Statistics
Linear Regression Calculator
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Runs in your browser
How to use it
Using the linear regression calculator
- 01
Paste paired data
One x,y pair per line, separated by a comma.
- 02
Read the fitted equation
y = mx + b is the prediction line; plug any x in to forecast y.
- 03
Judge fit with R²
Near 1 means points hug the line; low values warn against forecasting with it.
Good to know
Reading slope and intercept in context
The slope answers “per one unit increase in x, how much does y change?” The intercept is the predicted y at x = 0; meaningful only when zero lies within plausible data range. Always state units: “each extra ad dollar yields $2.34 more revenue” beats “m = 2.34”.
Extrapolation is where fits go to die
Least squares lines describe the observed range. Predicting far outside it assumes the trend persists unboundedly; rarely true. Confidence in predictions decays quickly toward the edges and beyond the data.
- Perfect fit → R² = 1.
- R² of 0.49 means the line explains 49% of y variation.
- Outliers tilt the line strongly; inspect residuals.
How it's calculated
The math behind this calculator
m = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)² b = ȳ − m·x̄ R² = Sxy² / (Sxx·Syy)Ordinary least squares chooses the line minimizing the summed squared vertical distances to your points. Closed-form solutions give the slope from the covariance of x and y over the variance of x, and the intercept follows so the line passes through the data’s centroid (x̄, ȳ).
R² reports the share of y-variance the line explains: 1 is a perfect fit, 0 means the line explains nothing beyond the mean. Regression requires x-values that actually vary; constant x is rejected.
Assumptions & limitations
- One “x,y” pair per line, comma-separated.
- At least two pairs with varying x values.
- Linearity assumed; check a scatter plot before trusting R².
Worked example
Fitting the points (1,2), (2,4), (3,6) yields y = 2x + 0 exactly; slope 2, intercept 0, R² = 1, a perfect linear relationship.
FAQ
Frequently asked questions
- Why is my intercept absurd?
- Probably x = 0 lies far outside your data, so the intercept is an extrapolation with inflated uncertainty. Center x before fitting if the intercept matters.
- Does correlation imply causation here?
- No; the line summarizes association. Confounders, reverse causation and coincidence all produce strong-looking fits.
- What if some lines have extra commas?
- Each line must parse as exactly two numeric fields; malformed lines return an error naming the offending line number.
- Can I fit curves?
- Not with this tool; it is strictly linear in the parameters. Transform variables (log, sqrt) first if theory suggests curvature.
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