Covariance Calculator

About Covariance Calculator

Covariance measures how two variables change together. Positive = move together, negative = move opposite, zero = independent.

How to use this calculator

1. Enter the values for the first dataset, one pair at a time if needed.
2. Enter the matching values for the second dataset in the same order.
3. Make sure both datasets have the same number of values.
4. Click calculate to get the covariance result.

The formula explained

The formula computes the sample covariance, which measures the average joint spread of two variables around their means. It uses the deviations from each mean, then divides by \(n-1\) to adjust for a sample.

• \(x_i\) = the i-th value in the first dataset
• \(y_i\) = the i-th value in the second dataset
• \(\bar{x}\) = the mean of the first dataset
• \(\bar{y}\) = the mean of the second dataset
• n = the number of paired values

Step by step method

1. Find the mean of the x-values and the mean of the y-values.
2. Subtract each mean from its corresponding value to get the deviations.
3. Multiply each pair of deviations, add the products, then divide by \(n-1\).

Worked example

Problem. Find the covariance of the paired data \((2, 3), (4, 5), (6, 7)\).

1. Compute the means: \(\bar{x} = (2+4+6)/3 = 4\), \(\bar{y} = (3+5+7)/3 = 5\).
2. Find the products of deviations: \((2-4)(3-5)=4\), \((4-4)(5-5)=0\), \((6-4)(7-5)=4\). Add them to get \(8\).
3. Divide by \(n-1 = 2\): \(Cov(X,Y)=8/2=4\).

Tips and common mistakes

• Covariance depends on the units of the data, so the size of the number is not always easy to compare across different problems.
• Be sure the two datasets are paired correctly, because changing the order changes the covariance.

Frequently asked questions