Linear regression calculator

The slope is the working heart of the model: it is the predicted change in the outcome for every one-unit rise in the predictor, stated in the real units of each variable. A slope of 2.5 on a model of exam score against hours studied says the model expects 2.5 more marks per extra hour. The intercept is the predicted outcome when the predictor is zero, which is only meaningful when a value of zero actually falls inside your data; for a predictor like age or income it is often just a mathematical anchor rather than a real-world prediction.

R-squared reports the share of the outcome's variance the line explains, from 0 to 1, so 0.40 means the predictor accounts for 40 percent of the variation. Resist judging a model by r-squared alone. A high value can hide a violated assumption, and what counts as good is entirely field-dependent: 0.3 is a strong result in human behaviour research and disappointing in a physics laboratory. Always pair r-squared with the slope, its significance, and the wider context, and remember that explaining variance is not the same as proving cause, a point our note on the correlation and causation distinction makes in full.

A fitted line is only trustworthy when its assumptions hold, and the t value beside the slope assumes all of them. The relationship must be genuinely linear; the residuals must be independent of one another, which time-ordered data often breaks; their spread must stay roughly constant across the range of the predictor, a condition called homoscedasticity; and they should be approximately normally distributed. A residual-versus-fitted plot exposes curvature and a fanning spread at a glance, while a normal probability plot checks the distribution of the errors.

Two further problems deserve a look. Influential outliers can tilt the whole line, so inspect any point with a large residual or high leverage before trusting the fit, and in multiple models multicollinearity between predictors inflates the standard errors and destabilises the coefficients. Confirm the residuals are well-behaved with a formal normality check on the errors, and read our full walkthrough of the linear regression assumptions for the diagnostic plots and the fixes when one is violated.

Simple linear regression uses a single predictor, which is rarely enough for a real research question. Multiple linear regression adds two or more predictors to the same equation, and the payoff is interpretive: each coefficient now measures the effect of its predictor while holding the others constant. That lets you separate the contribution of, say, study hours from prior attainment instead of blending them, and it lets you control for confounders that a simple model would let masquerade as the effect of interest.

Moving to multiple predictors changes how you read the output. Quote the adjusted r-squared rather than the raw value, because the plain r-squared can only rise as predictors are added, while the adjusted figure penalises predictors that earn their place. Watch for multicollinearity among the predictors, since highly correlated inputs make individual coefficients unstable and hard to interpret. The same least-squares logic and the same regression assumptions still apply; there are simply more relationships to check before the model can be trusted.

The deciding factor is the type of outcome. Linear regression predicts a continuous outcome such as a score, a price, or a measurement, and its slope reads as a change in units. When the outcome is categorical, typically a yes or no event like passed or failed, a straight line is the wrong tool, because it can predict impossible probabilities below zero or above one. Logistic regression solves this by modelling the log-odds of the event, keeping every prediction inside the valid 0-to-1 range.

The interpretation shifts accordingly. A logistic model reports odds ratios rather than slopes, where a value above one means the predictor raises the odds of the outcome and a value below one means it lowers them. If your dependent variable is binary, switch methods rather than forcing a linear fit, and use the effect size converter to put an odds ratio on a familiar scale. Our guide to making sense of logistic regression output covers reading those odds ratios and their confidence intervals in detail.

APA style expects the overall model and the individual predictors to be reported in a fixed order. State the model fit first with its F test, for example F(1, 48) = 22.13, p < .001, followed by the r-squared as the share of variance explained. Then report each predictor's unstandardised slope b, its standard error, the standardised beta, and the exact p-value, so a reader can see both the raw effect in real units and the comparable standardised effect. Round coefficients to two decimals and give exact p-values to three.

For a multiple regression, a results table is far clearer than prose: one row per predictor with the coefficient, standard error, beta, and significance, plus the model fit summarised beneath it. Write a short paragraph above the table interpreting the meaningful predictors in plain language rather than restating every number. Our walkthrough of writing statistics into an APA results section and our results formatting tool turn the raw output into the exact sentences and tables your committee expects.