Covariance is a fundamental statistical measure that quantifies the degree to which two random variables change together. It indicates the direction of the linear relationship between variables:

• A positive covariance implies that as one variable increases, the other tends to increase as well.
• A negative covariance suggests that as one variable increases, the other tends to decrease.
• A zero covariance indicates no linear relationship between the variables.

Definition

The covariance between two random variables $X$ and $Y$ is defined as the expected value (mean) of the product of their deviations from their respective means:

$$ \text{Cov}(X, Y) = \mathbb{E}\left[ (X - \mu_X)(Y - \mu_Y) \right] $$

Where:

• $\text{Cov}(X, Y)$ is the covariance between $X$ and $Y$.
• $\mathbb{E}$ denotes the expected value operator.
• $\mu_X = \mathbb{E}[X]$ is the mean of $X$.
• $\mu_Y = \mathbb{E}[Y]$ is the mean of $Y$.

Alternative Expression

By expanding the definition and applying the linearity properties of expectation, covariance can also be expressed as:

$$ \text{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] $$

Derivation:

1. Start with the definition:

$$ \text{Cov}(X, Y) = \mathbb{E}\left[ (X - \mu_X)(Y - \mu_Y) \right] $$

1. Expand the product inside the expectation:

$$ \text{Cov}(X, Y) = \mathbb{E}\left[ XY - X \mu_Y - \mu_X Y + \mu_X \mu_Y \right] $$

1. Use the linearity of expectation:

$$ \text{Cov}(X, Y) = \mathbb{E}[XY] - \mu_Y \mathbb{E}[X] - \mu_X \mathbb{E}[Y] + \mu_X \mu_Y $$

1. Recognize that $\mu_X = \mathbb{E}[X]$ and $\mu_Y = \mathbb{E}[Y]$:

$$ \text{Cov}(X, Y) = \mathbb{E}[XY] - \mu_Y \mu_X - \mu_X \mu_Y + \mu_X \mu_Y = \mathbb{E}[XY] - \mu_X \mu_Y $$

Thus, we arrive at:

$$ \text{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] $$

Interpretation

• Positive Covariance ($\text{Cov}(X, Y) > 0 $): Indicates that $X$ and $Y$ tend to increase or decrease together.
• Negative Covariance ($\text{Cov}(X, Y) < 0 $): Indicates that when $X$ increases, $Y$ tends to decrease, and vice versa.
• Zero Covariance ($\text{Cov}(X, Y) = 0 $): Suggests no linear relationship between $X$ and $Y$.

Important Note:

• If $X$ and $Y$ are independent, then $\text{Cov}(X, Y) = 0 $.
• However, a covariance of zero does not necessarily imply independence. Variables can be uncorrelated (zero covariance) but still dependent in a non-linear way.

Properties of Covariance

I. Symmetry:

$$ \text{Cov}(X, Y) = \text{Cov}(Y, X) $$

II. Linearity in Each Argument:

For constants $a$ and $b$, and random variables $X$, $Y$, and $Z$:

$$ \text{Cov}(aX + bY, Z) = a \text{Cov}(X, Z) + b \text{Cov}(Y, Z) $$

III. Covariance with Itself (Variance Relation):

The covariance of a variable with itself is the variance of that variable:

$$ \text{Cov}(X, X) = \text{Var}(X) $$

IV. Scaling:

If $a$ and $b$ are constants:

$$ \text{Cov}(aX, bY) = ab \text{Cov}(X, Y) $$

V. Addition of Constants:

Adding a constant to a variable does not affect the covariance:

$$ \text{Cov}(X + c, Y) = \text{Cov}(X, Y) $$

VI. Relationship with Correlation:

Covariance is related to the correlation coefficient $\rho_{XY}$:

$$ \rho_{XY} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} $$

Where $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$, respectively.

