Linear Regression Calculator
Example 1
A researcher collects data on study hours and test scores for 5 students: (2, 55), (4, 65), (6, 73), (8, 80), (10, 88) where x = hours studied and y = test score out of 100. Find the linear regression equation to predict scores.
Solution
Calculate the intercept: b₀ = ṙ − b₁ × x̄ = 72.2 − 4.05 × 6 = 72.2 − 24.3 = 47.9.
Answer
The regression equation is y = 4.05x + 47.9. For every additional hour studied, the predicted test score increases by 4.05 points.
Source: StatQuest: Linear Regression
Example 2
Using the regression equation y = 4.15x + 47.3 from the previous example, predict the test score for a student who studies for 7 hours. Also determine how many hours a student needs to study to achieve a score of 90.
Solution
For x = 7 hours: y = 4.05 × 7 + 47.9 = 28.35 + 47.9 = 76.25.
To find hours needed for y = 90: x = (90 − 47.9) / 4.05 = 42.1 / 4.05 = 10.40 hours.
The R² value indicates how well the model fits the data. With 5 data points and a strong linear trend, the model explains approximately 98% of the variance in test scores.
Answer
A student studying 7 hours is predicted to score 76. To achieve a score of 90, approximately 10.4 hours of study would be needed.
Source: StatQuest: R-squared
Example 3
A temperature and pressure experiment yields paired data: (10 C, 101.3 kPa), (20 C, 103.5 kPa), (30 C, 107.1 kPa), (40 C, 109.8 kPa), (50 C, 113.2 kPa). Compute R-squared and predict pressure at 35 C.
Solution
At x = 35: y = 0.301 × 35 + 97.95 = 108.5 kPa.
Answer
R-squared = 0.994 (excellent fit). Predicted pressure at 35 C: 108.5 kPa.
Example 4
An economist collects data on annual advertising spend (thousands) and quarterly revenue (thousands): (10, 150), (15, 210), (20, 245), (25, 310), (30, 365). Calculate residual standard error and predict revenue at $22,000 advertising spend.
Solution
Predicted y: at x=10: 150; x=15: 203; x=20: 256; x=25: 309; x=30: 362.
Residuals: 0, 7, -11, 1, 3. Sum of squared residuals = 180.
Answer
Residual standard error: 7.75 ($7,750). Predicted revenue at $22,000 advertising: y = 10.6 x 22 + 44 = 277.2 ($277,200).
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References
- [1]Wiley, Applied Regression Analysis, 3rd ed., 1998.
- [2]NIST, NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook
- [3]PennState Eberly College of Science, STAT 501: Regression Methods. https://online.stat.psu.edu/stat501
- [4]IUPAC, Harmonized Guidelines for Single-Laboratory Validation of Methods of Analysis, 2002.
Glossary
- Linear regression – A statistical method that models the relationship between an independent variable X and a dependent variable Y by fitting a straight line through observed data points using the least squares criterion.
- Dependent variable – The outcome variable whose variation is being explained or predicted, denoted as Y in the regression equation y = mx + b. It is also called the response variable.
- Independent variable – The predictor variable used to explain or predict changes in the dependent variable, denoted as X in the regression equation y = mx + b. It is also called the explanatory variable.
- Slope – The coefficient m in the regression equation y = mx + b, representing the change in Y for each one-unit increase in X. A positive slope indicates a direct relationship; a negative slope indicates an inverse relationship.
- Intercept – The constant b in the regression equation y = mx + b, representing the predicted value of Y when X equals zero. It is also called the y-intercept.
- Correlation coefficient (r) – A measure of the strength and direction of a linear relationship between two variables, ranging from −1 to +1. Values near ±1 indicate a strong linear relationship; values near 0 indicate a weak or no linear relationship.
- Coefficient of determination (R²) – The proportion of variance in the dependent variable that is predictable from the independent variable, ranging from 0 to 1. An R² of 0.85 means 85% of the variability in Y is explained by X.
- Residual – The vertical distance between an observed data point and the value predicted by the regression line at the same X coordinate. A positive residual means the point lies above the line; a negative residual means it lies below.
