If each of the observation `x_(1), x_(2), ...,x_(n)` is increased by 'a', where a is a negative or positive number, show that the variance remains unchanged.

The geometric mean of numbers observations x_1, x_2, x_3,….,x_n is

Mean of n observations x_(1),x_(2),.......,x_(n) is bar(x) . If an observation x_(q) ' then the new mean is

The average of n numbers x_(1),x_(2),x_(3),..,x_(n) is M. If x_(n) is replaced by x', then new average is

Let x_1,x_2,x_3,…x_n be n values of a variable x. If these values are changed to x_1+a, x_2+a,…x_n+a , where a∈ R, show that the variance remains unchanged.

If the mean of the numbers x_1, x_2, x_3,….x_n is overlinex , then the mean of the numbers x_i+2 , is , where 1 le i le n

Prove that the function f(x)= x^(n) is continuous at x= n, where n is a positive integer.

Given that bar(x) is the mean and sigma^(2) is the variance of n observation x_(1), x_(2), …x_(n). Prove that the mean and sigma^(2) is the variance of n observations ax_(1),ax_(2), ax_(3),….ax_(n) are abar(x) and a^(2)sigma^(2) , respectively, (ane0) .

If f(x)=(a-x^n)^(1/n)," where "a gt0 and n is a positive integer, show that f[f(x)]=x.

For (2n+1) observations x_(1), -x_(1),x_(2),-x_(2),..,x_(n),-x_(n) and 0, where all x's are distinct, let SD and MD denote the standard deviation and median, respectively. Then which of the following is always true ?

If each observation of a raw data whose variance is sigma^2 is increased by lambda then the variance of the new set is