Sample Covariance

When working with sample data, the sample covariance between two variables $X$ and $Y$ is calculated as:

$$ s_{XY} = \text{Cov}(X, Y) = \frac{1}{n - 1} \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y}) $$

Where:

• $n$ is the number of observations.
• $X_i$ and $Y_i$ are the $i$-th observations of variables $X$ and $Y$.
• $\bar{X}$ and $\bar{Y}$ are the sample means of $X$ and $Y$.

Note: The denominator $n - 1$ provides an unbiased estimate of the covariance for a sample drawn from a population.

Example: Calculating Covariance Step by Step

Let's calculate the covariance between two variables $X$ and $Y$ using the following dataset:

Step 1: Calculate the Sample Means

Compute the mean of $X$ and $Y$:

$$ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i = \frac{1 + 2 + 3}{3} = \frac{6}{3} = 2 $$

$$ \bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i = \frac{2 + 4 + 6}{3} = \frac{12}{3} = 4 $$

Step 2: Compute the Deviations from the Mean

Calculate $(X_i - \bar{X})$ and $(Y_i - \bar{Y})$:

Step 3: Calculate the Product of Deviations

Compute $(X_i - \bar{X})(Y_i - \bar{Y})$:

Step 4: Sum the Products of Deviations

Compute the sum:

$$ \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y}) = 2 + 0 + 2 = 4 $$

Step 5: Calculate the Sample Covariance

Use the sample covariance formula:

$$ s_{XY} = \text{Cov}(X, Y) = \frac{1}{n - 1} \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y}) $$

Since $n = 3 $:

$$ s_{XY} = \frac{1}{3 - 1} \times 4 = \frac{1}{2} \times 4 = 2 $$

Interpretation:

• The positive covariance of $2 $ indicates that $X$ and $Y$ tend to increase together.
• Since the data points lie perfectly on a straight line ($Y = 2X$), the covariance reflects a perfect positive linear relationship.

Step 6: Calculate the Variances (Optional)

For completeness, calculate the variances of $X$ and $Y$:

Variance of $X$

$$ s_{XX} = \text{Var}(X) = \frac{1}{n - 1} \sum_{i=1}^{n} (X_i - \bar{X})^2 $$

Compute $(X_i - \bar{X})^2 $:

Sum:

$$ \sum_{i=1}^{n} (X_i - \bar{X})^2 = 1 + 0 + 1 = 2 $$

Compute variance:

$$ s_{XX} = \frac{1}{2} \times 2 = 1 $$

Variance of $Y$

Similarly, compute $(Y_i - \bar{Y})^2 $:

Sum:

$$ \sum_{i=1}^{n} (Y_i - \bar{Y})^2 = 4 + 0 + 4 = 8 $$

Compute variance:

$$ s_{YY} = \text{Var}(Y) = \frac{1}{2} \times 8 = 4 $$

Step 7: Calculate the Correlation Coefficient (Optional)

The correlation coefficient $r_{XY}$ standardizes the covariance, providing a dimensionless measure of the strength and direction of the linear relationship:

$$ r_{XY} = \frac{s_{XY}}{\sqrt{s_{XX} \times s_{YY}}} = \frac{2}{\sqrt{1 \times 4}} = \frac{2}{2} = 1 $$

Interpretation:

• A correlation coefficient of $1$ indicates a perfect positive linear relationship between $X$ and $Y$.
• This makes sense since $Y = 2X$ in the dataset.

Plot:

Limitations of Covariance

I. Scale Dependence:

• Covariance values are affected by the units of measurement of the variables.
• For example, measuring height in meters vs. centimeters will change the covariance.

II. Comparison Difficulties:

• Because covariance is not standardized, comparing covariances across different datasets or variables with different scales is challenging.
• This is why the correlation coefficient, which standardizes covariance, is often used.

III. Not a Measure of Strength:

• The magnitude of covariance does not directly indicate the strength of the relationship.
• A large covariance could be due to large variances rather than a strong relationship.

IV. Linear Relationships Only:

• Covariance measures only linear relationships.
• It does not capture non-linear dependencies between variables.