Let r be the range and S2=1n−1∑i=1n xi−x¯2 be the SD  of a set of observations x1,x2,…,xn, then

We have r=maxxi−xj and S2=1n−1∑i=1n xi−x¯2 Now, xi−x¯2=xi−x1+x2+⋯+xnn2=1n2xi−x1+xi−x2+⋯+xi−xi−1+xi−xi+1+⋯+xi−xn≤1n2[(n−1)r]2          [∵]xi−xj∣≤r⇒xi−x¯2≤r2⇒∑i=1n xi-x¯2≤nr2⇒1n−1∑i=1n xi−x¯2≤nr2(n−1)⇒S2≤nr2(n−1)⇒S≤rnn−1