Coefficient of Determination Calculator

What the coefficient of determination measures

The coefficient of determination, written R squared or R2, answers one question: of all the up-and-down variation in the y values, how much does the fitted line account for? A value of 0.80 says 80 percent of the variance sits on the line and 20 percent scatters around it. A value of 0.00 says the line explains none of it, no better than guessing the average y for every point.

Statisticians frame it as a comparison of two errors. Predict every response with the mean of y and you rack up the total sum of squares, SStotal. Predict with the regression line instead and the leftover error shrinks to SSresidual. R squared is the share of that error the line removed: R2 = 1 - SSresidual / SStotal.

It is a goodness of fit number for least squares regression, not a correlation and not a proof that x causes y. The NIST/SEMATECH e-Handbook section on least squares presents it in exactly that role, as a way to judge a fitted line.

An error framing helps some readers. Suppose you had to guess every student's score with no information at all; the best available guess is the class mean, and your typical miss is measured by SStotal. Handing you the study hours and the fitted line cuts that miss down to SSresidual. R squared reports the size of the cut, so it can be read as how much the one variable bought you, from nothing at 0 to everything at 1.

Two formulas that give the same number

For a simple regression with one x variable there are two routes to the same value. The variance route uses the sums of squares. The correlation route squares the Pearson correlation r. Both give 0.9925 for the worked example below.

With more than one predictor the two routes part ways, and the variance definition is the one to use. Multiple R squared never drops when you add a predictor, which is why the adjusted version exists; this page sticks to the one-variable case, where the number is unambiguous.

Either route needs the raw pairs. With only a published r and no data, squaring it gives the one-predictor R squared on the spot, but SStotal and SSresidual cannot be recovered without the points themselves, so the two formulas are complements rather than substitutes: the sums of squares come from the data, and the squared correlation is a shortcut that works only when a single x is in the model.

A full worked example, six students

Six students log 1, 2, 3, 4, 5 and 6 hours of study and score 52, 55, 61, 64, 70 and 74 points on a test. The means are 3.5 hours and 62.67 points. Least squares gives the line score = 46.87 + 4.5143 x hours, where the slope comes from 79 / 17.5 = 4.5143 and the intercept from 62.67 - 4.5143 x 3.5 = 46.87.

Squaring each residual and adding the six squares gives SSresidual = 2.70. The total sum of squares around the mean of 62.67 comes to SStotal = 359.33. The coefficient of determination is 1 - 2.70 / 359.33 = 0.9925: study time accounts for 99.25 percent of the variance in scores, and 0.75 percent is left to sleep, luck and everything else. Residuals are shown rounded to two decimals, so their squares do not recompute exactly from the rounded column.

Reading an R squared value

A coefficient of determination is a share, running from 0 to 1 in the one-predictor case, and it carries no units. What counts as strong depends entirely on the setting, which is why the number should travel with context rather than a fixed grading scale.

The table shows how misleading a correlation can be on its own. A correlation of 0.5 sounds middling to decent, yet the line explains only 0.5 x 0.5 = 0.25 of the variance, one quarter. In lab calibration work an R squared above 0.99 is routine; in studies of human behavior 0.25 can be a real finding. Compare against the norm for your kind of data, not in the abstract.

Field context cuts both ways. A 0.49 from a 0.7 correlation is excellent for guessing whether rain is coming from a rising barometer but weak for predicting a metal part's length from its temperature in a controlled shop, and the same number printed next to a different dataset deserves a different verdict. The statistic is one input to that judgment, not the judgment itself.

R squared against the correlation r

For one predictor, R squared is exactly r squared, so the two carry the same information about fit. The differences are still worth knowing. The correlation keeps its sign: r = -0.9 for a strong downward trend and r = 0.9 upward both give R squared = 0.81, so the coefficient of determination alone cannot tell you which way the line slopes. The correlation is also the one with a direct visual reading, while the squared version is the one that says how much variance moved onto the line.

The student data checks out both ways: r = 0.99623, and 0.99623 x 0.99623 = 0.9925, matching the variance route to four decimals.

What R squared does not tell you

Three blind spots come up again and again. First, fit is not causation. Ice cream sales and drowning counts track each other through summer heat, and a high R squared would happily sit on top of that pairing without a causal link between the two.

Second, the number says nothing about the shape of the trend. A curved relationship scored against a straight line can still post a respectable R squared, which is why a picture belongs next to any regression; the scatter plot and the Boston University statistics module on correlation and regression both make that case with examples.

Third, one outlier can move the value a long way on its own. A single wild point at the edge of the x range can drag the slope and either inflate or sink R squared, so read the residual column before trusting the headline number. And when predictors are added one after another, R squared can only stay flat or climb, never fall, even if the new variable is noise; that gap is what the adjusted R squared fills in multiple regression.

Reporting it with the regression line

On its own, R squared says how much variance sits on the line, which is why it is normally reported beside the line itself. The student example reads naturally as a sentence: scores rise 4.51 points per study hour on the fitted line, and an R squared of 0.9925 says the line tracks nearly all of the variation across the six students. The slope and intercept give the predictions, R squared gives the trust you can put in them, and the residual column shows where that trust runs out.

Being unitless, the statistic compares across models that predict different things. A 0.6 from speed against braking distance and a 0.6 from salary against years of experience mean the same thing, a 60 percent variance share, even though one predicts meters per second and the other dollars. For the spread of a single variable with no predictor at all, the variance calculator and the standard deviation calculator handle the SStotal starting point, and the statistics calculator covers the summary stats around it.

A short reporting template keeps the trio straight: state the line, then R squared, then the residual scale. Written out, the example becomes score = 46.87 + 4.5143 x hours, R squared = 0.9925, with residuals of a point or less across the six students. Anyone reading that sentence gets the prediction rule, the variance share and the typical miss in one pass.

From R squared to prediction quality

R squared describes the line, but predictions carry their own yardstick, the standard error of the estimate. It comes from the same leftover sum: take SSresidual, divide by the degrees of freedom, n - 2 for a straight line, and take the square root. For the student example, 2.70 / 4 = 0.675, and the square root of 0.675 is about 0.82, so individual scores typically land within roughly 0.82 points of the line's prediction.

The two numbers answer different questions and work best as a pair. R squared says how much of the pattern the hours explain; the standard error says how far off a single prediction will usually be, in the units of y. A model can post a strong R squared and still be of little use for individual calls if the y values span a huge range, and a modest R squared can support workable predictions when the leftover spread is small compared with what the decision needs.