Summary: This linear regression calculator computes best-fit line and R squared with live steps, formulas and a chart. It accepts labeled numeric inputs, works offline through file:// and includes source-backed explanations for students, analysts and researchers.
Linear regression calculator for simple OLS best-fit line, slope, intercept, R squared, prediction and scatter plot. The calculator works offline, updates instantly and includes a worked example, plain-text formula, MathML, references and structured data.
Default values are loaded. Click any field and edit it; results and chart update automatically.
| Input or setting | Result or interpretation | Use this when |
|---|---|---|
| slope b | change in y per 1 unit x | best-fit straight line |
| intercept a | predicted y when x=0 | best-fit straight line |
| R^2=.60 | 60% of y variation explained | best-fit straight line |
slope b = sum((x_i-mean_x)(y_i-mean_y)) / sum((x_i-mean_x)^2); intercept a = mean_y - b*mean_x; yhat = a + b*xSimple linear regression estimates the straight line that minimizes squared residuals.
x=[1,2,3,4,5], y=[2,4,5,4,5]
yhat = 2.2 + 0.6x; R^2 = 0.6
This page uses the shared statistics core for distribution functions, quantiles and exact integer counting where needed. The formula is shown in plain text so screen readers and search engines can parse it reliably.
Use Load example in the calculator to reproduce this reference result.
{
"tool": "Linear Regression Calculator",
"input": "x=[1,2,3,4,5], y=[2,4,5,4,5]",
"output": "yhat = 2.2 + 0.6x; R^2 = 0.6",
"formula": "slope b = sum((x_i-mean_x)(y_i-mean_y)) / sum((x_i-mean_x)^2); intercept a = mean_y - b*mean_x; yhat = a + b*x"
}| Calculator | Example input | Expected output |
|---|---|---|
| Linear Regression Calculator | x=[1,2,3,4,5], y=[2,4,5,4,5] | yhat = 2.2 + 0.6x; R^2 = 0.6 |
What does p = 0.03 mean? If the null hypothesis and model assumptions were true, a result at least this extreme would occur about 3% of the time. The American Statistical Association cautions that a p-value alone does not measure effect size, practical importance or the probability that Hâ‚€ is true.3
For most classroom and professional reports, pair the calculator result with the question you are answering. A mean or median summarizes location, but spread explains consistency. A confidence interval estimates plausible values, while a hypothesis test evaluates compatibility with a null model. Regression and correlation describe association, so they should be reported with a chart and residual or outlier review. When a result is statistically significant, still ask whether the effect is large enough to matter in the real setting.
| Statistic | Small | Medium | Large | Use |
|---|---|---|---|---|
| Cohen's d | 0.2 | 0.5 | 0.8 | t-test effect size |
| Cramér's V | 0.1 | 0.3 | 0.5 | chi-square association |
| |r| | 0.10 | 0.30 | 0.50 | correlation strength |
| R² | 0.01 | 0.09 | 0.25 | variance explained |
The line of best fit is found using ordinary least squares, which picks the slope and intercept that minimise the sum of squared vertical distances between your data and the fitted line. The result is ŷ = b0 + b1x, where b1 = Σ(xi − x̄)(yi − ȳ) / Σ(xi − x̄)² and b0 = ȳ − b1x̄. Calculators and software handle this almost instantly. The line passes through the point (x̄, ȳ) and gives you a tool to predict y from any new x within the observed range.
The slope b1 measures how much ŷ changes for every one-unit increase in x. The intercept b0 is the predicted y when x equals zero. Compute the slope first: b1 = Σ(xi − x̄)(yi − ȳ) / Σ(xi − x̄)². Then find the intercept: b0 = ȳ − b1x̄. For example, if a regression of test score on study hours gives b1 = 5 and b0 = 50, students start at a baseline of 50 marks and gain 5 marks per extra hour studied.
R², or the coefficient of determination, tells you what fraction of the variation in y is explained by your regression line. An R² of 0.78 means 78% of the variability in y can be attributed to its linear relationship with x; the remaining 22% comes from other unmeasured factors or random noise. R² ranges from 0 to 1 — higher generally suggests a better fit. But beware: a high R² doesn't guarantee a good model. Outliers, non-linearity, or omitted variables can still cause problems even when R² looks great.
Plug your x value into the equation ŷ = b0 + b1x. Suppose your model is ŷ = 50 + 5x (where x is study hours and y is exam score) and a student studies 4 hours: ŷ = 50 + 5 × 4 = 70. That's a point estimate. For more honest reporting, build a prediction interval around it, which accounts for both the uncertainty in the regression line and the natural scatter of individual observations. Also, only predict within the range of x values you actually observed — extrapolation is risky.
A residual is the gap between an observed y and the value the regression line predicts: residual = yi − ŷi. If your line predicts 75 for x = 5 but you actually observed 80, the residual is +5. Positive residuals mean the actual point sat above the line; negative ones mean below. Plot the residuals against x or ŷ — random scatter around zero suggests a good model fit, while patterns (curves, fans, clusters) point to problems like non-linearity, missing variables, or non-constant variance.
Ordinary least squares (OLS) is the technique used to estimate the slope and intercept of the regression line. The idea: for every possible line you could draw through the data, compute the sum of squared vertical distances between each point and the line. OLS picks the line that makes that sum as small as possible. Squaring keeps the differences positive and penalises larger errors more heavily. The "ordinary" part means we treat each observation equally — no weighting. It's the workhorse method behind almost all introductory regression analyses.
Two common tests. First, the t-test on the slope: if its p-value is below 0.05, the slope is significantly different from zero, suggesting x has predictive value for y. Second, the F-test for the overall model: it tests whether the regression as a whole explains more variance than the null model with no predictors. In simple linear regression, both tests give the same conclusion. Also examine R² for practical effect size and check the residual plots — a significant model with bad residuals might still be unreliable.
Correlation measures the strength and direction of a linear association between two variables — a single number, r, between −1 and +1. It treats x and y symmetrically. Linear regression goes further: it fits an equation ŷ = b0 + b1x, allowing you to predict y from x. The two are deeply linked — r² = R² in simple linear regression — but regression specifies a direction (you're predicting y from x, not the reverse) and provides slope, intercept, residuals, and prediction tools that correlation alone cannot.