When finding the variance where did the '2' go in the second terrm of the third step and why do - Maths - Statistics

Question

When finding the variance where did the '2' go in the second
terrm of the third step and why do we mulitply another 'n' to last
term how do we get it? this is the answer to exercise 15 2
question2 provided by meritnation

Shridhar Kaushik · Accepted Answer

The said term is $\frac{1}{n} Σ_{i = 1}^{n} (\frac{n + 1}{2})^{2} .$

Since $(\frac{n + 1}{2})^{2}$ does not have i, so it can be taken out of the summation as summation is taken over i = 1, 2, ..., n.

So,

$\frac{1}{n} Σ_{i = 1}^{n} (\frac{n + 1}{2})^{2} = \frac{1}{n} (\frac{n + 1}{2})^{2} Σ_{i = 1}^{n} 1 = \frac{1}{n} (\frac{n + 1}{2})^{2} [1 + 1 + 1 + ... + 1 (n times)] = \frac{1}{n} (\frac{n + 1}{2})^{2} \times n [∵ 1 + 1 + ... + 1 n times will be n] = \frac{1}{n} \frac{(n + 1)^{2}}{4} \times n = \frac{(n + 1)^{2}}{4}$

Hope you will get this step now!

When finding the variance where did the '2' go in the second terrm of the third step and why do we mulitply another 'n' to last term how do we get it? this is the answer to exercise 15.2 question2 provided by meritnation....