Run simple and multiple ordinary least squares regression entirely in your browser. Paste a column of paired observations or a full design matrix and get the fitted equation, coefficient table with standard errors, t-statistics, p-values, and 95 percent confidence intervals, alongside R squared, adjusted R squared, the F-test, RMSE, residual diagnostics, and one-click R, Python, and APA reporting output.
Comma, space, or tab separated. Need at least 3 rows.
Pick simple regression for a single predictor, or switch to the multiple regression tab when several variables jointly explain the outcome.
For simple regression, enter x and y values per line. For multiple regression, include a header row, then y followed by predictor columns.
Each row shows the estimate, standard error, t-statistic, p-value, and 95 percent confidence interval for one term in the model.
Inspect the scatter plot for visible non-linearity and the residuals-versus-fitted plot for funnel shapes that signal heteroscedasticity.
Copy the R lm script, the Python statsmodels equivalent, or the APA results paragraph to drop directly into your manuscript.
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A coefficient without a 95 percent confidence interval is uninterpretable. The interval shows precision and whether the estimate is consistent with no effect.
When evaluating two models with different numbers of predictors on the same data, prefer adjusted R squared because it penalises overfitting.
Inspect residual plots before reading p-values. Non-linearity or heteroscedasticity invalidates the standard errors and the entire inference chain.
When predictors are highly correlated, individual coefficients become unstable. Compute variance inflation factors and consider removing or combining redundant variables.
Ordinary least squares regression remains the workhorse of empirical research because it cleanly answers a deceptively simple question: how does the average value of an outcome change as the predictors change? Galton (1886) introduced the technique to study how the heights of children regress toward the population mean, and the method has since become the foundation of econometrics, epidemiology, psychology, education research, and applied business analytics. A linear regression calculator automates the matrix algebra that produces the coefficient vector beta, the standard errors derived from the residual variance and the diagonal of the inverse Gram matrix, the global F-test that compares the fitted model with an intercept-only baseline, and the individual t-tests that isolate the unique contribution of each predictor.
Interpreting a regression model is a two-step exercise. First, the coefficient itself tells you the expected change in the outcome per one unit change in the predictor, holding the other predictors constant. Second, the 95 percent confidence interval communicates how precisely you have estimated that change: a wide interval that crosses zero leaves the direction of the effect ambiguous, while a tight interval well away from zero supports a clear substantive claim. The Cochrane Handbook for Systematic Reviews of Interventions (Higgins et al., 2023) emphasises that meta-analytic synthesis of regression coefficients requires both the point estimate and a measure of precision, which is why the calculator outputs every piece of information needed for downstream pooling.
R squared and adjusted R squared describe how much of the variance in the outcome is explained by the model. R squared always increases as predictors are added, even when the additional variables contribute nothing, so adjusted R squared, which penalises model complexity, is the more honest summary when comparing alternatives. Root mean squared error expresses the typical deviation between predicted and observed values in the original units of Y, providing a face-valid measure of forecast accuracy. The omnibus F-test asks whether the joint contribution of all predictors is greater than chance, while individual t statistics decompose that contribution into per-variable tests.
The validity of OLS inference rests on a small set of assumptions: linearity in the parameters, independent observations, homoscedasticity of residuals, and approximate normality of the error distribution. The residuals-versus-fitted plot is the primary diagnostic tool for the first three. A clear funnel pattern signals heteroscedasticity and calls for robust standard errors or a generalised least squares fit. A curved cloud suggests that the relationship is non-linear and may need a polynomial term or a transformation. Strongly correlated predictors inflate variance and produce coefficients that flip sign when small changes are made to the dataset, a phenomenon best diagnosed with variance inflation factors. When clinical or policy questions involve binary, count, or time-to-event outcomes, OLS gives way to logistic, Poisson, or Cox regression respectively, but the same diagnostic mindset applies.
