Enter paired X and Y values, as x, y pairs or as two lists, to get the sample covariance (divide by n − 1) and population covariance (divide by n). Every deviation product is shown in a step table, together with the means, the correlation coefficient and a scatter plot that shows which points push the covariance up or down.
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Covariance type
Data format
Separate x and y with a comma, space or tab. You can paste two spreadsheet columns; a header line is skipped.
Separated by commas, spaces or new lines
Enter your values to see the result.
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How to use the Covariance Calculator
Choose sample or population covariance. Use sample if your data is a sample from a larger group.
Enter your data: one x, y pair per line, or switch to Two lists and paste each column.
Press Calculate covariance; results also update as you type.
Read the covariance, its sign and the correlation coefficient.
Check the step table and the scatter plot, then copy the results or download the CSV.
Covariance formulas
Quantity
Formula
When to use
Sample covariance
sxy = Σ(x − x̄)(y − ȳ) ÷ (n − 1)
Data is a sample from a larger group
Population covariance
σxy = Σ(x − μx)(y − μy) ÷ n
Data covers the whole population
Shortcut (sample)
(Σxy − n·x̄·ȳ) ÷ (n − 1)
Quick hand calculation from Σxy
Correlation
r = sxy ÷ (sx · sy)
Unit-free strength, −1 to 1
Worked example
Take the pairs (1, 2), (2, 4), (3, 5), (4, 4), (5, 5). The means are x̄ = 15 ÷ 5 = 3 and ȳ = 20 ÷ 5 = 4.
x
y
x − x̄
y − ȳ
(x − x̄)(y − ȳ)
1
2
−2
−2
4
2
4
−1
0
0
3
5
0
1
0
4
4
1
0
0
5
5
2
1
2
Sum
6
Sample covariance = 6 ÷ (5 − 1) = 1.5. Population covariance = 6 ÷ 5 = 1.2. Both are positive, so larger x values go with larger y values.
Reading the result
Sign matters most. Positive means the variables rise together, negative means one rises as the other falls.
Size depends on units. Covariance of height in centimeters and weight in kilograms is 100 times larger than with height in meters, so compare strength with the correlation r instead.
Outliers have a big effect. One point far from both means adds a large product to the sum; look at the step table to spot it.
Linear only. Covariance near zero does not prove the variables are unrelated, only that there is no straight-line trend.
Frequently asked questions
What is covariance?
Covariance measures how two variables move together. If Y tends to be above its mean when X is above its mean, the covariance is positive; if Y tends to be below its mean when X is above, it is negative; if there is no linear pattern, it is near zero.
What is the covariance formula?
Sample covariance is cov(X, Y) = Σ(x − x̄)(y − ȳ) ÷ (n − 1). Population covariance uses the same sum divided by n. Here x̄ and ȳ are the means and n is the number of pairs.
Should I use sample or population covariance?
Use population covariance when your data includes every member of the group you care about. Use sample covariance, which divides by n − 1, when the data is a sample used to estimate the covariance of a larger population; the n − 1 (Bessel's correction) removes the bias of using the sample means.
What is the difference between covariance and correlation?
Covariance depends on the units of X and Y, so its size is hard to interpret: changing meters to centimeters multiplies it by 100. Correlation divides the covariance by both standard deviations, giving a unit-free number from −1 to 1 that shows the strength of the linear relationship.
Can covariance be negative?
Yes. A negative covariance means the variables tend to move in opposite directions, for example outdoor temperature and hot drinks sold. Covariance has no fixed range; it can be any positive or negative number.
What does a covariance of zero mean?
It means there is no linear relationship between X and Y. The variables can still be related in a non-linear way: points on a U-shaped curve can have a covariance of exactly zero.
How is covariance related to variance?
The covariance of a variable with itself is its variance: cov(X, X) = Σ(x − x̄)² ÷ (n − 1). Variances and covariances together form the covariance matrix used in statistics, finance and machine learning.
Are all the tools on FreeToolCabin tested for quality and accuracy?
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