A regression table looks intimidating — a block of coefficients, standard errors, t-statistics, p-values, and R-squared, all in tiny type. But every number answers a specific question, and exams ask you to pull out exactly those answers. This walkthrough reads a regression output line by line so you know what each cell means and which one to quote.
The Output We Will Read
A study models a student's final exam score using hours studied. The regression output:
R-squared: 0.64 Standard error of the estimate: 7.2 | Term | Coefficient | Std. Error | t-stat | p-value | |---|---|---|---|---| | Intercept | 52.0 | 4.10 | 12.68 | 0.000 | | Hours studied | 4.5 | 0.95 | 4.74 | 0.000 |
The fitted equation is: predicted score = 52.0 + 4.5 × (hours studied). The rest of this guide explains where every number comes from and what it tells you.
The Coefficients: The Heart of the Output
The coefficient column builds the prediction equation, and each coefficient has a precise interpretation.
The intercept, 52.0, is the predicted value of the outcome when every predictor equals zero — the predicted exam score for a student who studied 0 hours. Sometimes that is meaningful; sometimes zero is outside the realistic range and the intercept is just a mathematical anchor.
The slope coefficient on hours studied, 4.5, is the one that matters most. It is read as: for each one-unit increase in the predictor, the outcome changes by the coefficient, holding everything else constant. So each additional hour of study is associated with a 4.5-point increase in predicted exam score. The sign matters — a positive coefficient means the outcome rises with the predictor, a negative one means it falls.
Use the equation to predict: a student who studies 6 hours has a predicted score of 52.0 + 4.5 × 6 = 52.0 + 27.0 = 79.
Note the careful word associated with. Regression describes a relationship; it does not by itself prove that studying causes the score to rise.
Standard Error, t-Stat, and P-Value: Is the Coefficient Real?
A coefficient of 4.5 came from one sample. The next three columns test whether it reflects a real relationship or could be noise.
The standard error of a coefficient, 0.95 for hours studied, measures how much that coefficient would bounce around from sample to sample. A small standard error relative to the coefficient means a precise, stable estimate.
The t-statistic is the coefficient divided by its standard error: 4.5 ÷ 0.95 = 4.74. It measures how many standard errors the coefficient sits from zero. The test here is whether the true coefficient is zero — meaning no relationship — so a t-stat far from zero is evidence of a real effect.
The p-value, 0.000, is the result of that test. It answers: if the true coefficient were zero, how likely is a t-statistic this extreme? A p-value below your significance level (usually 0.05) means the coefficient is statistically significant — the predictor genuinely helps explain the outcome. Here 0.000 (meaning below 0.0005, not literally zero) says hours studied is a significant predictor of exam score. A coefficient with a p-value of, say, 0.62 would be one you cannot distinguish from zero.
R-Squared: How Much the Model Explains
R-squared measures how well the whole model fits. It is the proportion of the variation in the outcome that the model explains, running from 0 to 1.
Here R-squared = 0.64, so the model explains 64% of the variation in exam scores. The other 36% is due to factors not in the model — sleep, prior knowledge, test anxiety, luck — plus randomness.
Two cautions. First, a high R-squared does not mean the model is "correct" or that the relationship is causal; it only means the model tracks the data well. Second, do not chase R-squared blindly: adding more predictors can only raise it, even useless ones, which is why analysts also look at adjusted R-squared, a version that penalizes adding predictors that do not earn their place.
Standard Error of the Estimate
Separate from the coefficient standard errors, the standard error of the estimate (here 7.2) describes the typical size of a prediction error — how far actual values tend to fall from the regression line, in the units of the outcome.
For this model, predictions of exam scores are typically off by about 7.2 points. It is a plain-language measure of accuracy: smaller is better, and it is in the same units as the outcome, which makes it easy to communicate.
Getting Help
Regression output leans on two ideas covered elsewhere: every coefficient's significance test produces a p-value, and the "associated with, not causes" warning is the heart of correlation vs. causation. Read those alongside this guide and the table stops being a wall of numbers.
Conclusion
Reading regression output is a matter of knowing what each column answers. The coefficients build the prediction equation, and the slope is the change in the outcome per unit of the predictor. The standard error, t-stat, and p-value together test whether each coefficient is real — a p-value under 0.05 means significant. R-squared reports the share of variation the model explains, and the standard error of the estimate gives the typical prediction error in plain units. Read it column by column and any regression table becomes a set of clear answers.
