Covariance Calculator
Calculate sample and population covariance from paired data, with the full deviation-product table, both means and Pearson's correlation coefficient alongside.
Summary
Strong positive linear relationship
cov(X,Y) = Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1) = 6 / 4 = 1.5
r = cov(X,Y) / (sₓ · s_y) = 0.774597
Covariance carries the units of X times the units of Y, so its size only means something next to those units. Pearson r divides it by both standard deviations to give a unit-free value between −1 and +1, which is why the two are shown side by side. The least-squares slope cov/varₓ is 0.6.
Deviation products
| 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 |
| Σ (x − x̄)(y − ȳ) | 6 | |||
What is the Covariance Calculator?
The ByteTools Covariance Calculator measures how two variables move together. Paste your X values and your Y values, and it returns both the sample covariance with the n − 1 denominator and the population covariance with n, so whichever one your course asks for is already on screen.
- Sample covariance (÷ n−1) and population covariance (÷ n) together
- Pearson correlation coefficient r for a unit-free comparison
- Plain-English strength and direction reading
- Both means, both standard deviations and the least-squares slope
- Deviation-product table showing every term in the sum
- 100% private — the data never leaves your browser
How to use the Covariance Calculator
- 1
Paste your X values into the first box, separated by spaces, commas or new lines.
- 2
Paste the matching Y values into the second box in the same order.
- 3
Read the sample covariance, population covariance and Pearson r across the top.
- 4
Check both means and standard deviations in the summary panel.
- 5
Scan the deviation-product table to see each pair's contribution, then click Copy results.
About the Covariance Calculator
The ByteTools Covariance Calculator measures how two variables move together. Paste your X values and your Y values, and it returns both the sample covariance with the n − 1 denominator and the population covariance with n, so whichever one your course asks for is already on screen.
Because raw covariance carries the units of X times the units of Y, its size on its own says very little. That is why Pearson's correlation coefficient r appears right next to it — the same relationship expressed on a unit-free −1 to +1 scale, along with a plain-English reading of how strong the relationship is.
A deviation-product table shows each pair's contribution, listing x − x̄, y − ȳ and their product so you can follow the sum term by term. Every calculation runs locally in your browser and none of your data is uploaded.
Frequently asked questions
What is the difference between covariance and correlation?
They describe the same relationship, but covariance keeps the original units so its magnitude depends on the scale of your data. Correlation divides covariance by both standard deviations, producing a value between −1 and +1 that can be compared across completely different datasets.
Should I use the sample or population covariance formula?
Use the sample version with n − 1 when your data is a sample and you want to estimate the covariance of the wider population, which covers most real analyses. Use the population version with n only when your data really is the entire population you care about.
What does a negative covariance mean?
It means the two variables tend to move in opposite directions: when one is above its mean the other is usually below its. A positive covariance means they tend to rise and fall together, and a value near zero means there is no consistent linear pattern.
Does a high covariance prove one variable causes the other?
No. Covariance and correlation only measure how two variables move together, and that co-movement can come from coincidence, a shared cause or reverse causation. Establishing cause needs an experiment or a design that rules those alternatives out.
Why do X and Y need the same number of values?
Covariance is defined on pairs, so every x must have exactly one matching y in the same position. If the two lists are different lengths there is no way to know which values pair up, so the calculator asks you to fix it rather than guessing.
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