- Least squares method – The optimization technique used to fit a regression line by minimizing the sum of squared vertical distances (residuals) between observed data points and the line. This produces the best-fitting linear model for the data.
- Line of best fit – The straight line that best represents the relationship between two variables in a scatter plot, determined by the least squares method. It is also called the regression line and is expressed as y = mx + b.
How to Use?
- 1
Locate your data
Collect paired X and Y observations. Each X value must have a corresponding Y value.
- 2
Enter data pairs
Set the number of data points, then enter each paired X and Y value in the table rows — for example: (1, 2.1), (2, 3.9), (3, 6.2), (4, 7.8), (5, 10.1).
- 3
Predict a value (optional)
Enter an X value in the Predict Y field to compute the predicted Y from the regression line. Leave it blank to skip.
- 4
Review the results
The calculator displays the regression equation, slope, intercept, Pearson r, R², and the number of points. If you entered a prediction X, the predicted Y appears as a primary result.
- 5
Inspect the graph
A scatter plot shows your raw data points with the fitted regression line overlaid. Use it to visually assess model fit.
Frequently Asked Questions
Linear regression is a statistical method that models the relationship between an independent variable X and a dependent variable Y by fitting a straight line through observed data points. The line is fitted using the least squares criterion, which minimizes the sum of squared vertical distances between each point and the line. The result is an equation in the form y = mx + b, where m is the slope and b is the intercept. For example, with study hours as X and exam scores as Y, the regression line might be y = 7.2x + 47.9, meaning each additional study hour adds about 7.2 points to the predicted score.
R², the coefficient of determination, measures the proportion of variance in Y that is explained by X. A value of 0.85 means 85% of the variability in Y is accounted for by the regression model, while the remaining 15% is due to other factors or random variation. In controlled laboratory settings, R² values above 0.95 are common. In social science research, R² values between 0.30 and 0.50 are often considered meaningful because human behavior is influenced by many uncontrolled variables.
You need at least two matching X and Y values to compute a regression line. However, with only two points the line will always fit perfectly (R² = 1), which does not tell you anything about the strength of the relationship. For a meaningful regression analysis, aim for at least 10 data points. With more data, the slope and intercept estimates become more precise, and R² becomes a reliable measure of fit quality. The calculator will return an error if fewer than two valid points are provided or if the X and Y arrays have different lengths.
The slope tells you how much Y changes for each one-unit increase in X. A slope of 2 means Y increases by 2 units for every 1-unit increase in X. A negative slope indicates an inverse relationship where Y decreases as X increases. For example, if you regress fuel consumption against vehicle speed, a slope of -0.05 means each additional km/h reduces fuel economy by 0.05 km/L. The slope is in units of Y per unit X, so its practical meaning depends on the scale of both variables.
Pearson r measures the direction and strength of the linear correlation on a scale from -1 to +1. An r of +0.90 indicates a strong positive relationship, while -0.90 indicates a strong negative relationship. R² is the square of r and represents the proportion of variance explained, ranging from 0 to 1. R² removes the direction information and focuses on effect size. With r = 0.80, R² = 0.64, meaning 64% of the variance is explained. With r = -0.80, R² is still 0.64. Always report both r and R² for a complete picture.
This tool performs simple linear regression, which assumes a straight-line relationship between X and Y. If your data follows a curve, such as exponential growth or logarithmic decay, the linear regression will still compute a line, but R² will be low and the predictions will be inaccurate. For example, fitting a line to bacterial population data that doubles every hour will underpredict values at higher time points. In such cases, consider transforming the variables, such as taking the logarithm of Y, or using a dedicated non-linear regression tool.
Outliers have disproportionate influence on the regression line because the least squares method squares each residual. A point 10 units from the line contributes 100 to the sum of squared errors, while a point 1 unit away contributes only 1. This means a single extreme value can shift the slope and intercept substantially. If you accidentally enter 1000 instead of 100 for one data point, the regression line can change enough to reverse the direction of the relationship. Always review your raw data for entry errors and use the scatter plot in this calculator to spot points that fall far from the regression line before drawing conclusions.
