Question

The standard deviation of 20 observations is v6. If each observation is multiplied by3, find
the standard deviation and variance of the resulting observations.​

Answer 1

Answer:

Standard Deviation = 18

Variance = 324 .

Step-by-step explanation:

We are given that standard deviation of 20 observations is 6. If each observation is multiplied by 3 then standard deviation also get multiplied by 3.

To prove this let us consider a simple example;

Suppose;

              X                         X - $Xbar$                $(X - Xbar)^{2}$

              3                          3 - 6 = -3                      9

              5                          5 - 6 = -1                       1

              10                        10 - 6 = 4                      16

Mean of above data, $Xbar$ = $\frac{\sum X_i}{n}$ = $\frac{3+5+10}{3}$ = 6

Standard Deviation of above data, s = $\sqrt{\frac{\sum (X-Xbar)^{2} }{n-1}}$ = $\sqrt{\frac{26}{3-1}}$ = $\sqrt{13}$ .

Now, suppose each entry in above example is multiplied by 2;

        New X                 New (X - $Xbar$)           New $(X - Xbar)^{2}$

            6                        6 - 12 = -6                           36

           10                       10 - 12 = -2                            4

           20                      20 - 12 = 8                           64

Mean of above data = $\frac{6+10+20}{3}$ = 12

Standard Deviation of above data, s = $\sqrt{\frac{104}{3-1}}$ = $2\sqrt{13}$ .

This shows that when multiplying each observation by 2 the standard deviation also gets multiplied by 2.

Therefore, for our question when each observation is multiplied by 3 the standard deviation = 6 * 3 = 18   and

Variance = $(Standard Deviation)^{2}$ = $18^{2}$ = 324 .