Enter Paired Data
Use commas, spaces, semicolons, or line breaks between numbers. The calculator fits a simple linear model with one predictor and one response.
Example Data Table
This sample is already loaded into the form. It demonstrates how X and Y move together across six paired observations.
| Observation | X | Y |
|---|---|---|
| 1 | 2 | 5 |
| 2 | 4 | 8 |
| 3 | 6 | 9 |
| 4 | 8 | 14 |
| 5 | 10 | 17 |
| 6 | 12 | 20 |
Formula Used
Core regression equations
Slope: β₁ = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / Σ[(xᵢ − x̄)²]
Intercept: β₀ = ȳ − β₁x̄
Prediction: ŷ = β₀ + β₁x
Correlation: r = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / √(Σ[(xᵢ − x̄)²] × Σ[(yᵢ − ȳ)²])
Fit and error equations
R²: 1 − SSE / SST
SSE: Σ(yᵢ − ŷᵢ)²
RMSE: √(SSE / (n − 2))
Prediction interval: ŷ₀ ± z × √[MSE × (1 + 1/n + (x₀ − x̄)² / Σ(xᵢ − x̄)²)]
Confidence bounds on this page use common normal critical values for 90%, 95%, and 99% intervals.
How to Use This Calculator
1. Paste the X series into the independent variable box.
2. Paste the matching Y series into the dependent variable box.
3. Keep both lists in the same order and length.
4. Choose decimal precision and a confidence level.
5. Enter an optional X value for prediction output.
6. Click Calculate Regression to generate the model.
7. Review summary metrics, coefficients, and diagnostics.
8. Export the analysis using the CSV or PDF buttons.
Frequently Asked Questions
1. What does bivariate regression measure?
It estimates how one response variable changes as one predictor changes. The calculator returns the fitted line, correlation, fit strength, and error statistics for paired numeric observations.
2. How many observations should I enter?
Use at least two pairs to fit a line, but three or more pairs are better. Additional observations improve error estimates, intervals, and the stability of your slope and intercept.
3. What happens if all X values are the same?
Regression cannot be estimated when the predictor never changes. The slope denominator becomes zero, so the page stops and asks for X values with real variation.
4. What is the difference between correlation and R²?
Correlation shows the direction and strength of the linear relationship. R² shows the share of Y variance explained by X through the fitted line.
5. Why are residuals useful?
Residuals show the gap between actual and fitted values. They help you spot unusual observations, changing variance, and patterns that suggest the linear model may be incomplete.
6. What does the prediction section provide?
When you enter a target X value, the page returns the predicted Y, an approximate mean response interval, and an approximate prediction interval for a single future observation.
7. Can I paste values on separate lines?
Yes. The parser accepts commas, spaces, semicolons, and line breaks. That makes it easy to paste columns copied from spreadsheets, reports, lab files, or statistical notes.
8. When should I use this instead of multiple regression?
Use this page when you have one predictor and one response. If several predictors influence the outcome together, a multiple regression model is more appropriate.