Correlation Coefficient Calculator – Find Pearson r Online

Calculate the Pearson correlation coefficient between two variables with our free online calculator. Measure the strength and direction of linear relationships in your data.

X Values (Variable 1)

Y Values (Variable 2)

Examples:

Understanding Correlation

Correlation measures how two variables move together. When one goes up, does the other go up too? Go down? Or is there no pattern at all? The correlation coefficient, written as r, gives you a single number that captures this relationship.

The value of r always falls between -1 and +1. A positive r means the variables move in the same direction – as one increases, so does the other. A negative r means they move in opposite directions – as one increases, the other decreases. An r near zero means there's no linear relationship.

The closer r is to -1 or +1, the stronger the relationship. At exactly +1 or -1, the points fall perfectly on a straight line. But remember: correlation doesn't mean causation. Just because two things are correlated doesn't mean one causes the other – they might both be caused by something else entirely.

The Pearson Correlation Formula

r = (nΣxy - ΣxΣy) / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]

This formula calculates how much x and y vary together, relative to how much they vary individually.

Number of data points

Σx, Σy

Sum of x values, sum of y values

Σxy

Sum of each x multiplied by its paired y

Σx², Σy²

Sum of squared x values, sum of squared y values

Correlation coefficient (-1 to +1)

r²

Coefficient of determination (explained variance)

Interpreting r Values

±0.9 to ±1.0Very strong

±0.7 to ±0.9Strong

±0.5 to ±0.7Moderate

±0.3 to ±0.5Weak

0 to ±0.3Very weak or none

Important Caveats

Correlation ≠ Causation

Just because two variables are correlated doesn't mean one causes the other. There could be a third variable causing both, or it could be coincidence.

Only Linear Relationships

Pearson's r only measures linear (straight-line) relationships. Two variables could have a strong curved relationship and still have r ≈ 0.

Sensitive to Outliers

A single extreme value can dramatically change the correlation. Always look at a scatterplot to check for outliers.

Worked Examples

Example 1: Study Time and Test Scores

Does studying more lead to better grades? Let's check the correlation.

Hours (X): 1, 2, 3, 4, 5, 6, 7, 8

Scores (Y): 60, 65, 70, 72, 78, 82, 88, 92

n = 8 students

Σx = 36, Σy = 607

Σxy = 2,934

Σx² = 204, Σy² = 47,045

r = 0.986 (very strong positive correlation)

Students who study more tend to score higher. The relationship is almost perfectly linear.

Example 2: Price and Demand

Economics says higher prices reduce demand. Let's verify.

Price (X): 10, 15, 20, 25, 30, 35, 40

Demand (Y): 100, 88, 75, 65, 52, 42, 30

n = 7 price points

Σx = 175, Σy = 452

Σxy = 9,575

Σx² = 5,775, Σy² = 33,082

r = -0.993 (very strong negative correlation)

As price increases, demand decreases almost perfectly. This is the classic demand curve from economics.

Example 3: Shoe Size and IQ

A classic example of no correlation.

Shoe Size (X): 6, 7, 8, 9, 10, 11, 12

IQ (Y): 105, 98, 112, 95, 108, 102, 99

n = 7 people

r = -0.08 (essentially no correlation)

There's no relationship between shoe size and intelligence. The correlation is essentially zero, as expected.

Example 4: Height and Weight

Taller people tend to weigh more, but the relationship isn't perfect.

Height cm (X): 160, 165, 170, 175, 180, 185

Weight kg (Y): 55, 60, 68, 72, 80, 88

n = 6 people

r = 0.982 (very strong positive correlation)

Height and weight are strongly correlated, but not perfectly. Body composition, muscle mass, and other factors create variation.

Quick Fact

Karl Pearson (1857-1936), the British mathematician who developed the correlation coefficient, was one of the most influential statisticians in history. He founded the world's first university statistics department at University College London in 1911. Pearson also developed the chi-squared test, introduced the concept of standard deviation, and coined the term "histogram." His work laid the foundation for modern statistics, though some of his views on eugenics are now rightly criticized. The Pearson correlation coefficient remains the most widely used measure of linear association in science, business, and social research.

Frequently Asked Questions

What's the difference between correlation and causation?

Correlation means two variables tend to move together. Causation means one variable actually causes changes in the other. Ice cream sales and drowning deaths are correlated (both increase in summer), but ice cream doesn't cause drowning – hot weather causes both. Always look for alternative explanations before assuming causation.

Can correlation be greater than 1 or less than -1?

No, mathematically impossible. The correlation coefficient is bounded between -1 and +1. If you calculate an r outside this range, there's an error in your calculation. The formula is designed so that the numerator can never exceed the denominator in absolute value.

What does r-squared tell me?

R² (r-squared) tells you what percentage of the variation in Y can be explained by X. If r = 0.8, then r² = 0.64, meaning 64% of Y's variation is explained by its linear relationship with X. The remaining 36% is due to other factors or random variation.

How many data points do I need for a reliable correlation?

You can calculate r with just 2 points (you'll get ±1), but it's meaningless. For any reliability, aim for at least 10-15 data points. With 30+ points, you can start making statistical inferences. More data always gives more reliable results, especially for detecting weak correlations.

What if my data has outliers?

Outliers can dramatically distort correlation. A single extreme point can turn a strong correlation into a weak one, or vice versa. Always plot your data first. Consider calculating correlation with and without outliers, or use Spearman's rank correlation which is less sensitive to extreme values.

Can two variables have zero correlation but still be related?

Yes! Pearson's r only measures linear relationships. If Y = X² (a perfect parabola), the correlation would be zero because the relationship is curved, not straight. Always visualize your data with a scatterplot – you might discover strong non-linear patterns that r misses.

When should I use Spearman correlation instead?

Use Spearman's rank correlation when your data isn't normally distributed, has outliers, or is ordinal (ranked) rather than continuous. Spearman converts values to ranks before calculating correlation, making it more robust to extreme values and non-linear but monotonic relationships.

Other Free Tools

Search tools