A negative slope indicates an inverse relationship: as X increases, Y decreases. For example, in the relationship between vehicle speed and fuel efficiency, the slope beyond 60 km/h is typically negative because driving faster reduces kilometers per liter. If your regression slope is -0.08 and X is speed in km/h, each additional km/h reduces fuel economy by 0.08 km/L. Negative slopes are common in fields like pharmacology (drug dose vs response), environmental science (pollutant concentration vs distance from source), and economics (price vs demand).
Simple linear regression uses one predictor variable X to explain Y. Multiple linear regression uses two or more predictors simultaneously. For example, simple regression might model house price from square footage alone, while multiple regression would include square footage, number of bedrooms, location score, and age as predictors. This calculator handles the simple case with one X and one Y. For multiple predictors, you need dedicated statistical software like R, Python with statsmodels, SPSS, or Stata.
The residual standard error (RSE) measures the typical size of the residuals. It represents the average distance between observed Y values and the regression line and is in the same units as Y. If the RSE is 2.3 and your Y variable is test score, then about 68% of observed scores fall within +/-2.3 points of the regression line, assuming normally distributed residuals. A smaller RSE means more precise predictions. For calibration curves in analytical chemistry, the RSE directly influences the method detection limit: MDL = 3.3 x (RSE / slope).
Yes, you can use time as the X variable to model trends over time. For example, if you have monthly sales data for 24 months, you can enter months 1 through 24 as X and the sales figures as Y to estimate the trend. However, time series data often has autocorrelation, meaning observations close in time are more similar than those far apart. This violates the independence assumption of linear regression, which means p-values and confidence intervals may be too optimistic. For rigorous time series analysis, consider specialized methods like ARIMA models.
This calculator does not compute p-values directly, but the critical values table for Pearson r at alpha = 0.05 gives you a rough guide. If your observed r exceeds the critical value for your degrees of freedom (n minus 2), the correlation is statistically significant at the 0.05 level. For example, with 15 data points (df = 13), the two-tailed critical r value is approximately 0.514. If your calculated r is 0.60, you can reject the null hypothesis that there is no linear relationship. Remember that statistical significance depends on both the effect size and the sample size: a weak correlation can be significant with enough data.
Interpolation means predicting Y for an X value that falls within your observed data range. Extrapolation means predicting Y for an X value outside that range. Interpolation is generally reliable because the regression line is well-supported by nearby data points. Extrapolation is risky because the linear relationship may not hold outside the observed X range. If your X values span 10 to 30, predicting Y at X = 20 is interpolation, but predicting at X = 100 assumes the trend continues unchanged. The calculator lets you enter any prediction X, but label extrapolated predictions as uncertain.
Check four key assumptions. First, the relationship between X and Y should be roughly linear; plot the data to see if a straight line makes sense. Second, the residuals should be independent; this matters most for time-ordered data where consecutive points may be correlated. Third, the residuals should have constant spread across all X values; a fan-shaped pattern indicates heteroscedasticity. Fourth, the residuals should be approximately normally distributed. The scatter plot and regression line in this calculator help you assess linearity and spot outliers. If assumptions are violated, consider transforming variables or using a non-parametric method.
Adding more data points tends to make R² more reliable but does not automatically increase or decrease it. With very few points, R² can be misleadingly high. With two points, R² is always 1 regardless of the true relationship. As you add points, R² converges toward its true population value. Adding points near the extremes of X increases leverage and can stabilize the slope estimate. Adding points near the mean of X provides less information about the slope but improves the intercept estimate. In general, more data produces more trustworthy R² values.
Yes, unevenly spaced X values are perfectly fine for linear regression. The method does not require equal spacing between X values. What matters is that the observations are independent and the relationship is linear across the full X range. If your X values cluster in one region with only a few points at the extremes, those extreme points will have high leverage and strongly influence the slope. For reliable regression, aim for X values spread across the entire range of interest rather than clustered in one area. This gives you a more stable and generalizable model.
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