📈 Linear Regression Calculator
Fit a least-squares line to paired data and calculate slope, intercept, correlation, and R².
📊 Statistics📐 Formula included🧠 Practice questions✓ Free
Please enter valid values.
Formula and Explanation
Fit a least-squares line to paired data and calculate slope, intercept, correlation, and R².
| Concept | Formula or rule |
|---|---|
| Linear Regression Calculator | ŷ = a + bx, where b = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² |
How to Use This Calculator
- Enter the sample, counts, probabilities, or summary statistics requested above.
- Keep paired lists in the same order and separate values with commas.
- Select the correct test direction or sample/population option when available.
- Press Calculate and interpret the result together with the supporting statistics.
Worked Example
Example: The fitted line minimizes the sum of squared vertical residuals.
✓ Use the slope to estimate expected change in Y per one-unit increase in X.
Where This Is Used
School & coursework
Check homework, understand formulas, and verify step-by-step statistics exercises.
Research
Explore estimates, uncertainty, distributions, and relationships before using full statistical software.
Business & data analysis
Summarize variation, compare groups, and communicate data-driven findings clearly.
Common Mistakes to Avoid
⚠️
Extrapolating far beyond the data
The fitted relationship may not continue outside the observed range.
The fitted relationship may not continue outside the observed range.
⚠️
Ignoring residual patterns
A line can be inadequate even with moderate R².
A line can be inadequate even with moderate R².
⚠️
Treating association as causation
Regression alone does not establish a causal effect.
Regression alone does not establish a causal effect.
Practice Questions
1. If slope=3, how much does predicted Y change when X rises by 2?
Answer: 6
2. What does least squares minimize?
Answer: Sum of squared residuals.
3. Can R² be negative for ordinary regression with intercept?
Answer: No.
Frequently Asked Questions
What does the slope mean?▾
Expected change in predicted Y for a one-unit increase in X.
What is the intercept?▾
Predicted Y when X=0, which may or may not be meaningful.
What does R² show?▾
The proportion of Y variation explained by the fitted linear model.