Sum of squares (SS) is one of the most fundamental building blocks in statistics — it measures how much a data set's values vary from their own mean, in squared units. This calculator computes SS directly from a list of numbers, then shows how that single number feeds into variance, standard deviation, and — eventually — techniques like ANOVA and regression.

How the Sum of Squares Calculator works

The calculator first finds the mean of the entered data set, then computes each value's deviation from that mean (x − mean). Because deviations above and below the mean would otherwise cancel out when summed, each deviation is squared before adding — this is why the result is always zero or positive, and why its units are the square of the original data's units.

Summing every squared deviation produces the sum of squares, SS = Σ(x − mean)². Dividing SS by the count of values (N) gives the population variance, and taking the square root of that variance returns you to the original units as the standard deviation.

Inputs and what they mean

The only input is the data set itself — any list of numbers separated by commas, spaces, or new lines. There's no unit requirement because SS works the same way whether the values are test scores, measurements, or financial figures; just remember the result is in squared units of whatever the input represents.

A data set of a single repeated value (e.g. 5, 5, 5) will always produce SS = 0, since every deviation from the mean is zero. The more spread out the values are, the larger SS grows — and larger data sets tend to produce larger SS totals even at the same average spread, since more terms are being summed.

Limits and edge cases

Sum of squares on its own doesn't account for sample size, so it isn't directly comparable between data sets of different lengths — that's exactly why variance (SS divided by N or n − 1) exists as a size-adjusted alternative. This calculator reports population variance (dividing by N); if you're estimating a population's variance from a sample, use the dedicated Variance calculator's sample mode instead, which divides by n − 1.

SS also appears as a component inside more advanced techniques — ANOVA partitions total SS into between-group and within-group pieces (SST = SSB + SSW), and regression analysis splits it into explained and unexplained portions (SST = SSR + SSE). This calculator only computes the single-sample total SS; for group comparisons or regression diagnostics, use the ANOVA or Linear Regression calculators.