What formula does this calculator use?

It uses ordinary least squares (OLS). The slope is β₁ = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)², and the intercept is β₀ = ȳ − β₁·x̄. These minimise the sum of squared residuals Σ(yᵢ−ŷᵢ)².

What does R² mean and what is a good value?

R² (the coefficient of determination) is the proportion of variance in y explained by the linear relationship with x. It ranges from 0 to 1 (or 0% to 100%). There is no universal "good" threshold — a value of 0.95 is typical in controlled lab experiments, while 0.30 can be meaningful in social-science data where many factors drive the outcome.

What is RSE (residual standard error)?

RSE — also called the standard error of the regression — is the square root of SSE/(n−2). It estimates the typical distance between observed y values and the fitted line, expressed in the same units as y. Smaller RSE means better fit.

What do the t-values for slope and intercept mean?

Each t-value is the coefficient estimate divided by its standard error. Large |t| (roughly above 2 for n > 30) suggests the coefficient is significantly different from zero. Because this calculator is client-side, it does not look up the exact p-value from the t-distribution, but you can compare the t-values against standard critical values for your sample size.

Can I use this for multiple regression?

No. This tool fits one predictor (x) to one response (y) — simple linear regression. For two or more predictors you need multiple regression, which requires matrix operations and is a separate tool.

How is the ANOVA table constructed?

The total sum of squares SS_total = Σ(yᵢ−ȳ)² is split into regression SS (SSR = Σ(ŷᵢ−ȳ)²) and error SS (SSE = Σ(yᵢ−ŷᵢ)²). The F-statistic is MSR/MSE where MSR = SSR/1 and MSE = SSE/(n−2). A large F indicates the linear model explains significantly more variance than a flat mean line.

Linear Regression Calculator

Linear regression is the workhorse of data analysis. Given a set of (x, y) pairs, it finds the single straight line that minimises the total squared distance from every point to the line — the ordinary least-squares (OLS) line. Whether you are studying hours versus exam scores, advertising spend versus sales, temperature versus crop yield, or any other two numeric variables, this calculator gives you the complete statistical picture in seconds, entirely in your browser.

How it works

The calculator solves the normal equations for simple linear regression:

β₁ = SS_xy / SS_xx where SS_xy = Σ(xᵢ − x̄)(yᵢ − ȳ) and SS_xx = Σ(xᵢ − x̄)²

β₀ = ȳ − β₁ · x̄

Once β₀ and β₁ are known, every fitted value is ŷᵢ = β₁xᵢ + β₀ and every residual is eᵢ = yᵢ − ŷᵢ. From these the calculator computes:

Pearson r = SS_xy / √(SS_xx · SS_yy) — linear correlation, −1 to +1
R² = r² — fraction of variance in y explained by x
RSE = √(SSE / (n − 2)) — residual standard error, the typical prediction error in y-units
SE(β₁) = RSE / √SS_xx — uncertainty around the slope
SE(β₀) = RSE · √(Σxᵢ² / (n · SS_xx)) — uncertainty around the intercept
t-statistics for both coefficients: t = estimate / SE
Full ANOVA table with regression, error and total rows

The interactive scatter plot draws the fitted line over the data points and labels R² directly on the chart for immediate visual confirmation.

Worked example — study hours vs exam score

Using the built-in demo data (12 students, study hours 1–10, scores 52–97):

Statistic	Value
Slope β₁	5.0559
Intercept β₀	45.9779
R	0.9944
R²	0.9888
RSE	1.43

Interpretation: The regression line is ŷ = 5.06·x + 46.0. Each extra hour of study predicts roughly 5 more marks. R² = 0.989 means study hours explain 98.9% of the variance in exam scores for this sample — an almost perfectly linear relationship. The RSE of 1.43 marks tells you the model’s typical prediction error is less than 1.5 marks.

Prediction: A student who studies 7.5 hours is predicted to score ŷ = 5.06 × 7.5 + 46.0 = 83.9 marks.

Formula reference

Slope        β₁ = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)²
Intercept    β₀ = ȳ − β₁·x̄
Fitted       ŷᵢ = β₁·xᵢ + β₀
Residual     eᵢ = yᵢ − ŷᵢ
SSE          Σeᵢ²
RSE          √(SSE/(n−2))
SE(β₁)       RSE / √SS_xx
SE(β₀)       RSE · √(Σxᵢ²/(n·SS_xx))
r            SS_xy / √(SS_xx · SS_yy)
R²           r²

All computations happen in your browser. No data is ever sent to a server.