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Coefficient Of Determination Calculator | Report R² & Regression

Measure how well a line explains your data with R-squared, the Pearson correlation and the regression equation, computed from paired x-y values. See the relationship strength and the fitted line's formula — the standard regression summary for coursework and data analysis.

Last updated: June 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Quick reference

StatisticMeaning
rDirection & strength of linear association (−1 to 1)
R²Share of y-variance explained by x (0 to 1)
SlopeChange in y per unit of x
InterceptPredicted y when x = 0

info Coefficient of Determination Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

info Private by design

Everything runs locally in your browser. No uploads, no accounts, no tracking.

info Works everywhere

Fully responsive and mobile-friendly — calculate on any device, any time.

info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

R²Interpretation (rule of thumb)
0.81–1.00Very strong fit — x explains most of y
0.49–0.80Strong fit
0.25–0.48Moderate fit
Below 0.25Weak — x alone predicts y poorly

lightbulb Worked example

Let's say you are using the Coefficient of Determination Calculator. Computes R², Pearson correlation and the regression line from paired x-y data, with interpretation. Enter the values that match your situation into the input fields and press calculate — using realistic numbers makes the result directly useful for you.

Result: The calculator instantly applies the formula R² = r² where r = Σ(x−x̄)(y−ȳ) ÷ √[Σ(x−x̄)² Σ(y−ȳ)²] and returns the result with appropriate precision.

What this means: Read the result in the context of what you are measuring. The step-by-step breakdown lets you confirm the math and understand which input most affects the outcome.

Formula & methodology

Formula: R² = r² where r = Σ(x−x̄)(y−ȳ) ÷ √[Σ(x−x̄)² Σ(y−ȳ)²]

The Coefficient of Determination Calculator is built on a well-established calculation method. It uses the formula R² = r² where r = Σ(x−x̄)(y−ȳ) ÷ √[Σ(x−x̄)² Σ(y−ȳ)²] to turn your inputs into a reliable result. Computes R², Pearson correlation and the regression line from paired x-y data, with interpretation. The steps are shown on the page so you can follow the reasoning from input to output.

  1. Compute the means of x and y.
  2. Sum the cross-products Σ(x−x̄)(y−ȳ) and the squared deviations of each variable.
  3. r is the cross-product sum over the geometric mean of the two deviation sums; squaring gives R².

Worked example: x = 1…5, y = 2, 4, 5, 4, 5 → r = 6 ÷ √(10 × 6) = 0.7746, so R² = 0.60: the regression line ŷ = 0.6x + 2.2 explains 60% of the variation in y.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do I calculate the coefficient of determination?
Enter the required values into the input fields and press the calculate button. The tool applies the formula R² = r² where r = Σ(x−x̄)(y−ȳ) ÷ √[Σ(x−x̄)² Σ(y−ȳ)²] and returns the result with a short explanation of each step, so you can check the working yourself.
What does an R² of 0.6 mean?
60% of the variation in y is accounted for by the linear relationship with x; the remaining 40% is due to other factors or noise. By rule of thumb that's a strong fit for social data, modest for physical data.
What is the difference between r and R²?
The Coefficient of Determination Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Computes R², Pearson correlation and the regression line from paired x-y data, with interpretation. There is no limit on usage, and it works on any device.
Can R² be negative?
Yes, in one common situation: when the linear regression model fits the data worse than a horizontal line at the mean. The usual formula R² = r² is always between 0 and 1, but the adjusted or out-of-sample R² can go negative. A negative value signals that the straight-line model is a poor fit and a different form may be needed.
Does a high R² mean x causes y?
The Coefficient of Determination Calculator is a free online tool that computes r², pearson correlation and the regression line from paired x-y data, with interpretation. You enter a few values, and it returns an accurate result together with a step-by-step explanation, so it works as both a calculator and a quick reference for the underlying concept.
What does the Coefficient of Determination Calculator do?
Computes R², Pearson correlation and the regression line from paired x-y data, with interpretation. The calculator takes your inputs, applies the standard calculation, and returns a clear result so you can make an informed decision without doing the math by hand.
What formula does the Coefficient of Determination Calculator use?
The Coefficient of Determination Calculator applies the standard formula: R² = r² where r = Σ(x−x̄)(y−ȳ) ÷ √[Σ(x−x̄)² Σ(y−ȳ)²]. The tool walks through each step of the calculation so you can verify the numbers yourself.
Is the Coefficient of Determination Calculator free and private?
Yes — the Coefficient of Determination Calculator. There is no sign-up, no paywall, no trial, and no limit on how many calculations you can run. CalcaTools covers its costs with non-intrusive display advertising, so the calculator itself never asks for payment or restricts any feature.
How accurate is the Coefficient of Determination Calculator?
The Coefficient of Determination Calculator applies the standard formula: R² = r² where r = Σ(x−x̄)(y−ȳ) ÷ √[Σ(x−x̄)² Σ(y−ȳ)²]. It is tested against published worked examples before launch and uses native double-precision arithmetic, so results are accurate to typical precision for the values you enter — with no intermediate rounding that could distort the answer.

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