21. Let {x1, x2, . . . , xn} be a set of real numbers and define x¯ = 1 n .n i=1 xi, s2 = 1 n − 1 .n i=1 (xi − x)¯ 2 . Prove that at least a fraction 1 − 1/k2 of the xi’s are between x¯ − ks and x¯ + ks. Sketch of a Proof: Let N be the number of x1, x2, . . . , xn that fall in A = [ ¯x − ks, x¯ + ks]. Then s2 = 1 n − 1 .n i=1 (xi − x)¯ 2 ≥ 1 n − 1 . xi,∈A (xi − x)¯ 2 ≥ 1 n − 1 . xi,∈A k2 s2 = n − N n − 1 k2 s2 . This gives (N − 1)/(n − 1) ≥ 1 − (1/k2). The result follows since N/n ≥ (N − 1)/(n − 1).