Correlation Calculator

What Is Correlation?

Correlation describes how two quantitative variables change together. If taller people tend to weigh more, height and weight are positively correlated; if cars with bigger engines tend to go fewer miles per gallon, engine size and fuel economy are negatively correlated. Correlation needs paired data: each observation gives one X value and one Y value for the same person, day or item.

What Is the Pearson Correlation Coefficient?

The Pearson correlation coefficient, r, is a single number that summarizes the strength and direction of a linear (straight-line) relationship. It always lies between −1 and +1. The sign gives the direction, and the size, |r|, gives how closely the points follow a straight line. r has no units, so it does not change if you convert centimetres to inches or swap which variable you call X and which Y.

Positive, Negative and Zero Correlation

Positive correlation (r > 0): as X increases, Y tends to increase, and the points rise from left to right. r = +1 means every point lies exactly on an upward straight line.

Negative correlation (r < 0): as X increases, Y tends to decrease, and the points fall from left to right. r = −1 means every point lies exactly on a downward straight line.

r = 0 means no linear correlation, not necessarily no relationship. Points in a perfect U-shape, such as y = x² for x from −3 to 3, have r = 0 even though Y is completely determined by X. Always look at the scatter plot as well as the number.

How to Calculate the Correlation Coefficient

1. List the pairs and compute x², y² and xy for each.

2. Add up each column to get Σx, Σy, Σx², Σy² and Σxy, and count the pairs n.

3. Numerator: nΣxy − (Σx)(Σy).

4. Denominator: √([nΣx² − (Σx)²] × [nΣy² − (Σy)²]).

5. Divide: r = numerator / denominator.

Example with X = 1, 2, 3, 4, 5 and Y = 2, 4, 6, 8, 10: Σx = 15, Σy = 30, Σx² = 55, Σy² = 220, Σxy = 110, n = 5. Numerator = 5 × 110 − 15 × 30 = 100. Denominator = √((5 × 55 − 225)(5 × 220 − 900)) = √(50 × 200) = 100. So r = 100 / 100 = 1, a perfect positive correlation. The calculator does this for your data and shows each total.

How to Interpret the Correlation Coefficient

Read the sign first: positive means the variables tend to rise together, negative means one tends to fall as the other rises. Then read the size. A common rule of thumb for |r| is: below 0.20 very weak, 0.20 to 0.39 weak, 0.40 to 0.59 moderate, 0.60 to 0.79 strong, and 0.80 or more very strong. These bands are conventions, not laws; in physics a correlation of 0.9 may be disappointing, while in social science 0.3 can be important.

r² (the coefficient of determination) is the share of the variation in Y that is linearly associated with X in the sample: r = 0.8 gives r² = 0.64, or 64%. It describes the fit of a straight line, not how much X causes Y.

Also consider the sample size. With few pairs, r is unstable, and with 2 pairs it is always exactly ±1.

Correlation Does Not Mean Causation

A strong correlation means two variables tend to move together in a straight-line pattern. It does not show that one causes the other. Possible explanations include:

Direct causation: X really does affect Y.

Reverse causation: Y affects X instead.

A confounding variable: a third factor drives both; ice cream sales and sunburn both rise with hot weather.

Coincidence: with enough variables, some will correlate by chance.

Establishing cause and effect needs study design, such as randomized experiments, and subject knowledge, not a correlation coefficient alone.

When Should You Use Pearson Correlation?

Pearson correlation is designed to measure the strength and direction of a linear relationship between two quantitative variables. Check that:

The data are paired numerical observations, one X and one Y for each case.

The relationship looks roughly like a straight line on the scatter plot. If it curves, r understates it; a U-shaped pattern can give r close to 0.

Both variables vary. If every X (or every Y) is the same, r cannot be calculated.

There are no extreme outliers driving the result. One unusual point can raise or lower r substantially; the calculator flags points whose removal would change r by 0.2 or more.

If the relationship is consistently rising or falling but curved, or the data are ranks, Spearman's rank correlation is often a better choice.

Outliers and Correlation

Because Pearson r uses the actual values, a single extreme point can dominate it. A scattered cloud with r close to 0 can show a clearly positive r after one far-away point is added, and one stray value can weaken an otherwise strong pattern. Before reporting r, look at the scatter plot, check unusual values for data-entry errors, and consider reporting r with and without them. Try it in the interactive demo above.