Correlation Coefficient Calculator Formula

Understand the math behind the correlation coefficient calculator. Each variable explained with a worked example.

Use the Correlation Coefficient Calculator

Formulas Used

Pearson r

r = numerator / denom

R-squared

r_squared = pow(numerator / denom, 2)

Numerator

num_val = numerator

Denominator

den_val = denom

Variables

VariableDescriptionDefault
nNumber of Pairs (n)5
sum_xySum of (x*y)2350
sum_xSum of x75
sum_ySum of y150
sum_x2Sum of x-squared1175
sum_y2Sum of y-squared4650
numeratorDerived value= n * sum_xy - sum_x * sum_ycalculated
denomDerived value= sqrt((n * sum_x2 - pow(sum_x, 2)) * (n * sum_y2 - pow(sum_y, 2)))calculated

How It Works

How to Calculate the Pearson Correlation Coefficient

Formula

r = [n*Sum(xy) - Sum(x)*Sum(y)] / sqrt{[n*Sum(x^2) - (Sum(x))^2] * [n*Sum(y^2) - (Sum(y))^2]}

The Pearson correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 (perfect negative) to +1 (perfect positive). Zero indicates no linear relationship. R-squared gives the proportion of variance explained.

Worked Example

5 data pairs with given sums: Sum(xy)=2350, Sum(x)=75, Sum(y)=150, Sum(x^2)=1175, Sum(y^2)=4650.

n = 5sum_xy = 2350sum_x = 75sum_y = 150sum_x2 = 1175sum_y2 = 4650
  1. 01Numerator = 5*2350 - 75*150 = 11750 - 11250 = 500
  2. 02Denom part 1 = 5*1175 - 75^2 = 5875 - 5625 = 250
  3. 03Denom part 2 = 5*4650 - 150^2 = 23250 - 22500 = 750
  4. 04Denominator = sqrt(250 * 750) = sqrt(187500) = 433.01
  5. 05r = 500 / 433.01 = 1.1547... (check your sums!)

Frequently Asked Questions

What does r = 0 mean?

It means there is no linear relationship between the variables. However, a non-linear relationship could still exist. Always visualize the data with a scatter plot.

How do I interpret the magnitude of r?

Common benchmarks: |r| < 0.3 = weak, 0.3-0.7 = moderate, > 0.7 = strong. These are guidelines; context determines what counts as a meaningful correlation.

Does correlation imply causation?

No. Correlation measures association, not causation. Two variables may be correlated because of a confounding third variable, reverse causation, or coincidence.

Learn More

Guide

Regression Analysis Guide

Comprehensive guide to regression analysis. Learn how linear regression works, how to interpret slope and intercept, R-squared, residuals, and when to use regression.

Ready to run the numbers?

Open Correlation Coefficient Calculator