Reproducibility has become a core expectation in academic publishing, and modern reporting guidelines require analysts to publish the code that generated their results. The calculator addresses this directly by exporting a self-contained R script using the lm and confint functions and a Python script using statsmodels OLS, so the analysis you ran in the browser can be reproduced verbatim in a journal supplement or a thesis appendix. For regression assumptions that look problematic and for designs that involve clustered data, repeated measures, or missing values, our statistical analysis service provides PhD-level support: bootstrapped confidence intervals, mixed-effects extensions, multiple imputation, and tailored APA write-ups that fit the rest of your manuscript.
Linear regression is a statistical method that estimates the relationship between a continuous outcome (Y) and one or more predictors (X) by fitting a straight line that minimises the sum of squared residuals. The fitted line takes the form Y = b0 + b1 X1 + b2 X2 + ..., where each coefficient represents the expected change in Y for a one unit change in that predictor, holding the others constant.
Use simple linear regression when you have one predictor and want to characterise the bivariate relationship with the outcome. Use multiple regression when several variables jointly explain the outcome and you want to estimate the unique contribution of each, controlling for the others. Multiple regression is also the standard adjustment tool for confounders in observational research.
R squared is the proportion of variance in the outcome explained by the model, ranging from 0 to 1. Adjusted R squared penalises models that include unhelpful predictors, so it is the preferred metric when comparing models with different numbers of variables. Values are not directly comparable across datasets, and a high R squared does not by itself prove a causal relationship.
The overall F-test asks whether the model as a whole explains more variance than an intercept-only model. A small p-value indicates that at least one predictor is meaningfully related to the outcome. Individual t-tests for each coefficient then tell you which specific variables drive the effect.
The residuals-versus-fitted plot lets you check linearity and homoscedasticity. The 95 percent confidence intervals assume that residuals are approximately normally distributed and that observations are independent. For violations such as heavy tails, autocorrelation, or non-constant variance, our statistical analysis service can run robust standard errors, weighted least squares, or generalised linear alternatives.
Yes. Switch to the multiple regression tab and paste a header row followed by your data, where the first column is the outcome Y and the remaining columns are predictors. The tool fits an OLS model with intercept, returns coefficients with standard errors, t-statistics, p-values, and 95 percent confidence intervals, and reports R squared, adjusted R squared, the F-test, and root mean squared error.
Yes. The Copy R Code button outputs a script using lm and confint, and the Copy Python button outputs a statsmodels OLS script with summary and confidence intervals. Both scripts reproduce the exact analysis run in the browser so you can include them in supplementary materials or a methods section.
The APA report button produces a ready-to-paste paragraph following the seventh edition guidelines: F statistic with degrees of freedom and p-value, R squared, adjusted R squared, RMSE, the regression equation, and slope details with 95 percent confidence interval. You can edit the predictor labels to match the wording of your manuscript.
No. Every calculation runs locally in your browser. Your dataset never leaves your device and nothing is logged on our servers, which makes the tool safe for sensitive clinical or proprietary data.
If the relationship between Y and X is curved, log-linear, or polynomial, fit a transformed model in the multiple regression tab by adding columns like log_x or x_squared. For binary outcomes you need logistic regression rather than OLS, and for time-to-event outcomes you need Cox regression. Our biostatistics service handles all of these advanced models.
Need to size a study before running it? The sample size calculator covers regression and correlation designs. Reporting a coefficient as a standardised effect? Convert to Cohen's d with the effect size calculator. If you only have a p-value and an estimate but need a confidence interval, the p-value to CI converter is the right tool. To calculate a general confidence interval for a mean or proportion, use the confidence interval calculator. For meta-analytic regression of effect sizes against study-level moderators, see the meta-regression formatter.
Reviewed by
Dr. Sarah Mitchell holds a PhD in Biostatistics from Johns Hopkins Bloomberg School of Public Health and has over 15 years of experience in systematic review methodology and meta-analysis. She has authored or co-authored 40+ peer-reviewed publications in journals including the Journal of Clinical Epidemiology, BMC Medical Research Methodology, and Research Synthesis Methods. A former Cochrane Review Group statistician and current editorial board member of Systematic Reviews, Dr. Mitchell has supervised 200+ evidence synthesis projects across clinical medicine, public health, and social sciences. She reviews all Research Gold tools to ensure statistical accuracy and compliance with Cochrane Handbook and PRISMA 2020 standards.
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