Coefficient table
How this is calculated
Regression Calculators
A regression calculator finds the straight line that best fits a set of paired data points, then uses that line to describe and predict the relationship between two variables. This regression calculator runs a simple linear regression: you enter your x values (the predictor) and y values (the outcome), and it returns the regression equation, the strength of the relationship, and whether that relationship is statistically significant.
How to use the regression calculators
Enter one (x, y) pair per line in the data box, separating the two numbers with a comma, space, or tab. The results update the moment you type, so there is no button to press. You can also load a sample dataset to see how it works, then replace it with your own numbers. Any line that does not contain two numbers is ignored automatically, so pasting from a spreadsheet usually works without cleanup.
What the regression calculator gives you
The headline output is the regression equation in the form ŷ = b₀ + b₁x. The intercept (b₀) is the predicted value of y when x is zero, and the slope (b₁) tells you how much y is expected to change for each one-unit increase in x. A positive slope means y rises as x rises; a negative slope means y falls as x rises.
Alongside the equation, the calculator reports the correlation coefficient (r), which ranges from −1 to +1 and measures how tightly the points cluster around the line. It also reports R², the coefficient of determination, which is the proportion of variation in y that the line explains. An R² of 0.90 means the model accounts for 90 percent of the variability in your outcome. The standard error of the estimate (Sₑ) describes the typical distance between the observed points and the fitted line in the units of y.
To test whether the relationship is real or could be due to chance, the calculator runs a t-test on the slope and reports a two-sided p-value. A p-value below 0.05 is the conventional threshold for calling the slope statistically significant.
The formulas behind the calculator
This tool uses ordinary least squares, the standard method taught in statistics courses. The slope is calculated as the sum of cross products divided by the sum of squares of x: b₁ = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)². The intercept is then b₀ = ȳ − b₁x̄. The correlation coefficient is r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² Σ(y − ȳ)²], and R² is simply r squared for a simple linear model. The "How this is calculated" section inside the tool shows each of these formulas with your own numbers substituted in, so you can follow the arithmetic step by step or check your work for a homework problem.
Reading the charts
The scatter plot shows your raw data points with the line of best fit drawn through them, so you can see at a glance how well a straight line describes the data. The donut chart shows R² as a percentage of variance explained. The residual plot shows the difference between each observed value and the value the line predicts; ideally these residuals scatter randomly around zero with no clear pattern, which suggests a straight line is a reasonable fit.
Making predictions
Once a line is fitted, you can predict y for any x using the slider or the input box. Keep in mind that predicting far outside the range of your data is extrapolation and becomes increasingly unreliable, so the tool flags when your chosen x falls outside the data you entered.
When to use linear regression
Simple linear regression is appropriate when you expect a roughly straight-line relationship between one predictor and one outcome, both measured on a continuous scale. It is widely used to study how study time relates to exam scores, how advertising spend relates to sales, or how temperature relates to energy use. If the relationship is curved, if you have several predictors, or if your outcome is a category rather than a number, you will need a different model such as polynomial, multiple, or logistic regression.
Frequently asked questions
What is a good R² value? It depends on the field. In tightly controlled experiments an R² above 0.9 is common, while in social science or behavioral data an R² of 0.3 may still be meaningful. R² should always be read alongside the p-value and the scatter plot rather than on its own.
What is the difference between correlation and regression? Correlation measures the strength and direction of a relationship with a single number, r. Regression goes further and gives you an equation you can use to predict one variable from another.
How many data points do I need? You need at least three points for this calculator to estimate the standard error and run the significance test, but more points give more reliable results.
