Enter Paired Data
Use one pair per line. Examples: 2,5 or 2 5 or 2;5
Example Data Table
| Point | X | Y | Comment |
|---|---|---|---|
| 1 | 1 | 2 | Initial paired observation |
| 2 | 2 | 3 | Y rises with X |
| 3 | 3 | 5 | Strong upward movement |
| 4 | 4 | 4 | One point sits below trend |
| 5 | 5 | 6 | Trend resumes upward |
| 6 | 6 | 7 | Useful for prediction practice |
Formula Used
Regression line: y = b + mx
Slope: m = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)²]
Intercept: b = ȳ - m x̄
Correlation: r = Σ[(x - x̄)(y - ȳ)] / √(Σ[(x - x̄)²] Σ[(y - ȳ)²])
Coefficient of determination: R² = r²
The calculator uses least squares regression. It finds the line that minimizes squared residuals between actual y values and fitted y values.
How to Use This Calculator
Enter paired observations in the data box, one pair per line.
Add an optional prediction x value to estimate a future y value.
Select the number of decimal places for displayed results.
Press Find Regression Line to show results above the form.
Review slope, intercept, correlation, residuals, and fitted values.
Use the chart to compare observed points with the best fit line.
Download CSV or PDF copies for reporting, review, or coursework.
Frequently Asked Questions
1. What does the regression line represent?
It shows the straight line that best summarizes how y changes as x changes across your paired observations. The calculator uses least squares to minimize squared residuals.
2. What is the difference between slope and intercept?
Slope measures how much y changes when x increases by one unit. Intercept is the estimated y value when x equals zero.
3. Why is correlation included?
Correlation shows the strength and direction of the linear relationship. Values near 1 or -1 indicate a stronger straight-line pattern.
4. What does R² tell me?
R² estimates how much of the variation in y is explained by the fitted line. Higher values usually mean the line describes the data better.
5. Can I use spaces instead of commas?
Yes. Each row can use commas, spaces, or semicolons. The parser reads the first two numbers on every non-empty line.
6. Why do I need at least two different x values?
A regression line needs variation in x. If all x values are identical, slope cannot be calculated because the denominator becomes zero.
7. What are residuals?
Residuals are actual y values minus fitted y values. They help you see how far each observation sits above or below the line.
8. When should I export the results?
Export when you need a record for assignments, reports, audits, or presentations. The CSV works well for spreadsheets, while the PDF suits sharing.