| x | y |
|---|---|
| 1 | 2.1 |
| 2 | 2.9 |
| 3 | 3.8 |
| 4 | 4.2 |
| 5 | 5.1 |
| 6 | 6.2 |
| 7 | 6.8 |
| 8 | 8.1 |
| 9 | 9.0 |
| 10 | 10.2 |
Regression model
We fit a simple linear model:
If intercept is disabled, b0 = 0.
Least squares estimates
b0 = \bar{y} − b1\bar{x}
ANOVA partition
SSE = Σ(y−\hat{y})²
SSR = SST − SSE
- Paste your dataset with x in column one and y in column two.
- Select the correct delimiter and enable header if needed.
- Choose whether to include an intercept in the model.
- Click Calculate to generate ANOVA and regression outputs.
- Use CSV or PDF export buttons in the results area.
Why ANOVA is paired with regression
Regression estimates the line that best explains how y changes with x. ANOVA then tests whether that explanation is stronger than random noise by splitting total variation into SSR (explained) and SSE (unexplained). For simple linear regression, the regression degrees of freedom are 1, the error degrees of freedom are n−2, and the F statistic is F = MSR/MSE. A small p-value indicates the slope is unlikely to be zero. In simple regression, this matches the slope t-test because F = t².
How coefficients are computed
The calculator uses least squares to compute the slope and intercept. The slope is b1 = Sxy/Sxx, where Sxx measures x-spread and Sxy measures joint movement of x and y. The intercept is b0 = ȳ − b1x̄, unless you disable intercept (then b0 = 0). Standard errors come from the residual variance MSE, enabling t-tests and confidence intervals for the coefficients.
Interpreting fit statistics
R² reports the proportion of y-variation explained by the model: R² = SSR/SST. Adjusted R² accounts for sample size and helps prevent overconfidence in small datasets. The root mean square error (RMSE) summarizes typical prediction error on the y-scale. Use the fitted values and residuals table to spot patterns; a strong model usually shows residuals that bounce randomly around zero. Adjusted R² reduces inflation by penalizing extra parameters and is computed from 1 − (MSE/MST) using the appropriate degrees of freedom.
Assumptions and diagnostic checks
ANOVA p-values rely on common regression assumptions: linear relationship, independent observations, constant error variance, and approximately normal residuals. This calculator reports residual summaries to support quick checks. If residual spread increases with x, consider transforming y or using weighted regression. If there are extreme points, compare results with and without those rows to understand sensitivity.
Practical workflow for decision making
Start by cleaning your data: remove blanks, confirm numeric units, and verify that x truly precedes y. Run the model, review the coefficient sign and magnitude, then confirm significance using the F-test and slope t-test. Finally, export CSV for audit trails or PDF for reporting. For forecasting, prioritize RMSE and residual behavior over p-values alone, especially when n is small. When reporting to stakeholders, often include the coefficient confidence intervals to show uncertainty and avoid treating the fitted line as an exact rule.
1) What does the F-test mean in this report?
The F-test compares explained variance to unexplained variance. If the p-value is small, the regression explains significantly more variation than random error, implying a non-zero slope.
2) Why do I need at least 3 rows?
With an intercept, simple regression estimates two parameters (slope and intercept). You need at least one error degree of freedom, so n ≥ 3 is required.
3) When should I disable the intercept?
Disable the intercept only if theory demands the line passes through the origin and your measurements are on a true ratio scale. Otherwise, keeping the intercept usually reduces bias.
4) Can I use this for multiple regression?
This tool is designed for one predictor x and one response y. For multiple predictors, you need a multivariate regression model with a larger ANOVA table and matrix-based estimation.
5) What should I do if residuals show a pattern?
A pattern suggests nonlinearity or changing variance. Try transforming variables, adding a nonlinear term in a different tool, or segmenting data. Confirm improvements with lower RMSE and cleaner residuals.
6) Why can the slope be significant but predictions still poor?
Statistical significance can occur with large samples even for small effects. Prediction quality depends on effect size and noise; RMSE and residual spread are better indicators for forecasting accuracy.