Calculator Input
Enter one x,y pair per line. Comma, tab, semicolon, or space works.
Example Data Table
| Example Row | X | Y |
|---|---|---|
| 1 | 1 | 2 |
| 2 | 2 | 3 |
| 3 | 3 | 5 |
| 4 | 4 | 4 |
| 5 | 5 | 6 |
| 6 | 6 | 8 |
| 7 | 7 | 9 |
Formula Used
Slope: b₁ = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)²]
Intercept: b₀ = ȳ - b₁x̄
Predicted value: ŷ = b₀ + b₁x
Residual: e = y - ŷ
Sum of squared errors: SSE = Σe²
Standard error of regression: SE = √[SSE / (n - 2)]
This calculator uses simple linear regression. It fits one straight line to paired x and y values.
The standard error of regression is also called the residual standard error or standard error of estimate.
Smaller values suggest the observations cluster more closely around the regression line.
How to Use This Calculator
- Enter one x,y pair per line in the paired data box.
- Use commas, spaces, tabs, or semicolons as separators.
- Add an optional target x value for a prediction.
- Choose the number of decimals you want displayed.
- Set custom labels for the x axis, y axis, and dataset.
- Click the calculation button to generate the regression results.
- Review the summary cards, fitted line chart, and residual chart.
- Download a CSV or PDF report for sharing or archiving.
Frequently Asked Questions
1. What does the standard error of regression show?
It shows the typical size of the residuals. In practical terms, it measures how far observed y values tend to fall from the fitted regression line.
2. Is a smaller standard error better?
Usually yes. A smaller value means the model’s predictions sit closer to the actual observations. It suggests less unexplained variation remains after fitting the line.
3. Why are at least three data pairs required?
Simple linear regression estimates two parameters: slope and intercept. You need more than two points to estimate residual variation and compute the regression standard error.
4. What happens if all x values are the same?
Regression fails because the slope denominator becomes zero. A line cannot be fitted when x has no spread, so the calculator asks for varied x values.
5. Does this calculator handle nonlinear relationships?
No. This page is designed for simple linear regression only. If the true pattern curves strongly, the standard error may look large even with valid data.
6. What is the difference between RMSE and regression standard error?
RMSE divides squared error by n. Regression standard error divides by n - 2, which adjusts for the estimated slope and intercept in simple linear regression.
7. Can I use decimal or negative numbers?
Yes. The parser accepts decimal values, negative values, and scientific notation. Each line still needs one x value and one y value.
8. Why should I inspect the residual chart?
The residual plot helps you spot patterns, curvature, and changing spread. Random scatter around zero usually supports the straight line assumption more strongly.