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Generated on: July 20, 2026 at 09:33 PM

Covariance Calculator

Inputs

X values1, 2, 3, 4, 5
Y values2, 4, 5, 4, 5

Results

Sample covariance1.5
\begin{aligned} s_{xy} &= \dfrac{\sum_{i}(x_i - \bar{x})(y_i - \bar{y})}{n - 1} \\ &= \dfrac{{\text{?}}}{5 - 1} \\ &= 1.5 \end{aligned}
Population covariance1.2
\begin{aligned} \sigma_{xy} &= \dfrac{\sum_{i}(x_i - \bar{x})(y_i - \bar{y})}{n} \\ &= \dfrac{{\text{?}}}{5} \\ &= 1.2 \end{aligned}
Mean of X3
Mean of Y4
Count5

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Mathematics

Covariance Calculator

Calculate the sample and population covariance of two data sets from paired X and Y values. Enter two comma-separated lists of equal length to see both covariances, the means, and a step-by-step derivation.

Inputs

Data

Results

\begin{aligned} s_{xy} &= \dfrac{\sum_{i}(x_i - \bar{x})(y_i - \bar{y})}{n - 1} \\ &= \dfrac{{\text{?}}}{5 - 1} \\ &= 1.5 \end{aligned}

Details

\begin{aligned} \sigma_{xy} &= \dfrac{\sum_{i}(x_i - \bar{x})(y_i - \bar{y})}{n} \\ &= \dfrac{{\text{?}}}{5} \\ &= 1.2 \end{aligned}

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What covariance measures

Covariance measures how two variables change together. If larger values of one variable tend to accompany larger values of the other, the covariance is positive; if larger values of one accompany smaller values of the other, it is negative. A covariance near zero indicates little linear relationship.

For paired observations (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ), covariance is built from the paired deviation products — each x value's distance from the mean of X multiplied by the matching y value's distance from the mean of Y:

Σ(xi − x̄)(yi − ȳ)

Dividing this sum by n gives the population covariance; dividing by n − 1 gives the sample covariance.

Population versus sample covariance

There are two formulas, differing only in the denominator:

  • Population covariance: σxy = Σ(xi − x̄)(yi − ȳ) / n
  • Sample covariance: sxy = Σ(xi − x̄)(yi − ȳ) / (n − 1)

Use the population form when your data cover every member of the group you care about. Use the sample form — the default in most statistical software — when the data are a sample drawn from a larger population. The n − 1 divisor, known as the Bessel correction, offsets the slight underestimate that arises because the means x̄ and ȳ are estimated from the same sample.

Worked example

Suppose a small study records hours studied (X) and quiz scores (Y) for five students:

  • X = 1, 2, 3, 4, 5
  • Y = 2, 4, 5, 4, 5

Step 1 — means. x̄ = (1 + 2 + 3 + 4 + 5) / 5 = 3 and ȳ = (2 + 4 + 5 + 4 + 5) / 5 = 4.

Step 2 — deviation products. Compute each (xi − x̄)(yi − ȳ):

  • (1 − 3)(2 − 4) = (−2)(−2) = 4
  • (2 − 3)(4 − 4) = (−1)(0) = 0
  • (3 − 3)(5 − 4) = (0)(1) = 0
  • (4 − 3)(4 − 4) = (1)(0) = 0
  • (5 − 3)(5 − 4) = (2)(1) = 2

Step 3 — sum. Σ(xi − x̄)(yi − ȳ) = 4 + 0 + 0 + 0 + 2 = 6.

Step 4 — divide. Population covariance = 6 / 5 = 1.2. Sample covariance = 6 / (5 − 1) = 6 / 4 = 1.5.

The positive result confirms that hours studied and quiz scores tend to rise together in this data.

How covariance relates to correlation

Covariance tells you the direction of a linear relationship but not its strength in any comparable way, because its size depends on the units and scale of the data. The correlation coefficient fixes this by dividing the covariance by the product of the two standard deviations, rescaling it to always fall between −1 and +1.

In short: covariance is the un-normalized measure, and correlation is its standardized, unit-free counterpart. When you want to judge how tightly two variables move together, compute the correlation coefficient; when you need the raw measure — for example, to assemble a covariance matrix — use covariance directly.

Interpreting the sign and magnitude

The sign of the covariance is meaningful on its own:

  • Positive: the variables tend to move in the same direction.
  • Negative: the variables tend to move in opposite directions.
  • Near zero: little linear relationship.

The magnitude, by contrast, is not directly interpretable, because it carries the product of the two variables' units. A covariance measured in kilogram-centimeters cannot be compared to one measured in dollars-per-year. To compare strengths across data sets, always convert to the correlation coefficient.

Frequently Asked Questions (FAQ)

What is the difference between sample and population covariance?

Both measure how two variables move together, but they differ in the denominator. Population covariance divides the sum of paired deviation products by n, treating the data as the complete population. Sample covariance divides by n − 1 instead — the Bessel correction — which compensates for the fact that a sample tends to underestimate the true spread when the population means are estimated from the same data.

Use the population form only when the data cover every member of the group you care about. In most practical work the data are a sample drawn from a larger population, so the sample covariance (÷ n − 1) is the appropriate choice, and it is the value most statistical software reports by default.

How is covariance related to the correlation coefficient?

The correlation coefficient is a standardized covariance. It divides the covariance by the product of the two standard deviations, which rescales the result to always lie between −1 and +1. That makes correlation comparable across data sets with different units and spreads, whereas covariance is not.

Because of this, covariance tells you the direction of the relationship (positive, negative, or none) but not its strength in any absolute sense — a large covariance can simply mean the variables have large units. If you want a unit-free measure of how tightly two variables are related, compute the correlation coefficient instead.

What does the sign of the covariance tell me?

A positive covariance means the two variables tend to move in the same direction: when X is above its mean, Y also tends to be above its mean. A negative covariance means they move in opposite directions — X above its mean pairs with Y below its mean. A covariance near zero indicates little linear relationship between the variables.

The magnitude, however, is not directly interpretable on its own because it depends on the units and scale of the data. To judge how strong the relationship is, look at the correlation coefficient rather than the raw covariance value.

What units does covariance have?

Covariance carries the product of the two variables’ units. If X is measured in kilograms and Y in centimeters, the covariance is expressed in kilogram-centimeters. This is why the raw number is hard to interpret and cannot be compared directly between different pairs of variables.

The correlation coefficient solves this by dividing out the units, producing a dimensionless number between −1 and +1. When you need a value you can compare or interpret as a strength, use correlation; use covariance when you specifically need the un-normalized measure, for example as an entry in a covariance matrix.

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