If s and R are respectively the standard deviation and range of set of n values of a variable x, then prove that `R^2/(2n) le s^2`.

If standard deviation of a ravdom variable x is 2 then var(5x) is

Prove that the standard deviation calculated from two values x_(1) and x_(2) of a vriable x is equal to half their difference.

In a right -angled triangle, prove that r+2R=S.

If S_1,S_2 and S_3 be respectively the sum of n, 2n and 3n terms of a G.P., prove that S_1(S_3-S_2)=((S_2)-(S_1))^2

If S_1, S_2 and S_3 be respectively the sum of n, 2n and 3n terms of a G.P .Prove that S_1(S_3-S_2)=(S_2-S_1)^2 .

If the mean and variance of one set of values be barx_1 and s_1^2 and those of another set be barx_2 and s_2^2 respectively and each set has values 2, then prove that the variance (s^2) of the combined set of values is given by 4s^2 = 2(s_1^2 + s_2^2) + d^2 where d = (barx_1 - barx_2) .

The mean and standard deviation for two values x_1 and x_2 (x_1 < x_2) of a variable x are respectively 25 and 4. find x_1 and x_2 .

Answer the following questions (Alternatives are to be noted): If r be the correlation coefficient between two variables x and y then prove that -1 le r le1 .

If S_(1), S_(2), S_(3) be respectively the sums of n, 2n and 3n terms of a G.P., prove that, S_(1)(S_(3) - S_(2)) = (S_(2) - S_(1))^(2) .

Prove that if x_1 and x_2 two values of a variable x, so its standard deviation is (1)/(2)(x_1-x_2),(x_1gtx_2)