Enter Paired Data
| Row | x | y | Action |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 |
Calculation and Display Options
Formula Used
How to Use This Calculator
- Enter paired x and y values in matching rows.
- Add rows or paste values from a spreadsheet.
- Choose precision, confidence, and equation display options.
- Enter an optional x-value for a prediction.
- Select Calculate Regression to generate all results.
- Review the equation, charts, residuals, and statistical tests.
- Copy, print, or download the results as needed.
Example Data
| Study Hours | Exam Score | Expected Use |
|---|---|---|
| 1 | 52 | Observed point |
| 2 | 55 | Observed point |
| 3 | 61 | Observed point |
| 4 | 66 | Observed point |
| 5 | 72 | Observed point |
| 6 | 78 | Observed point |
Understanding the Results
The slope describes the expected y-change for each x-unit. Its sign shows the relationship direction. Large absolute slopes indicate faster predicted change.
The intercept estimates y when x equals zero. It may lack practical meaning outside observed data. Always consider the measured data range carefully.
R² measures the variation explained by the fitted line. Values nearer one indicate stronger linear fit. Correlation still does not establish causal relationships.
Residuals compare observed and predicted y-values. Random residual patterns support a linear model. Structured patterns may suggest another model fits better.
Frequently Asked Questions
What is a least-squares regression line?
It is the line minimizing squared vertical residuals. It summarizes a linear relationship between paired variables. The method balances positive and negative errors.
How is the regression slope calculated?
The slope uses covariance divided by x variation. It measures predicted y-change per x-unit. A negative result indicates an inverse relationship.
What does the regression intercept mean?
The intercept is predicted y when x equals zero. This value may be outside observed data. Interpret it only when zero is meaningful.
What does R² measure?
R² measures explained variation in the response variable. Higher values indicate closer linear agreement. It does not prove the model is appropriate.
What is a residual?
A residual equals observed y minus predicted y. Positive residuals sit above the fitted line. Negative residuals sit below the fitted line.
Can this calculator predict future values?
It predicts y from any entered x-value. Predictions inside the sample range are interpolations. Outside predictions require stronger caution and assumptions.
What happens when all x-values are equal?
The regression slope becomes undefined without x variation. No unique nonvertical least-squares line exists. Enter at least two distinct x-values.
How do outliers affect the regression equation?
Outliers can strongly alter slope and intercept estimates. High-leverage points may dominate the fitted line. Review residuals and the scatter plot carefully.
Does a strong correlation prove causation?
No, correlation only describes statistical association between variables. Confounding factors may explain the observed relationship. Controlled evidence is needed for causal conclusions.