Coefficient Of Determination Calculator - R Squared & Goodness of Fit

A coefficient of determination calculator finds R squared for a simple linear regression by fitting a least-squares line through your paired observations and reporting how much of the scatter in Y that line explains. You paste two matched lists, X and Y, and the tool returns R squared, the Pearson correlation r, and the regression, residual, and total sums of squares.

• • Lab and homework checks: Students verify a hand-computed R squared against an automated result on a small dataset.
• • Model fit reporting: Researchers summarize how well a single predictor explains an outcome before writing up results.
• • Business trend review: Analysts gauge whether a straight-line trend actually tracks a metric such as weekly sales.
• • Teaching examples: Instructors show how changing a few points moves R squared and the sums of squares.

The coefficient of determination is the proportion of the dependent variable's variance that the model accounts for. It is the standard goodness-of-fit number you meet the moment a regression leaves the simple correlation stage.

Because this calculator targets the single-predictor case, its R squared equals the square of Pearson's r. That makes it a natural next step after you have explored the raw association between two variables.

Use it whenever you have fit or plan to fit a straight line and you want one comparable percentage that says how much of the movement in Y the line captures.

If you only need the strength and direction of the linear link between two variables, the correlation calculator returns Pearson r, Spearman rho, or Kendall tau with a p-value.

The coefficient of determination calculator fits the best straight line to your points by least squares, then splits the total spread of Y into the part the line explains and the part it leaves behind. R squared is the explained share.

• x̄, ȳ: Sample means of the X and Y datasets.
• Sxx, Syy: Sum of squared deviations of X and Y from their means.
• Sxy: Sum of cross-products of the deviations; the covariance numerator.
• SS_tot: Total sum of squares, Syy, the raw spread in Y.
• SS_reg: Regression sum of squares, Sxy² / Sxx, the spread the line explains.
• SS_res: Residual sum of squares, SS_tot − SS_reg, the scatter left over.

The calculator first parses X and Y into matched finite numbers and requires at least three complete pairs. It then computes the means, the deviation sums, and Pearson r.

From r it squares to get R squared, and it also builds the three sums of squares so you can see the decomposition directly rather than trusting a single rounded percentage.

If X or Y has no spread, or the lists are mismatched, R squared is undefined because there is no line to fit or no variation to explain.

According to NIST/SEMATECH e-Handbook of Statistical Methods, the sample correlation coefficient r measures the strength of the linear relationship between two variables and is the basis for R-squared in simple linear regression

Because R squared builds on the shared deviation product, the covariance calculator helps you inspect the numerator that Pearson r and R squared both rely on.

Four ideas sit behind every result from a coefficient of determination calculator. Knowing them keeps you from over-reading a single percentage.

These concepts turn R squared from a mystery percentage into a measurable split between what the model captures and what it does not.

Keep the df in mind: with very few points even a modest R squared can come from an unstable line.

Since R squared compares explained variation to total variation, the variance calculator explains the spread that SS_total measures.

Two pasted lists are all you need. Follow the steps and then read the result against your question.

1. 1 Enter Dataset X: Paste the independent values, one per observation, separated by commas or spaces.
2. 2 Enter Dataset Y: Paste the dependent values in the same order so the first Y lines up with the first X.
3. 3 Review R squared: Read the coefficient of determination as both a decimal and a percent to see the explained share.
4. 4 Inspect the sums of squares: Check SS_reg, SS_res, and SS_tot to see how the total variation splits.
5. 5 Note sample size and df: Confirm n and the residual degrees of freedom before trusting the fit.
6. 6 Interpret with context: Compare the value to what is normal in your field rather than to a fixed rule.

A teacher records study hours (1,2,3,4,5) and quiz scores (2,3,5,4,6). Pasting them returns R squared = 0.81, meaning study time explains 81% of the score spread for this small class.

After you measure fit with R squared, the linear regression calculator fits the slope and intercept of the same least-squares line.

A dedicated R squared tool saves the arithmetic and surfaces the pieces a single percentage hides.

• • Full decomposition: You see SS_reg, SS_res, and SS_tot, not just a final rounded number.
• • Fewer hand errors: Squaring deviations and cross-products by hand is where students slip; the tool removes that risk.
• • Clear undefined handling: Zero-variance or mismatched inputs return an explicit undefined instead of a misleading value.
• • Teaching transparency: Changing a point updates every sum, which helps learners see what moves the fit.
• • Report-ready output: Decimal, percent, and df give you the figures most write-ups ask for.

The benefit is less about speed and more about seeing the whole variance split at once.

That view is what separates a real fit assessment from a number pasted into a table.

When you want to compare group means instead of a fitted line, the ANOVA calculator partitions variance the same way R squared does.

R squared moves with your data and with how you set up the model. Watch these before drawing conclusions.

• • R squared says nothing about causation; it only measures how well a line fits the sample you entered.
• • Adding predictors to a model generally raises R squared even when the new terms are meaningless, so a high value is not proof of a good model.

Treat R squared as a description of fit, not as a verdict on the underlying relationship.

Pair it with a look at residuals and the subject-matter reasoning before you call a model useful.

According to Wikipedia: Coefficient of determination, R2 is the proportion of the variance in the dependent variable that is predictable from the independent variable(s)

According to LibreTexts Introductory Statistics, the least-squares regression line minimizes the sum of squared residuals, which is the SS_res that the coefficient of determination compares against the total spread

To summarize means, spread, and shape before modeling, the statistics calculator gives the descriptive stats that feed a regression.