CalcaTools

Coefficient of Determination Calculator

Computes R², Pearson correlation and the regression line from paired x-y data, with interpretation.

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

functions Shows the working, not just the answer

For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

calculate Accurate to the spec

Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

school Free for classroom use

Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

tips_and_updates Pair with the spoke articles

Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

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

Formula & methodology

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

  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.

Frequently asked questions

How do I calculate the coefficient of determination?
Compute Pearson's r from the paired data and square it. For x = 1–5 and y = 2,4,5,4,5: r = 0.7746, so R² = 0.60 — 60% of y's variance is explained by x.
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?
r (−1 to 1) gives the direction and strength of the linear association; R² (0 to 1) is its square and reads as percent of variance explained. r = −0.9 and r = 0.9 both give R² = 0.81.
Can R² be negative?
Not when computed as r² from correlation. In regression software it can go negative only for models without an intercept or models worse than a horizontal line — a sign the model is misspecified.
Does a high R² mean x causes y?
No — R² measures fit, not causation. Ice-cream sales and drownings correlate strongly through summer weather; only a controlled design can establish cause.

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