Question

Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1 is added to each number, the variance of the numbers so obtained is

Answer 1

The variance of the number is 8.25

• The given numbers are 1,2,3,4,5,6,7,8,8,10
• If 1 is added to the numbers it will become as 2,3,4,5,6,7,8,9,10,11

•    we know that    $s^{2} =$∑$\frac{x_{I} ^{2} }{n}$ -(∑$\frac{x_{i} }{n}) ^{2}$

           where s ^2 is the variance

•         ∑$x_{i} ^{2}=2^{2}+3^{2}+4^{2}+. . . . +11^{2}$

•        ∑$x_{i} ^{2}=\frac{n(n+1)(2n+1)}{6}-1^{2}$    

                       = 11 (12)(23)/6  - 1

                      =  506-1  =505

•        ∑$\frac{x_{i} ^{2} }{n}$ = $\frac{505}{10}$=50.5

             ∑$x_{i}=2+3+4+. . . .+11$

                   = 10(2(2)+(9)(1)/2

                  = 65

•      ∑$\frac{x_{i} }{n}$ = 65/10 = 6.5

•          now

                     

                    $s^{2} =$∑$\frac{x_{I} ^{2} }{n}$ -(∑$\frac{x_{i} }{n}) ^{2}$

                         = 50.5 - (6.5)(6.5)

                         = 50.5-42.25

                         =8.25