How this instrument works
The sum of squares (SS) — more precisely the sum of squared deviations from the mean — measures total variability in a data set before it gets divided down into variance or standard deviation. For every value, subtract the mean, square the result (so negative and positive deviations don't cancel out), and add all those squares together. A tight, clustered data set produces a small SS; a widely scattered one produces a large SS, and the units are whatever your original units were, squared.
SS matters mainly as a building block, not a final answer on its own: divide it by n for the population variance, or by n-1 for the sample variance, and take the square root of either to get a standard deviation. It's also the quantity decomposed in analysis of variance (ANOVA), where a total sum of squares gets split into a part explained by group differences and a part left over as unexplained error — the same underlying arithmetic idea scaled up to comparing several groups at once.
There's a second, algebraically equivalent way to compute it that avoids finding the mean first: SS = Σx² - (Σx)²/n, summing the squares of the raw values and subtracting a correction term. Both routes give the identical answer for the same data — the first is easier to reason about conceptually, the second is what many calculators and older textbooks use because it needs only one pass through the data instead of two.
- Paste or type your numbers into the Data set field — commas, spaces or new lines all work.
- Read Sum of squared deviations (SS) — the total of every value's squared distance from the mean.
- Use this figure directly if you're building your own variance or ANOVA calculation by hand and need the intermediate SS term.
- Enter at least two values; SS is defined but uninformative for a single point, since there's nothing to deviate from besides itself.
Worked example — the eight quiz scores
Take {2, 4, 4, 4, 5, 5, 7, 9}, n = 8. The mean is (2+4+4+4+5+5+7+9)/8 = 40/8 = 5. The deviations from the mean are -3, -1, -1, -1, 0, 0, 2, 4; squaring each gives 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32.
So this instrument reports SS = 32.0000. Checking with the algebraic shortcut: Σx² = 4+16+16+16+25+25+49+81 = 232, and (Σx)²/n = 40²/8 = 1600/8 = 200, so SS = 232 - 200 = 32 — the same answer, confirming both routes agree. From here, sample variance would be 32/7 ≈ 4.57 and population variance 32/8 = 4.00.
Questions
What's the difference between sum of squares and variance?
Sum of squares (SS) is the running total of every squared deviation from the mean — a raw, un-averaged figure. Variance divides that total by the number of values (population variance, dividing by n) or by one less than that (sample variance, dividing by n-1), turning the total into a per-value average. SS is the numerator; variance is SS scaled down.
Why square the deviations instead of just adding them up?
Because the deviations from the mean always sum to exactly zero on their own — positive and negative differences cancel out perfectly by the definition of the mean. Squaring makes every deviation positive before summing, so spread above and below the mean both contribute to the total instead of cancelling.
What are the units of the sum of squares?
Whatever your original data's units are, squared — if you entered measurements in centimetres, SS is in square centimetres, because every deviation was squared. This is exactly why standard deviation (the square root of variance) is more directly comparable to the original data than SS or variance are.
Does this sum of squares relate to ANOVA?
Yes — the sum of squares computed here is conceptually the same quantity ANOVA calls the 'total sum of squares' when applied to one combined group. ANOVA extends the idea by splitting that total into a portion explained by differences between groups and a portion left over as unexplained variation within groups, but the underlying squared-deviation arithmetic is identical.
Why does the instrument offer two equivalent formulas?
Both SS = Σ(x - x̄)² and SS = Σx² - (Σx)²/n compute the exact same number for the same data — they're algebraically equivalent, not two different statistics. The first is easier to understand conceptually since it shows each deviation directly; the second needs only running totals of x and x², which is why many hand calculators and older references use it.