Calculator
Example data table
| Y | X1 | X2 |
|---|---|---|
| 12 | 1 | 4 |
| 15 | 2 | 5 |
| 14 | 2 | 4 |
| 18 | 3 | 6 |
| 20 | 4 | 7 |
| 22 | 5 | 8 |
| 24 | 6 | 9 |
| 27 | 7 | 10 |
Formula used
Multiple linear regression models an outcome as a weighted sum of several predictors:
y = β₀ + β₁x₁ + β₂x₂ + … + βₖxₖ + ε
In matrix form, the least‑squares estimates are:
β̂ = (XᵀX)⁻¹ Xᵀ y
How to use this calculator
- Prepare data with one row per observation.
- Put the outcome in the first column (Y).
- Add one or more predictor columns (X1, X2, …).
- Paste data or upload a CSV file.
- Choose settings like intercept and standardization.
- Click Calculate, then download CSV or PDF.
Data structure and variable roles
Multiple regression needs a clear row‑based dataset. Put the dependent variable in the first column (Y) and list predictors as X1, X2, and X3. Keep units consistent and avoid mixing totals with rates. Remove rows with missing numbers or replace them before running the model. Encode categories as separate 0/1 columns. A practical minimum is 10–20 observations per predictor for stable estimates, especially when predictors are correlated.
Model outputs and statistical tests
After calculation, the tool returns coefficients (β), standard errors, t‑statistics, and p‑values for each predictor. The intercept β₀ represents the expected Y when all X values are zero. R² reports explained variance, while adjusted R² penalizes unnecessary predictors. Use the overall F‑test to confirm whether the set of predictors improves fit versus a constant‑only model. The fitted equation can be used to predict Y for new X values.
Fit diagnostics and stability checks
Diagnostics help you judge reliability. Residual standard error summarizes typical prediction deviation in Y units, while RMSE and MAE provide complementary error views. Check VIF values to detect multicollinearity; VIF above 5 suggests unstable coefficients. A high condition number also indicates near‑linear dependence among predictors and can inflate standard errors. Review residual plots for curvature and heteroscedasticity, and identify high‑leverage points that can dominate the solution.
Coefficient interpretation and uncertainty
Interpretation should focus on effect size and uncertainty. A coefficient βᵢ indicates the expected change in Y for a one‑unit increase in Xᵢ, holding other predictors fixed. Compare standardized betas when predictors use different scales, or enable standardization in settings. Confidence intervals add context: if a 95% interval crosses zero, the effect may be practically small even when the point estimate looks large. Prefer domain‑meaningful units and consider whether a change of one unit is realistic.
Export workflow and documentation
For reporting, capture both model quality and assumptions. Summarize sample size, predictor list, R²/adjusted R², and key coefficients with p‑values and confidence intervals. Review residual patterns for nonlinearity and outliers before sharing results. Use the CSV export for spreadsheets and the PDF export for a printable appendix that preserves coefficients, tests, and fit statistics. When comparing models, keep the same response and record removed predictors. This supports consistent, auditable decisions.
FAQs
What CSV format does the calculator accept?
Paste or upload comma‑separated rows with numeric values. The first column is Y, and the remaining columns are predictors. Headers are optional, but avoid mixed text and numbers in data rows.
How many predictors can I include?
Include as many as your sample size supports. A common guideline is at least 10 observations per predictor, and more when predictors are correlated. Too many predictors can overfit and weaken inference.
Should I keep the intercept enabled?
Usually yes, because it prevents systematic bias in residuals. Disable the intercept only when theory requires the regression line to pass through the origin and your variables are measured from a true zero.
What does a high VIF mean?
VIF measures how much multicollinearity inflates a coefficient’s variance. Values above about 5 suggest the estimate may be unstable. Consider removing a redundant predictor, combining variables, or collecting more diverse data.
Can I standardize variables here?
Yes. Standardization converts predictors to z‑scores, making coefficients comparable across scales. Use standardized betas for relative importance, but keep unstandardized coefficients for real‑unit predictions and business reporting.
Do significant p‑values prove causation?
No. P‑values indicate evidence against a zero coefficient under model assumptions. Causal claims require study design support, like randomization or strong controls. Always check residual behavior and external validity before concluding cause.