Correlation coefficient from covariance:
formula and worked example

Why covariance is rescaled into a correlation

Covariance tells you whether two variables tend to rise and fall together, but its size is hard to interpret. It carries the units of both variables multiplied together, such as rupees times kilograms, and it grows if you simply measure in a different unit. A covariance of 15 might be strong in one data set and trivial in another.

The correlation coefficient fixes this by dividing out the spread of each variable. The result is unit-free and capped at -1 and 1, which lets you compare the strength of association between completely different pairs of variables.

It is used in finance to compare how assets move together, in science to relate measurements, and in marketing to link spending with sales. In all cases it reports linear association, not cause.

The formula and its inputs

Divide the covariance by the product of the two standard deviations.

• Covariance: the covariance between x and y, 15 in the example.
• Std x: the standard deviation of x, 5 in the example.
• Std y: the standard deviation of y, 6 in the example.

The ordering of the calculation does not matter, since multiplication and division commute. All three must be computed under the same convention, either sample-based or population-based. Mixing them, for instance a sample covariance with population standard deviations, biases the answer by a factor that depends on the sample size.

Worked example: covariance 15, spreads 5 and 6

Two variables have a covariance of 15, and their standard deviations are 5 and 6.

1. 1Multiply the standard deviations: 5 × 6 = 30.
2. 2Divide the covariance by that product: 15 ÷ 30 = 0.5.

The correlation is 0.5. Before accepting it, check that the answer lies between -1 and 1, and that 15 is smaller in size than 30, which is the largest covariance these spreads could produce.

What a correlation of 0.5 does and does not say

A value near 1 means the two variables move almost in lockstep along a straight line, a value near -1 means they move in opposite directions, and a value near 0 means no linear pattern. At 0.5, there is a moderate positive tendency: higher x values go with higher y values more often than not, but with plenty of scatter.

Squaring r gives the share of variation in one variable that a straight-line relationship with the other accounts for. Here 0.5² = 0.25, or a quarter. That is a sobering way to read a figure that sounds moderately strong.

Descriptive labels for ranges, such as weak, moderate or strong, are conventions that vary between fields. A correlation of 0.5 is considered useful in some social sciences and disappointing in a laboratory measurement comparison.

How the covariance moves the correlation

Holding both spreads fixed, the correlation is directly proportional to the covariance. A 10% rise in covariance gives a 10% rise in r, from 0.5 to 0.55.

This proportionality holds only inside the valid range. A covariance of 30 would give a perfect correlation of 1, and any figure above that is impossible with these standard deviations. Change either standard deviation and r moves in the opposite direction.

Errors that distort a correlation

• Using variances in place of standard deviations. Variances are the squares, so the denominator becomes far too large and r far too small.
• Mixing sample and population conventions between the covariance and the standard deviations.
• Reading a high correlation as proof of cause. Two things can move together because of a third factor, or by coincidence.
• Ignoring outliers and non-linear patterns. A curved relationship can have a correlation near zero even though the variables are tightly linked.

A check on a reported figure

A report states a covariance of 40 between two series with standard deviations of 4 and 5. Is the implied correlation believable?

1. 1Multiply the standard deviations: 4 × 5 = 20.
2. 2Divide: 40 ÷ 20 = 2.
3. 3The result exceeds 1, which is impossible, so something in the report is wrong.

Likely culprits are a variance quoted as a standard deviation, a unit mismatch, or a covariance computed on different data from the spreads.