quadratsummen

Quadratsummen is the German term for sums of squares, a foundational concept in statistics and in algebra. It refers to the sum of squared deviations or magnitudes, used to quantify dispersion, variation, or the quality of a model’s fit. In its standard form for a data set x1, x2, ..., xn, the total sum of squares relative to the sample mean is SST = sum over i of (x_i - x̄)^2. If data are not centered, the quantity sum of x_i^2 can be used, and it relates to SST through SST = sum of x_i^2 - n x̄^2.

In regression analysis, the total sum of squares is decomposed into components that reflect different sources of variation: the explained sum of squares SSR = sum over i of (ŷ_i - ȳ)^2, and the residual (or error) sum of squares SSE = sum over i of (y_i - ŷ_i)^2, with SST = SSR + SSE. These quantities are nonnegative and provide a measure of how well a model captures the observed variation. The sample variance s^2 can be obtained as S / (n - 1), where S is a sum of squares such as SST or SSE, depending on the context.

Quadratsummen are widely used in analysis of variance (ANOVA), goodness-of-fit testing, regression diagnostics, and other statistical procedures to assess dispersion, model adequacy, and the contribution of different factors to total variability. In addition to statistical contexts, the term may also appear in broader algebraic settings as sums of squared numbers relative to a reference value, typically a mean.

Overall, quadratsummen provide a compact, additive way to quantify squared deviations and are central to estimating variability and model performance.