Linear Regression Calculator (Least Squares)

The slope m tells you how much y changes for each one-unit increase in x. For example, m = 2.3 means y rises by 2.3 units when x increases by 1. If m is negative (e.g., m = −1.5), y decreases as x increases — an inverse relationship. The slope has units equal to (units of y) / (units of x), so if x is hours and y is dollars, m is dollars per hour.

R² = 0.85 means that 85% of the total variability in your y values is explained by the linear relationship with x. The remaining 15% is unexplained — measurement noise, omitted variables, or non-linear effects. According to NIST/SEMATECH guidelines, R² ≥ 0.80 is considered a very strong fit in most applied fields. However, a model with R² = 0.85 can still produce poor predictions if residuals show systematic patterns. Always combine R² with a residual plot.

A commonly cited rule of thumb is at least n = 10–20 observations for simple linear regression. With n < 5 the estimates are highly unstable. For inference (hypothesis testing on the slope), n = 10 ensures the t-distribution approximation is reasonable. For reliable prediction, n = 30+ is preferred. For machine-learning baselines and industrial calibrations, n = 100+ is the working minimum.

Technically yes — R² can be negative if you compute it for a model that was not fit by OLS on your specific data (e.g., you imposed a fixed slope from external knowledge, or you evaluate on a test set with a model trained on different data). A negative R² means the model fits worse than simply predicting the mean ȳ for every observation. When you use standard OLS on the same data you fit on, R² is always between 0 and 1 by construction.

Simple linear regression uses one predictor (x): ŷ = mx + b. Multiple linear regression uses two or more: ŷ = b₀ + b₁x₁ + b₂x₂ + … Adding more predictors almost always increases R², even if irrelevant — which is why adjusted R² penalizes unnecessary predictors in multiple regression. Multiple regression also introduces multicollinearity (predictors correlated with each other) and higher overfitting risk when predictors outnumber observations.

No. In social sciences and economics, R² values of 0.30–0.50 are common and meaningful because human behavior is noisy. In physics or chemistry, R² < 0.99 might indicate a measurement problem. Context matters: an overfitted model on small data can show R² ≈ 1.0 yet predict new observations poorly. Always use cross-validation (train/test split, or k-fold) for an honest assessment of predictive power.

OLS makes four classical assumptions (acronym LINE): (1) Linearity — the true relationship is linear in the parameters. (2) Independence — observations are not correlated with each other (critical violation: time-series autocorrelation). (3) Normality of residuals — residuals are approximately normal (important for inference, less critical for point estimates). (4) Equal variance (homoscedasticity) — residual variance is constant across all x values. Check assumptions via residual plots, Q-Q plots, Durbin-Watson (autocorrelation), and Breusch-Pagan (heteroscedasticity) tests.

In simple linear regression (one predictor), R² = r², where r is the Pearson correlation coefficient. For example, r = 0.90 gives R² = 0.81. The sign of r tells you the direction (positive or negative slope) which R² alone does not — R² is always ≥ 0 regardless of direction. This relationship breaks down in multiple regression. Use r when you want a symmetric association metric; use R² when you want the fraction of variance explained.

Use it with caution. OLS minimizes squared residuals, giving disproportionate weight to extreme points. A single outlier can shift m dramatically and inflate or deflate R². Best practice: (1) always plot a scatter diagram first; (2) identify outliers via residual plots, Cook's distance, or studentized residuals; (3) investigate before removing — is the outlier a data-entry error or a genuine observation? (4) consider robust alternatives (Theil-Sen estimator, Huber regression) for a sanity check.

Use log-transformations when the relationship is multiplicative (e.g., economic elasticities, incomes spanning orders of magnitude, or when residuals fan out with x). Common transforms: log y on log x (elasticity model), log y on x (exponential growth), y on log x (logarithmic growth). Use polynomial regression (y = a + bx + cx² + …) when a scatter plot shows clear curvature. Limit polynomial degree to 2 or 3 to avoid wild oscillations (Runge's phenomenon). If the relationship is more complex, consider spline regression or theory-derived non-linear forms.

Correlation (Pearson r) measures the strength and direction of a linear association symmetrically — it tells you whether x and y move together but produces no predictive equation. Regression fits the equation ŷ = mx + b for prediction — it is asymmetric (regressing y on x gives a different line than regressing x on y). Use correlation when both variables are exchangeable and you only want to quantify association. Use regression when one variable is naturally the predictor and you want to predict or estimate effect sizes.

A 95% confidence interval for the slope like [1.8, 2.6] means that if you repeated the sampling and regression many times, 95% of such intervals would contain the true slope. The p-value for the slope tests H₀: m = 0 (no linear relationship). A small p-value (typically p < 0.05) means you reject H₀ and conclude there is evidence of a linear association. Note: statistical significance is not the same as practical significance — a tiny slope can have a tiny p-value with enough data. Always report the point estimate and confidence interval, not just the p-value.

Standard post-regression checks: (1) Scatter plot — visually confirm the relationship looks linear and check for outliers. (2) Residual plot (residuals vs. fitted values ŷ) — random scatter around zero is good; systematic curves or fans indicate non-linearity or heteroscedasticity. (3) Q-Q plot of residuals — checks normality assumption needed for inference. (4) Cook's distance — flags influential observations that disproportionately affect the estimates. (5) Cross-validation — split data into train/test sets and compare predicted vs. actual values on the test set to estimate true prediction error.