Regression Coefficient Input Form
Enter paired observations as comma, space, semicolon, or line separated values.
Example Data Table
| Observation | X | Y |
|---|---|---|
| 1 | 1 | 2 |
| 2 | 2 | 4 |
| 3 | 3 | 5 |
| 4 | 4 | 4 |
| 5 | 5 | 5 |
| 6 | 6 | 7 |
Formula Used
Mean values
x̄ = Σx / n and ȳ = Σy / n
Centered sums
Sxx = Σ(x - x̄)², Syy = Σ(y - ȳ)², and Sxy = Σ(x - x̄)(y - ȳ)
Regression coefficient of Y on X
byx = Sxy / Sxx
Intercept of Y on X
ayx = ȳ - byxx̄
Regression coefficient of X on Y
bxy = Sxy / Syy
Intercept of X on Y
axy = x̄ - bxyȳ
Correlation and fitted line diagnostics
r = Sxy / √(Sxx·Syy), R² = r², and ŷ = ayx + byxx
Residuals are e = y - ŷ. The standard error of estimate uses √(SSE / (n - 2)) when at least three observations exist.
How to Use This Calculator
- Enter the paired X and Y values in matching order.
- Choose labels for each variable to personalize equations.
- Set the number of decimal places for displayed results.
- Optionally enter an X value to predict Y.
- Optionally enter a Y value to predict X.
- Click the calculate button to generate coefficients and diagnostics.
- Review the result cards, equations, table, and graph.
- Use the export buttons to save CSV or PDF output.
Frequently Asked Questions
1. What does the regression coefficient measure?
It measures the expected change in one variable for a one unit change in the other. In this calculator, you get both Y on X and X on Y coefficients, which are usually different unless the data are perfectly linear with equal standardized spread.
2. Why are there two regression coefficients?
Regression is directional. Predicting Y from X minimizes vertical errors, while predicting X from Y minimizes horizontal errors. Because the objective changes, the slope changes too. Both are useful when prediction direction matters.
3. How is this different from correlation?
Correlation shows strength and direction of linear association on a standardized scale from minus one to one. A regression coefficient is a slope in original units, so it directly describes change and prediction.
4. What does R² tell me?
R² shows the fraction of variation in Y explained by the fitted line using X. Higher values indicate a tighter linear fit, but a high R² alone does not prove causation or confirm that the model is appropriate.
5. Why do I need paired observations?
Each X value must correspond to its matching Y value from the same observation. If the order is mismatched, the covariance structure changes and the estimated coefficients, residuals, and plotted line will be misleading.
6. What happens when one variable has no variance?
Regression cannot be computed in that direction because the slope formula divides by the centered sum of squares. If all X values match, Y on X fails. If all Y values match, X on Y fails.
7. Can I use negative and decimal values?
Yes. The calculator accepts integers, decimals, and negative numbers. Separate values with commas, spaces, semicolons, or line breaks, and keep the X and Y series the same length.
8. What does the residual table help me inspect?
It helps you compare observed Y values against fitted Y values, check the size and direction of prediction errors, and spot large residuals that may suggest unusual points or a poor linear model.