Quick answer
Covariance is cov = Σ(x − x̄)(y − ȳ) ÷ (n − 1) for a sample or ÷ n for a population. For x = 1 to 5 and y = 2, 4, 5, 4, 5 the sample covariance is 1.5 and the population covariance 1.2.
Enter equal-length x and y lists. The calculator returns the sample and population covariance.
Related tools: Least Squares Regression Calculator: Fit the least-squares regression line to paired data and read off the slope, intercept and a prediction. Correlation Coefficient Calculator: Pearson correlation of two variables. Linear Regression Calculator: Least-squares line y = a + bx through paired data.
How to use the Covariance Calculator
- Type x values, for example 1, 2, 3, 4, 5.
- Type y values, for example 2, 4, 5, 4, 5.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| x values | Text, such as 1, 2, 3, 4, 5 | — |
| y values | Text, such as 2, 4, 5, 4, 5 | — |
Formula and method
Formula
cov = Σ(x − x̄)(y − ȳ) ÷ (n − 1) (sample) or ÷ n (population)
IEEE-754 double precision; shown to at most 6 decimal places.
Computed from deviations from the means.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: x values = 1,2,3,4,5; y values = 2,4,5,4,5.
| Result | Value |
|---|---|
| Sample covariance | 1.5 |
| Population covariance | 1.2 |
The first line is the headline result (sample covariance). Below it the covariance Calculator lists population covariance, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
What it tells you
A positive covariance means the variables tend to rise together. Its size depends on units, so it can't be compared across data sets; correlation standardises it: r = cov ÷ (sₓ s_y).
Sample or population
Use n − 1 for a sample (the unbiased estimate) and n for a full population. Here the sum of cross-products is 6, so 6 ÷ 4 = 1.5 and 6 ÷ 5 = 1.2.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- y values: entering "1" shows "Enter at least 2 numbers." beside that field and no result.
Choosing between the Covariance Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Least Squares Regression Calculator](/statistics/least-squares-regression/) | x values and y values | Regression line |
| [Correlation Coefficient Calculator](/statistics/correlation-coefficient-calculator/) | x values and y values | Correlation r |
| [Linear Regression Calculator](/statistics/linear-regression-calculator/) | x values and y values | Regression line |
| [R-Squared Calculator](/statistics/r-squared-calculator/) | x values and y values | R² |
| [Multiple Linear Regression Calculator](/statistics/multiple-regression-calculator/) | response values and predictor 1 | Coefficients (intercept, b₁, b₂ …) |
Background: regression and correlation
Least squares chooses the line that minimises the sum of the squared vertical distances from the points. The slope is the sum of the cross-products of the deviations divided by the sum of the squared x deviations, and the intercept is the mean of y minus the slope times the mean of x.
For simple regression R² is the square of the correlation coefficient r, the share of the variation in y that the line explains. R² has no sign, so the sign of the slope tells you whether the relationship rises or falls. A strong fit does not prove one variable causes the other.
Assumptions and limitations
- Equal-length lists
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for y values to confirm this.
Frequently asked questions
Can covariance be negative?
Yes, when one rises as the other falls.
What is the covariance of x with itself?
The covariance of x with itself is its variance, so cov(x, x) = s².
Does zero covariance mean independence?
Not necessarily: variables can be dependent in a non-linear way (such as y = x² on symmetric data) and still have zero covariance.
What does the Covariance Calculator calculate?
Sample and population covariance. It returns Sample covariance and Population covariance.
What formula does the Covariance Calculator use?
cov = Σ(x − x̄)(y − ȳ) ÷ (n − 1) (sample) or ÷ n (population). Computed from deviations from the means.
What do I need to enter in the covariance Calculator?
You enter x values and y values.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Equal-length lists.
What is an example of a covariance result?
With x values 1,2,3,4,5 and y values 2,4,5,4,5, the sample covariance is 1.5.
Related calculators
- Least Squares Regression Calculator — Fit the least-squares regression line to paired data and read off the slope, intercept and a prediction.
- Correlation Coefficient Calculator — Pearson correlation of two variables.
- Linear Regression Calculator — Least-squares line y = a + bx through paired data.
- R-Squared Calculator — Coefficient of determination for a simple linear fit.
- Multiple Linear Regression Calculator — Least-squares coefficients and R² for up to three predictors.
- Standard Deviation Calculator — Sample or population standard deviation with variance and mean.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.