What is the difference between sample covariance and Pearson's correlation coefficient r?

## The quantity underneath both Suppose you have n paired observations `(x1, y1), ..., (xn, yn)` -- say height and weight for n people. Centre each column on its own sample mean and multiply the two deviations for each person: `(xi - xbar) * (yi - ybar)` A person who is above average on both variables contributes a positive product. Someone below average on both also contributes a positive product, because a negative times a negative is positive. Someone above average on one and below on the other contributes a negative product. Sum the products and divide: `cov(x, y) = sum_i (xi - xbar) * (yi - ybar) / (n - 1)` That is the **sample covariance**. The `n - 1` divisor (rather than n) is the same correction used for the sample variance: two means were estimated from the same data, so one degree of freedom is spent. Note that covariance with itself is just the variance: `cov(x, x) = var(x)`. ## Why the covariance number is hard to read The sign of a covariance is informative: positive means the two variables tend to sit on the same side of their means, negative means opposite sides, zero means the cross-products cancel. The **magnitude** is not readable, for two reasons. 1. **It carries units.** A covariance of height (cm) with weight (kg) is measured in centimetre-kilograms. There is no intuition for whether 120 cm-kg is big. 2. **It rescales with the units.** Convert the same people to inches and pounds. Each height is multiplied by about 0.394 and each weight by about 2.205, so every deviation and therefore every cross-product is multiplied by about 0.394 * 2.205 = 0.869. The covariance changes even though nothing about the people changed. In general, for `x' = a + b*x` and `y' = c + d*y`, `cov(x', y') = b * d * cov(x, y)`: shifts do nothing, scalings multiply through. Because of this you cannot compare a covariance across datasets, across variable pairs, or even across unit choices for the same pair. ## Pearson's r as standardised covariance Pearson's product-moment correlation divides the covariance by the two sample standard deviations: `r = cov(x, y) / (sx * sy)` The standard deviations carry the same units as their variables, so units cancel completely and r is a pure number. Under the same rescaling, `sx' = |b| * sx` and `sy' = |d| * sy`, so the b and d factors cancel in the ratio and `r' = sign(b * d) * r`. Rescaling by positive constants leaves r untouched; only flipping the direction of one variable flips the sign. An equivalent way to see it: r is the mean product of z-scores, `r = sum(zx_i * zy_i) / (n - 1)` where `zx_i = (xi - xbar) / sx`. Standardise both columns first and covariance and correlation become the same number. ## Reading the value r lies in -1 to 1, and the bound follows from the Cauchy-Schwarz inequality. r = 1 means every point lies exactly on a straight line with positive slope; r = -1 means the same with negative slope; r = 0 means no linear tendency in this sample. Strength is the absolute value: r = -0.8 is a stronger linear association than r = 0.5, just in the opposite direction. Common rough language calls |r| around 0.1 weak, 0.3 moderate and 0.5 or more strong, but those thresholds are field-dependent and should never be treated as rules. ## What each is good for Covariance is the building block: it is what you accumulate when you assemble a covariance matrix over many variables, and it is what enters variance formulas for sums of variables. Correlation is the reporting quantity: comparable, unitless, bounded, and immediately interpretable as tightness around a straight line. One consequence worth stating out loud: a large covariance does **not** imply a large r. If both variables have huge spread, the covariance can be enormous while the points still form a loose cloud, and r comes out small. The reverse holds too: two tightly-linked variables with tiny spread can have a near-zero covariance and an r of 0.95. ## In an interview Write both formulas, say the words "covariance has units, r does not", and give the unit-conversion example. Then close the loop: r is covariance divided by both standard deviations, which is the same as the covariance of the standardised columns.