Suppose three tests are administered to a random sample of college students. Let X₁,..., X_N be observation vectors in R³ that list the three scores of each student, and for j = 1, 2, 3, let x_j denote a student's score on the jth exam. Suppose the covariance matrix of the data is Let y be an "index" of student performance, with y = cx₁x₁ + c₂x₂ + c₃x₃ and c₁² + c₂² + c₃² = 1. Choose c₁, c₂, c₃ so that the variance of y over the data set is as large as possible. The eigenvalues of the sample covariance matrix are λ = 3, 6, and 9.

S = 5 2 0 2 6 2 0 2 7