Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 9 / a / Solution

Use the corrected decompositionThe hint's minus sign before $({\mu }_j-w_j)$ is a typographical error. Squaring, summing, and using ${\sum }_i(y_{ij}-{\mu }_j)=0$ for each coordinate gives the centroid sum-of-squares identitySince $m>0$ and $w\ne \mu$, the final term is strictly positive. HenceThus the centroid uniquely minimizes the sum of squared Euclidean distances to the observations.