How this calculator works
Raw sum of squares adds squared observations without moving the origin. A corrected sum of squares instead centres observations on their own mean and measures spread. The implementation uses centred differences directly, avoiding subtraction of two large, nearly equal raw totals. Paired mode additionally returns the centred cross-product used in covariance and the sum of squared differences between supplied pairs. The latter becomes a residual sum of squares only when one list contains observations and the other the corresponding predictions.
Formula
Raw SS = sum(x_i²). Corrected SS = sum[(x_i - mean X)²]. Cross-product = sum[(x_i - mean X)(y_i - mean Y)]. Paired difference SS = sum[(x_i - y_i)²].
Worked example
For 1, 2, 3, raw SS is 1 + 4 + 9 = 14 and corrected SS is 1 + 0 + 1 = 2. For matched X = 1, 3 and Y = 2, 6, corrected totals are 2 and 8, the cross-product is 4 and squared differences total 10.
Advertisement
Frequently asked questions
Which sum belongs in sample variance?
Use corrected SS, then divide by n - 1. Raw SS generally includes a contribution from the mean.
Can X and Y have different lengths?
No. Paired calculations require a one-to-one correspondence between observations.
What units do sums of squares use?
A squared total uses the square of its observation unit. A cross-product uses X units times Y units.