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Linear Regression Calculator

Calculate the least-squares regression line, slope, intercept, and R² for a set of paired data points.

Result

Regression Equation
y = 2.0343x + -0.0533
R-Squared
0.9979
Slope
2.0343
Y-Intercept
-0.0533

About the Linear Regression Calculator

The Linear Regression Calculator fits a least-squares line through a set of paired X and Y values, returning the equation, R-squared, slope, and intercept that best describe a straight-line relationship between two variables. It's the standard tool for turning a scatter of data points into a predictive equation and a measure of how well that line actually fits.

How It Works

Enter comma-separated X values and their matching Y values. The calculator finds the mean of each list, computes how X and Y values jointly and individually vary around those means, and uses those sums to solve for the slope and intercept that minimize the total squared vertical distance between the line and every data point. It also reports R-squared, built from the same underlying deviation sums as the Pearson correlation coefficient.

Slope b = sum((x - x-mean)(y - y-mean)) / sum((x - x-mean)^2); Intercept a = y-mean - b times x-mean; r = sum((x - x-mean)(y - y-mean)) / sqrt(sum((x - x-mean)^2) times sum((y - y-mean)^2)); R-squared = r^2.

Formula & Methodology

To calculate by hand, find the mean of the X values and the mean of the Y values. For each pair, multiply the deviation of X from its mean by the deviation of Y from its mean, and sum those products - that total, divided by the sum of squared X deviations, is the slope. Subtract the slope times the X mean from the Y mean to get the intercept. Squaring the correlation coefficient built from the same deviation sums gives R-squared, the share of Y's variation the line explains.

Examples

Near-Perfect Linear Trend

Six X values (1 through 6) paired with Y values (2.1, 3.9, 6.2, 7.8, 10.1, 12.3) produce the equation y = 2.0343x - 0.0533 with an R-squared of about 0.998, indicating the line explains nearly all of the variation in Y.

Study Hours vs. Test Score

Five students studying 1 through 5 hours score 50, 55, 65, 70, and 80 on a test. The fitted line is y = 7.5x + 41.5 with an R-squared of about 0.987, suggesting each additional hour of study is associated with roughly 7.5 more points.

Advantages

  • Produces both the predictive equation and a fit-quality measure (R-squared) in one calculation, rather than requiring separate tools.
  • Uses the exact least-squares formulas underlying regression software, so results match what a spreadsheet or statistics package would return.
  • Works from raw paired data with no need to pre-calculate means or sums by hand.

Common Mistakes

  • Using the fitted line to predict Y for X values far outside the range of the original data, where the linear relationship may no longer hold.
  • Treating a high R-squared as proof that X causes Y, when it only shows that a straight line fits the observed data well.
  • Fitting a line to data with an obviously curved or cyclical pattern, where a straight line is the wrong model regardless of how the slope and intercept come out.

Edge Cases to Watch For

  • Requires at least 2 paired points with equal counts of X and Y values, or the calculator returns an error.
  • If every X value is identical, the denominator of the slope formula is zero and no line can be fit - the calculator detects this and reports it rather than dividing by zero.
  • The fitted line describes a linear relationship only; a high R-squared confirms the data fits a straight line well but says nothing about whether the true relationship is curved or whether X causes Y.
  • Because the method minimizes squared distances, a single far-outlying point can pull the slope and intercept substantially, especially with a small number of data points.

Common Use Cases

  • Researchers and students quantifying the relationship between two measured variables for a report or assignment.
  • Analysts building a simple predictive model, such as estimating one variable from another based on historical paired data.
  • Anyone checking whether an apparent trend in a scatter of data points is actually a strong linear relationship or a weak, noisy one.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

What does R² tell you?

R-squared (from 0 to 1) measures how much of the variation in Y is explained by the linear relationship with X - an R² of 0.95 means 95% of Y's variation is explained by the fitted line, while a low R² suggests X alone doesn't predict Y well with a straight-line model.

How is the regression line calculated?

The least-squares method finds the slope and intercept that minimize the sum of squared vertical distances between each data point and the line - it's the same underlying calculation as the Pearson correlation coefficient, just expressed as a predictive equation instead of a single correlation number.

Related calculators

Confidence Interval CalculatorCalculate the confidence interval for a sample mean.Sample Size CalculatorCalculate the minimum sample size needed for a survey or study at a given confidence level and margin of error.Margin of Error CalculatorCalculate the margin of error for a survey result given sample size and confidence level.Correlation Coefficient CalculatorCalculate the Pearson correlation coefficient between two paired sets of numbers.

Conclusion

Least-squares regression gives a precise, reproducible answer to what line best fits a dataset, along with R-squared to show how much confidence that fit deserves. It works best as a first step, followed by a visual check of the data for curvature or outliers the equation alone won't reveal.