Sum of Squares Calculator

Two quantities, one name

Search for a sum of squares calculator and you will find tools that answer Σx² and tools that answer Σ(x − x̄)², mostly without saying which.

Σx² is the literal reading. Square each value, add them up. It is what the phrase means in a sum-of-squares identity from algebra, in a least-squares normal equation, and in the denominator of a correlation formula written out longhand.

Σ(x − x̄)² is the statistical one. Subtract the mean first. This is the quantity meant whenever the context is variance, standard deviation, regression or analysis of variance — and it is what an ANOVA table’s “SS” column holds. Its other names are the corrected sum of squares and the total sum of squares.

They differ by exactly n·x̄², which is called the correction for the mean. On data centred at zero the two coincide exactly — the second preset has a mean of precisely zero, and both sums come to 228.

On data far from zero they are not close. Ten test scores in the sixties to nineties give a raw sum of 66,310 and a corrected sum of 700: a factor of 94.7. If a tool hands you one of those and you needed the other, nothing about the number will look wrong.

So this tool prints both, always, with the gap between them. The disambiguation is the useful part, and it costs nothing.

SST = SSR + SSE needs an intercept

The decomposition appears in every regression chapter as though it were arithmetic. It is a consequence, and it has a condition.

The algebra leaves a cross-product. Writing y − ȳ as (ŷ − ȳ) + (y − ŷ) and squaring gives SSR + SSE plus 2Σ(ŷ − ȳ)(y − ŷ). The identity holds only if that last term is zero.

It is zero because of the intercept. Least squares with an intercept produces a normal equation that says exactly Σ(y − ŷ) = 0 — the residuals sum to zero — and a second saying the residuals are uncorrelated with x. Together those kill the cross-product.

Force the line through the origin and both equations are gone. On the first preset the residuals then sum to 1.835 rather than zero, and SSR + SSE comes to 202.75 against an SST of 165.86 — out by 36.89. Every individual term is still computable and still looks plausible.

The same happens with weighted least squares and with some robust fits, for the same reason: whatever removes the plain intercept condition removes the identity with it.

Even when the origin is the right constraint, the identity still fails. The third preset is data that genuinely passes through zero, and fitting it without an intercept is entirely reasonable — the gap is 0.65 rather than 36.89, but it is not zero. A small failure is still a failure, and it is the reason software switches to a different R² definition for these models rather than reporting the usual one.

R² below zero, and the other R² software quietly substitutes

Once the identity is gone, the two usual formulas for R² stop agreeing, and one of them stops being bounded.

1 − SSE/SST has no lower bound. It compares your model against the flat line at ȳ. A model forced through an origin the data is nowhere near can leave far more error than that flat line would, and the ratio goes past 1.

The second preset makes it vivid. Five points between 105 and 125, forced through the origin: R² comes out at −36.80. With an intercept the same data scores 0.992149. The fit is not slightly worse; it is worse than knowing nothing but the mean.

SSR/SST is the other formula, and it is not the same number here. While the identity holds the two are identical. Without it they diverge, and SSR/SST can exceed 1 — on that same preset SSR is 12,066 against an SST of 242.

Which is why statistical software silently changes the definition. For a no-intercept model, R, SAS and Excel all report an uncentred R², computed as 1 − SSE/Σy² rather than 1 − SSE/SST. That measures against zero rather than against the mean, it is nearly always flattering, and it is not comparable with an ordinary R² from a model that has an intercept.

The practical rule is short. Do not compare an R² across models with and without an intercept, and be suspicious of a no-intercept R² above 0.99 — on data far from zero that is frequently what the uncentred formula returns for a poor fit. The R² calculator covers the other ways the statistic goes negative.

The computational formula is a numerical trap

Textbooks give Σ(x − x̄)² = Σx² − n·x̄² as a shortcut, and it is exactly correct in real arithmetic. In floating point it is a way to lose most of your precision.

It subtracts two nearly equal large numbers. On the test-scores preset it is 66,310 minus 65,610 to get 700. Every leading digit the two share is a digit of the answer that cancels away, and the relative error in the result is amplified by roughly the ratio of the operands to the difference — a factor of 94.7 here.

On realistic data it can leave nothing. Measurements around 100,000 with a spread of 0.1 give operands agreeing to ten significant figures; a double carries about sixteen, so the answer arrives with six. A variance computed that way can come out negative, which is the classic symptom.

