What is the difference between B and beta (β) in regression?

B is the unstandardised coefficient: the change in Y for a one-unit change in X, in the original units. β is standardised (in SD units), so you can compare which predictor has the stronger effect.

What is a good R² value?

It depends on the field. In social science and survey research, R² of .10 to .30 is common and useful. Use adjusted R² when comparing models with different numbers of predictors.

How do I check multicollinearity?

Look at the VIF for each predictor. Below 5 is usually fine; above 10 means predictors overlap heavily and coefficients become unstable. Remove or combine highly related predictors.

How many participants do I need for multiple regression?

A common rule of thumb is N ≥ 50 + 8k to test the overall model and N ≥ 104 + k to test individual predictors, where k is the number of predictors (Green, 1991).

LINEAR REGRESSION CALCULATOR

Linear regression calculator: simple and multiple regression.

Paste your data with column names, pick the outcome and predictors, and get R², adjusted R², the F test, a full coefficients table with B, SE, β, t, p, and 95% CIs, VIF, residual diagnostics, and an APA-style sentence.

DATA

One row per participant

BROWSER-ONLY

Paste columns from Excel, SPSS, or Google Sheets with a header row of variable names. Values can be separated by tabs, spaces, or commas.

DATA TABLE

OUTCOME (Y)

Predictors (X):

Linear regression formulas, assumptions, and APA reporting

Simple vs multiple linear regression

Simple linear regression predicts one numeric outcome (Y) from one predictor (X): Y = b₀ + b₁X. Multiple regression adds more predictors: Y = b₀ + b₁X₁ + b₂X₂ + … Each coefficient is the expected change in Y for a one-unit increase in that predictor, holding the other predictors constant.

How the coefficients are estimated

b = (XᵀX)⁻¹Xᵀy, SE(bⱼ) = √[MSE · (XᵀX)⁻¹ⱼⱼ], t = bⱼ ÷ SE(bⱼ), df = N − k − 1

Ordinary least squares chooses the coefficients that minimise the sum of squared residuals. The standardised coefficient β = B × SDₓ ÷ SDᵧ lets you compare predictors measured on different scales.

R², adjusted R², and the F test

R² = SS_regression ÷ SS_total, adj. R² = 1 − (1 − R²)(N − 1) ÷ (N − k − 1), F = (SS_reg ÷ k) ÷ MSE

R² is the proportion of variance in the outcome explained by the model. Adjusted R² penalises extra predictors and is better for comparing models. The F test checks whether the model as a whole predicts better than the mean. Cohen's f² = R² ÷ (1 − R²) gives .02 small, .15 medium, and .35 large.

Assumptions to check

Linearity: the residuals-vs-fitted plot shows no curve
Independence of residuals: Durbin-Watson close to 2 (roughly 1.5 to 2.5)
Homoscedasticity: the spread of residuals is similar across fitted values (no funnel shape)
Normality of residuals: check a histogram or Q-Q plot (use the Shapiro-Wilk calculator on the residuals)
No multicollinearity: VIF below 5 (some texts allow up to 10)
No highly influential cases: Cook's distance below 1
Sample size: a common rule is N ≥ 50 + 8k for testing the overall model (Green, 1991)

How to report it (APA style)

“A multiple linear regression was run to predict exam score from study hours and sleep. The model was significant, F(2, 12) = 27.22, p < .001, R² = .82, adjusted R² = .79. Study hours (B = 3.26, β = .76, t = 6.19, p < .001) and sleep (B = 4.29, β = .44, t = 3.57, p = .004) were both significant predictors.” In a table, report B, SE, β, t, p, and the 95% CI for each predictor.

References

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Lawrence Erlbaum.
Durbin, J., & Watson, G. S. (1951). Testing for serial correlation in least squares regression. Biometrika, 38(1–2), 159–177.
Cook, R. D. (1977). Detection of influential observation in linear regression. Technometrics, 19(1), 15–18.
Green, S. B. (1991). How many subjects does it take to do a regression analysis? Multivariate Behavioral Research, 26(3), 499–510.
Field, A. (2018). Discovering statistics using IBM SPSS Statistics (5th ed.). SAGE.

Linear Regression Calculator

Run simple or multiple OLS regression from pasted columns, with SPSS-style model summary, ANOVA and coefficients tables, assumption diagnostics, a residual plot, and a ready APA sentence.

Test which variables predict an outcome in a thesis or survey

Get B, β, t, and p for each predictor in an SPSS-style table

Check multicollinearity, independence, and influential cases

KEY FUNCTIONS

Simple and multiple regression with any number of predictorsR, R², adjusted R², standard error of the estimate, F testCoefficients with B, SE, β, t, p, 95% CI, and VIFDurbin-Watson, Cook's distance, and a residual plotAPA 7 report sentence and regression equation

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