Linear regression is a statistical method that predicts one number (like exam score) from one or more other numbers (like hours studied) by fitting a straight line through your data — and interpreting it means reading four things: the slope, the intercept, the R², and the p-value. If you Googled this because you ran a regression and your output is a wall of numbers you don't understand, you're in the right place — this guide translates every piece into plain English.
Key Takeaways
- Linear regression predicts a continuous outcome (the dependent variable) from one or more predictors (independent variables) by fitting the best-fitting straight line through your data points.
- The slope (B) tells you how much the outcome changes for each one-unit increase in the predictor — the single most important number in your output.
- R² tells you the percentage of variation in the outcome your model explains, ranging from 0 (useless) to 1 (perfect).
- A p-value below .05 means the relationship is statistically significant — unlikely to be due to chance alone.
- Use simple linear regression for one predictor and multiple linear regression for two or more predictors.
What is linear regression in simple terms?
Linear regression answers the question: "If I know X, what's my best guess for Y?" Imagine plotting hours studied against exam scores for 50 students. The dots roughly trend upward — more study, higher scores. Linear regression draws the single straight line that sits closest to all those dots, giving you a formula to predict any student's score from their study hours.
That line is defined by two numbers you'll see in every output:
- Intercept (constant): the predicted value of Y when X is zero.
- Slope (B or b): how much Y changes for each one-unit increase in X.
The formula is simply: Y = intercept + (slope × X). Everything else in your regression output exists to tell you how trustworthy that line is.
When should I use linear regression?
Use linear regression when your outcome variable is continuous (a number that can take many values — test scores, blood pressure, income, reaction time) and you want to predict or explain it using one or more other variables.
Some concrete examples from real research:
- Predicting systolic blood pressure from age and BMI (medicine).
- Predicting depression scores from hours of sleep and social support (psychology).
- Predicting house price from square footage and number of bedrooms (economics).
If your outcome is a category instead (pass/fail, disease/no disease), you need logistic regression, not linear. And if you're just testing whether two group means differ, a t-test or ANOVA is the better fit — see our guide on choosing the right statistical test.
Simple vs multiple linear regression
The difference is just the number of predictors.
| Feature | Simple linear regression | Multiple linear regression |
|---|---|---|
| Predictors | One | Two or more |
| Question answered | Does X predict Y? | Which of several variables predict Y, and how much? |
| Example | Does study time predict exam score? | Do study time and sleep and anxiety predict exam score? |
| Key extra output | — | Each predictor gets its own slope and p-value |
| Watch out for | Outliers, non-linear trends | Multicollinearity (predictors overlapping too much) |
Most real theses use multiple regression because outcomes rarely depend on just one thing.
How do you interpret linear regression output?
Here's the part everyone gets stuck on. Every regression output gives you four things worth reading, in order of importance.
1. The slope (B) — the effect
The slope is your headline result. If the slope for hours studied is 4.2, it means: for every extra hour a student studies, their predicted exam score rises by 4.2 points, holding other variables constant. A negative slope means the outcome goes down as the predictor goes up.
2. The p-value — is it real?
Next to each slope is a p-value. If p < .05, the relationship is statistically significant — meaning it's unlikely you'd see a slope this large by random chance if the true relationship were zero. A slope of 4.2 with p = .003 is a finding you can report; the same slope with p = .41 is probably noise.
3. R² — how good is the model?
R² (R-squared) is the proportion of variation in the outcome your predictors explain. An R² of .34 means your model explains 34% of the variation in exam scores — the other 66% comes from things you didn't measure. In social sciences, R² values of .10–.30 are common and perfectly publishable; don't expect .90.
4. The intercept — the baseline
The intercept is Y when all predictors equal zero. It's often not meaningful on its own (a student who studies zero hours may be outside your data range), but it's needed to complete the prediction formula.
A worked example with real numbers
Suppose we surveyed 60 undergraduates and ran a simple regression predicting exam score (0–100) from hours studied per week. StatRyx returns:
- Slope (B) = 4.15, SE = 1.02
- Intercept = 41.3
- R² = .22
- F(1, 58) = 16.35, p < .001
Here's what each number means:
- The prediction formula is: Exam score = 41.3 + (4.15 × hours studied). A student studying 10 hours has a predicted score of 41.3 + 41.5 = 82.8.
- The slope of 4.15 means each additional study hour adds about 4 points to the predicted score.
- R² = .22 means study time explains 22% of the variation in exam scores — meaningful, but clearly other factors matter too.
- F(1, 58) = 16.35, p < .001 confirms the model as a whole is statistically significant.
How do I report linear regression in APA 7 format?
APA 7 wants the F statistic, degrees of freedom, p, and R² for the overall model, then the unstandardised slope (B), its standard error, and confidence interval for each predictor. Here's the write-up for the example above:
A simple linear regression was conducted to predict exam score from weekly study hours. The model was statistically significant, F(1, 58) = 16.35, p < .001, and explained 22% of the variance in exam scores (R² = .22). Study hours significantly predicted exam score, B = 4.15, 95% CI [2.11, 6.19], p < .001.
Note the APA conventions: test statistics and p are italicised, and there's no leading zero before the decimal in p and R² values (write p < .001, not p < 0.001).
Do I need to check assumptions first?
Yes — linear regression rests on assumptions, and skipping them is the most common thesis mistake. The four to check are: a linear relationship between predictor and outcome, independence of observations, homoscedasticity (constant spread of residuals), and normally distributed residuals. If your residuals fan out or curve, your slope and p-value may be misleading, and you might need a transformation or a different model.
Checking these by hand with residual plots is tedious. StatRyx runs the assumption checks automatically, flags violations in plain English, and tells you whether your model is safe to report — the kind of guardrail that turns SPSS from a two-hour ordeal into a two-minute one.
Stop calculating this by hand — run it free in StatRyx → Try StatRyx
Frequently Asked Questions
What is the difference between R and R² in regression?
R is the correlation between your predicted values and the actual outcome, ranging from −1 to +1. R² is simply R squared, and it's the more useful figure because it directly tells you the *proportion of variance expl