What Is Square Root Regression?
Square root regression fits a curve that rises quickly at first and then slows. The model is y = a + b√x, and it works by replacing each x with its square root and fitting an ordinary straight line to the result.
Paste x values that are 0 or more and the matching y values. The calculator returns the equation, the coefficients a and b, R², RMSE and an optional prediction. It also shows the R² of a plain straight line on the same data so you can see if the curve really helps.
Use the model when growth shows diminishing returns but does not level off. If your data reach a ceiling, an exponential or logistic model will fit better.
How to Use the Square Root Regression Calculator
- Paste x values that are 0 or more and the matching y values.
- Optionally enter an x to predict y.
- Compare R² with the straight-line R² shown in the results.
Square Root Regression Formula
Model
Slope on √x
Intercept
Square Root Regression Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| a | Intercept of the model | |
| b | Coefficient of √x | |
| √x | Square root of x, the transformed predictor | x must be 0 or more |
| R² | Share of the variation in y explained | |
| RMSE | Root mean squared error of the fit |
Square Root Regression Worked Example
With the example values loaded in the calculator above, the steps are:
- Transform each x to √x.
- Fit a straight line of y on √x by least squares: slope b = Sxy ÷ Sxx = 2.10866, intercept a = ȳ − b·mean(√x) = 0.616408.
- The model is y = a + b√x = 0.616408 + 2.10866·√x.
- R² = 0.997733, so the model explains 99.77% of the variation in y.
Result: Square root regression equation = y = 0.616408 + 2.10866·√x; b (coefficient of √x) = 2.10866; R² = 0.997733.
When a square root model makes sense
Use it when y grows with x but with diminishing returns, and the growth does not reach a ceiling. Examples include distance versus time for some motion, or the response of a sensor to input.
If your data level off to a maximum, a logistic or exponential approach model fits better. Check the residual plot and the straight-line comparison.
How the transformation works
The model is not a straight line in x, but it is a straight line in √x. So the calculator takes √x for every point and runs simple linear regression on those values. The R² reported is the R² of that straight-line fit.
Where Square Root Regression Is Used
Diminishing-returns data
Fit a response that rises quickly then slows, such as output against effort.
Physics and engineering
Model relationships that follow a square root law.
Model comparison
Test whether √x fits better than a straight line before choosing a model.
Common Square Root Regression Mistakes to Avoid
- Check that x is never negative.
- A high R² on a few points does not prove the shape. Look at the plot.
Square Root Regression Terms and Definitions
- Transformation
- Replacing a variable with a function of it, such as √x, so a curve becomes a line.
Square Root Regression FAQs
What is square root regression?
It fits y = a + b√x to data by least squares, giving a curve that rises quickly and then slows.
Can x be negative?
No. The square root of a negative number is not real, so x must be 0 or more.
How do I know the square root model is better than a line?
Compare R² and look at the plot. The calculator shows the straight-line R² next to the square root R².
Is the fit linear regression?
It is linear regression on √x, so the usual least squares formulas apply.
How do I predict y?
Compute a + b times the square root of your new x. Enter it in the optional box to have it done for you.
What if my data level off?
Try an exponential or logistic model, since a square root curve keeps rising.