This tool computes the deviations directly. One pass for the mean, a second for the squared deviations. It costs an extra traversal and it never cancels, which is the same choice the variance calculator makes.

Welford’s method is the single-pass alternative, updating the mean and the corrected sum together as values arrive. It is as stable as the two-pass version and is what you want when the data cannot be held in memory or arrives as a stream.

The shortcut is not wrong to teach. It makes the relationship between the two sums visible, which is genuinely useful — it is the same n·x̄² this page opens with. It is wrong to implement, and the distinction between an identity and an algorithm is worth keeping.

The same split, one level up

An ANOVA table is the identical decomposition applied to groups rather than to a fitted line, which is why the arithmetic looks familiar.

Total splits into between and within. SS_total= Σ(y − ȳ)² over every observation; SS_between= Σnᵢ(ȳᵢ − ȳ)² over the group means; SS_within is what is left. Between plays the role of SSR and within the role of SSE.

And it holds for the same reason. Each group mean is the least-squares fit within that group, so the deviations from it sum to zero and the cross-product vanishes — the group means are playing the part the intercept played.

Dividing by degrees of freedom turns sums into mean squares, and their ratio is the F statistic. That is the whole of a one-way ANOVA: a sum-of-squares split, divided by its degrees of freedom, compared as a ratio.

Eta-squared is SS_between/SS_total, which is R² under another name — the share of the total sum of squares the grouping accounts for. The two literatures use different words for the same quantity, which is worth knowing when reading across them.

The split generalises further. Two-way designs add an interaction term, repeated measures split the within part again, and every one of those tables is the same cross-product argument applied recursively. Getting the one-variable case exact is what makes the rest readable.

Why the divisor is n − 1, and then n − 2

A sum of squares becomes a variance by dividing by degrees of freedom, and the divisor changes depending on what was estimated to produce it.

Σ(x − x̄)² has n − 1 degrees of freedom. The deviations sum to zero by construction, so once n − 1 of them are known the last is determined. There are n numbers and one constraint.

That is the whole of Bessel’s correction. Dividing by n gives the population variance, which is exactly right if x̄ is the true mean and biased downward otherwise — because Σ(x − x̄)² is the smallest such sum over any centre, and the true mean is not the sample mean.

A regression residual sum has n − 2. Two parameters were estimated, so two constraints were imposed: the residuals sum to zero and are uncorrelated with x. The residual standard error divides by n − 2 for the same reason variance divides by n − 1.

A quadratic costs one more. Three parameters, so n − 3, and a k-predictor regression has n − k − 1. That is the counting behind adjusted R², which charges for each parameter and can therefore fall when one is added.

With n = 2 the residual sum is exactly zero and has zero degrees of freedom. A line through two points fits perfectly, R² is 1, and the residual standard error is 0/0. That is not a good fit; it is a model with nothing left to be wrong about.

Which is the practical warning. Degrees of freedom are how much information remained after the fitting, and a model with few of them will report a flattering sum of squares for reasons that have nothing to do with the data. This tool prints n and the divisor alongside every figure so that count stays visible.

Sources and methodology

References for the decomposition and for computing it without losing precision.

• NIST/SEMATECH e-Handbook — the ANOVA sum-of-squares split and its degrees of freedom
• Welford, “Note on a Method for Calculating Corrected Sums of Squares and Products”, Technometrics 4(3), 1962
• Chan, Golub & LeVeque, “Algorithms for Computing the Sample Variance”, The American Statistician 37(3), 1983 — what the textbook shortcut costs
• Variance Calculator — the corrected sum divided by its degrees of freedom
• R² Calculator — the ratio this decomposition is built to produce, and how it goes negative
• How we calculate — methodology

Method. The corrected sum is computed from the deviations in a second pass rather than as Σx² − n·x̄², because the shortcut subtracts two nearly equal numbers and loses a digit of precision for every digit they share. The regression decomposition is computed term by term with SST − SSR − SSE reported rather than assumed to be zero, so the identity can be watched failing when the intercept is removed. The suite asserts that the identity holds on every one of 500 random intercept fits and on none of the corresponding through-the-origin fits, that raw minus corrected equals n·x̄² exactly, and pins the presets: a gap of 36.889412 and residuals summing to 1.835294 without an intercept, and an R² of −36.799324 on data offset from zero. That engine is verified on every change against 188 assertions. The count and the per-case breakdown are published on the formula verification page.

Related calculators

Where this goes next:

More in Math, or browse all calculators.