Linear Model Fit Calculator

Calculator Form

Enter paired data and fitting options

X values

Use commas, spaces, or line breaks.

Y values

Y values must match the X count.

Prediction X values

Optional extra X values for forecasting.

Decimal places

Fit option

Force model through origin

Unchecked fits slope and intercept. Checked sets intercept to zero.

Example Data Table

Sample paired dataset for testing

Observation	X	Y
1	1	2.1
2	2	2.9
3	3	4.2
4	4	5.1
5	5	5.8
6	6	7.2
7	7	8.1
8	8	9.0

Formula Used

Core regression equations

Linear model: y = a + bx

Slope: b = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)²]

Intercept: a = ȳ - b x̄

Prediction: ŷ = a + bx

Residual: e = y - ŷ

R²: 1 - SSE / SST

SSE: Σ(y - ŷ)²

MSE: SSE / n

RMSE: √MSE

MAE: Σ|y - ŷ| / n

When the origin option is enabled, the intercept is fixed at zero and the slope becomes b = Σ(xy) / Σ(x²).

How To Use

Steps for practical use

Enter X values in the first box.
Enter matching Y values in the second box.
Add optional prediction X values for forecasting.
Choose decimal precision for displayed results.
Enable the origin option only when theory supports zero intercept.
Click Fit Linear Model to compute the regression.
Review slope, intercept, R², RMSE, residuals, and prediction outputs.
Export the table as CSV or PDF for reporting.

FAQs

Common questions

1. What does this calculator estimate?

It estimates a straight-line relationship between one predictor and one response variable. The tool returns slope, intercept, fit quality, residual errors, and predicted values.

2. What does R² mean here?

R² measures how much variation in Y is explained by the fitted line. Values closer to 1 usually indicate a stronger linear relationship.

3. When should I force the line through the origin?

Use that option only when theory or measurement design clearly requires zero response at zero input. Otherwise, let the model estimate the intercept.

4. What do residuals show?

Residuals show the difference between actual and predicted Y values. They help reveal bias, unusual observations, and non-linear patterns the line cannot explain.

5. Can I paste values with spaces or new lines?

Yes. The calculator accepts commas, spaces, semicolons, and line breaks. Each X value must still align with one Y value.

6. What if all X values are identical?

A usable linear model cannot be fitted because the slope denominator becomes zero. The tool will show an error instead of misleading output.

7. Why are RMSE and MAE both shown?

MAE gives average absolute error, while RMSE penalizes larger errors more strongly. Together, they describe model accuracy from different perspectives.

8. Is this the same as multiple regression?

No. This page fits one predictor against one response. Multiple regression needs several predictors and a different model structure.