Covariance measures how two variables change together. A positive covariance means they tend to increase together; a negative covariance means one tends to increase when the other decreases. Unlike correlation, covariance is not scaled, so its magnitude depends on the units of the variables.

What is the difference between sample and population covariance?

Sample covariance divides by n minus 1 (Bessel's correction) to give an unbiased estimate of the population covariance when you only have a sample of data. Population covariance divides by n and is used when you have data for the entire population.

How is covariance related to correlation?

Pearson correlation r is simply covariance divided by the product of the standard deviations of x and y. This standardisation removes the effect of the units and scales r to always lie between -1 and +1, which makes it easier to compare relationships across different data sets.

Covariance Calculator

Covariance measures the direction of the linear relationship between two variables by showing how they move together relative to their respective means. When both variables tend to be above their means at the same time, covariance is positive. When one tends to be above its mean while the other is below, covariance is negative. A covariance near zero indicates little or no linear relationship. Unlike correlation, covariance is not constrained to a fixed scale, so a covariance of 1.5 could represent a weak relationship if the variables have large standard deviations, or a strong one if they are tightly distributed. This is why covariance is often used as a building block for other statistics, such as the Pearson correlation coefficient, rather than as a standalone measure. The key distinction between sample and population covariance is the denominator: population covariance divides by n (the number of pairs), while sample covariance divides by n minus 1. The n minus 1 version applies Bessel's correction to give an unbiased estimate when your data are a sample drawn from a larger population. Enter your x and y values as comma-separated lists of equal length and the calculator returns both versions of covariance, the mean of each variable, and the number of pairs used in the calculation. It suits students, researchers, and data analysts who need to check the direction and raw magnitude of association between two continuous variables before computing correlation or building a regression model.

X values (comma-separated)

Y values (comma-separated)

1.5000

sample covariance

Population covariance1.2000

Mean of x3.0000

Mean of y4.0000

Pairs (n)5

How it works

The calculator parses the two comma-separated lists and takes the shorter length as n. It computes the mean of x and the mean of y. For each pair it calculates (xi minus x-bar) times (yi minus y-bar) and sums those products. Population covariance is that sum divided by n. Sample covariance divides by n minus 1 instead, applying Bessel's correction to reduce bias when working with a sample rather than the whole population.

Worked example

With x values 1, 2, 3, 4, 5 and y values 2, 4, 5, 4, 5, the mean of x is 3 and the mean of y is 4. The sum of (xi minus 3) times (yi minus 4) across all five pairs is (1-3)(2-4) + (2-3)(4-4) + (3-3)(5-4) + (4-3)(4-4) + (5-3)(5-4) = 4 + 0 + 0 + 0 + 2 = 6. Population covariance is 6 / 5 = 1.2000. Sample covariance is 6 / 4 = 1.5000. These match the default values pre-filled above.

Related calculators

Correlation Coefficient Calculator: standardise covariance to get Pearson r on a -1 to +1 scale.
Coefficient of Variation Calculator: compare variability as a proportion of the mean.
Standard Error Calculator: estimate the precision of a sample mean.
Percent Error Calculator: measure deviation from an accepted reference value.

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