Regression and correlation Updated November 2026

Correlation coefficient (Pearson r)

The Pearson correlation coefficient r measures the strength and direction of a straight-line relationship between two variables, from -1 to 1: r = sum((x - x-bar)(y - y-bar)) / sqrt(sum((x - x-bar)^2) * sum((y - y-bar)^2)). Values near 0 mean no linear relationship.

Syntax

r = Sxy / sqrt(Sxx * Syy)

Part	What it is
`Sxy`	sum of (x - x-bar)(y - y-bar)
`Sxx`	sum of (x - x-bar)^2
`Syy`	sum of (y - y-bar)^2

Examples

x: 1, 2, 3, 4, 5  (x-bar = 3)
y: 2, 4, 5, 4, 5  (y-bar = 4)
Sxy = (-2)(-2) + (-1)(0) + 0 + (1)(0) + (2)(1) = 6
Sxx = 10, Syy = 6
r = 6 / sqrt(60)

Result: r = 0.775

Excel: =CORREL(A1:A5, B1:B5) or =PEARSON(A1:A5, B1:B5)

x: 1, 2, 3, 4, 5; y: 2, 4, 6, 8, 10 (exact line, slope 2)

Result: r = 1

x: 1, 2, 3; y: 3, 2, 1

Result: r = -1

Test whether r = 0.775 from n = 5 is significant
t = r * sqrt(n - 2) / sqrt(1 - r^2) = 0.775 * 1.732 / 0.632

Result: t = 2.12, df = 3, two-sided p = 0.12

Common mistakes

Reading r = 0.8 as 80% of the variation explained; that is r^2 = 0.64.
Reporting a strong r as proof of cause; correlation can come from a third variable or coincidence.
Using Pearson r for a curved relationship; it only measures straight-line association, so plot the data first.

Practice Correlation coefficient (Pearson r)

These drills check your answer on live data, free, no account needed.

Correlation coefficientData on monthly ad spend and sales at regional stores (x = Ad spend ($k), y = Sales ($k)): (9, 87.8), (10, 98.7), (6, 81.3), (12, 99.1), (15, 111.9), (6, 78.6). What is the correlation coefficient r? Round to 3 decimal places.
Testing a correlationIn a sample of 10 stores, the correlation between staff hours and customer satisfaction scores is r = 0.24. Find the t statistic for H0: ρ = 0, then decide (two-sided) at α = 0.1. Round to 2 decimal places.

Questions

What is a strong correlation?

Rules of thumb: |r| below 0.3 weak, 0.3 to 0.7 moderate, above 0.7 strong. The right threshold depends on the field; in physics 0.7 is weak, in social science it is strong.

What is the difference between correlation and covariance?

Covariance is in the units of x times y and can be any size. Correlation is covariance divided by both standard deviations, which makes it unitless and bounded between -1 and 1.

When should I use Spearman instead of Pearson?

When the relationship is monotonic but not linear, when the data is ranked, or when outliers distort Pearson r. Spearman is Pearson r computed on the ranks.

CovarianceHow to calculate sample and population covariance, a worked example with exact numbers, how covariance becomes correlation, and Excel COVARIANCE.S and .P.
Linear regressionHow to fit a regression line y = a + bx: slope b = Sxy / Sxx, intercept a = y-bar - b x-bar, a worked example, prediction and Excel SLOPE and INTERCEPT.
R-squaredR-squared is the share of variation in y explained by the regression, 1 - SSE / SST. Worked example with exact numbers, link to r, Excel RSQ and its limits.
OutliersHow to detect outliers with the 1.5 IQR fence rule and the z-score rule, worked examples with exact fences, and what to do with outliers once found.