Correlation & Linear Regression Calculator

Correlation & Linear Regression Calculator

This free correlation and linear regression calculator is built for business, economics, and data analytics use cases where you need to understand the relationship between two numeric variables. Paste in your paired x, y data — such as advertising spend and sales, price and demand, years of experience and salary, or study hours and exam scores — and instantly get the Pearson correlation coefficient (r), the least-squares regression line equation (y = mx + c), and the R² value, all calculated together with a live scatter plot and full step-by-step working shown below.

Correlation and regression are two of the most widely used statistical tools in business analytics, econometrics, marketing research, finance, and the social sciences. Correlation tells you whether and how strongly two variables move together, while regression goes a step further and gives you an actual equation you can use to predict one variable from the other. This calculator handles both at once, since they're built from the exact same underlying sums.

Pearson Correlation Coefficient (r) Formula

The Pearson correlation coefficient measures the strength and direction of the linear relationship between two variables, x and y. It always falls between −1 and +1. This calculator uses the raw-score (computational) formula, which avoids having to calculate deviations from the mean for every point:

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

Here, n is the number of paired observations, Σxy is the sum of each x multiplied by its paired y, Σx and Σy are the simple sums of each variable, and Σx² and Σy² are the sums of each variable squared. A result of r = +1 means a perfect positive linear relationship (as x increases, y increases proportionally), r = −1 means a perfect negative linear relationship (as x increases, y decreases proportionally), and r = 0 means no linear relationship at all. Values in between are interpreted by their magnitude: generally, |r| below 0.3 is considered weak, 0.3–0.7 moderate to moderately strong, and above 0.7 a strong linear relationship — though the right threshold can vary by field.

Linear Regression Formula (y = mx + c)

Simple linear regression fits a straight line through the data using the ordinary least squares (OLS) method — the method that minimizes the sum of the squared vertical distances between each observed y value and the line's predicted value. The resulting line has the equation y = mx + c, where m is the slope and c is the y-intercept:

Slope: m = (nΣxy − ΣxΣy) / (nΣx² − (Σx)²)

Intercept: c = ȳ − m·x̄, where x̄ and ȳ are the means of x and y

Once you have m and c, you can plug in any x value to predict the corresponding y value — this is the core idea behind using regression for forecasting in business and economics, such as predicting sales from ad spend, or costs from production volume.

What is R² (Coefficient of Determination)?

R², the coefficient of determination, is simply the square of the correlation coefficient: R² = r². It represents the proportion of the variation in y that can be explained by its linear relationship with x, expressed as a value between 0 and 1 (or 0% to 100%). An R² of 0.80, for example, means 80% of the variability in y is explained by x through the fitted line, while the remaining 20% is due to other factors, noise, or a relationship that isn't purely linear. In business and economics contexts, R² is often used to judge how reliable a regression model is before using it for forecasting or decision-making — a low R² suggests the linear model may not be capturing the real relationship well, even if the slope looks meaningful.

How to Use This Calculator

Enter your paired data with one (x, y) observation per line, separating the two numbers with a comma, space, or tab — for example, '3, 12' on one line and '5, 18' on the next. The calculator needs at least two pairs, though results are more meaningful with several. It instantly returns the Pearson correlation coefficient, the regression equation, R², the slope, the intercept, and the means of both variables, along with a scatter plot showing your data points and the fitted regression line.

• Use the Pearson correlation coefficient (r) when you only want to know how strongly two variables are related, and in which direction.
• Use the linear regression equation (y = mx + c) when you want to predict or forecast one variable based on the other.
• Use R² when you want to know how much of the variation in y is actually explained by the linear model, before trusting its predictions.
• Remember that correlation and regression only capture linear relationships — a strong non-linear pattern can produce a low r even when x and y are clearly related.

Worked Example

Consider five paired observations: (1, 2), (2, 4), (3, 5), (4, 4), (5, 5). Here, n = 5, Σx = 15, Σy = 20, Σxy = 66, Σx² = 55, and Σy² = 86.

Correlation: r = (5×66 − 15×20) / √[(5×55 − 15²)(5×86 − 20²)] = (330 − 300) / √[(275 − 225)(430 − 400)] = 30 / √(50 × 30) = 30 / √1500 ≈ 0.7746, indicating a moderately strong positive linear relationship.

Regression: slope m = (5×66 − 15×20) / (5×55 − 15²) = 30 / 50 = 0.6. Intercept c = ȳ − m·x̄ = 4 − 0.6×3 = 2.2. So the regression line is y = 0.6x + 2.2.

R² = r² = 0.7746² ≈ 0.6, meaning about 60% of the variation in y is explained by its linear relationship with x. Enter this same data set into the calculator above to see every one of these steps computed automatically.

Correlation vs Regression vs Causation

It's important to remember that correlation and regression describe association, not causation. A strong positive r or a high R² only tells you that two variables tend to move together in a predictable linear way — it doesn't prove that changes in x directly cause changes in y. In business and economics, this distinction matters: advertising spend and sales might correlate strongly, but a third factor (like seasonality or overall market growth) could be driving both. This calculator, like any correlation and regression tool, gives you the statistical relationship — interpreting whether that relationship is causal is a separate analytical step that depends on domain knowledge, experimental design, or more advanced techniques like controlled experiments or multiple regression.