Question

36.
The mean and standard deviation of 20 observations are found
to be 10 and 2 respectively. On rechecking, it was found that an
observation 8 was incorrect. Calculate the correct mean and
standard deviation in each of the following cases:
() If wrong item is omitted.
(ii) If it is replaced by 12​

Answer 1

Answer:

(i) Correct Mean = 10.105  ,  Correct Standard deviation = 1.998

(ii) Correct Mean = 10.2  ,  Correct Standard deviation = 1.9894

Step-by-step explanation:

We are given that the mean and standard deviation of 20 observations are found  to be 10 and 2 respectively. And On rechecking, it was found that an  observation 8 was incorrect.

(i) If wrong item is omitted ;

Mean formula is given by, $Xbar$ = $\frac{\sum X}{n}$

Mean of 20 observations = $\frac{\sum X}{20}$

So, $\sum X$ = 10 * 20 = 200.

Now if wrong item of 8 is omitted, then ;

Correct $\sum X$ = 200 - 8 = 192  and number of observation is 20 - 1 = 19

So, Correct Mean = 192 ÷ 19 = 10.105

Now, Standard deviation formula, S.D. = $\sqrt{\frac{\sum X^{2} - n*Xbar^{2} }{n-1} }$

S.D. of 20 observations = $\sqrt{\frac{\sum X^{2} - 20*10^{2} }{20-1} }$

So, $\sum X^{2}$ = $(2^{2}*19) + (20*100)$ = 2076

Now, if wrong item of 8 is omitted, then ;

Correct $\sum X^{2}$ = 2076 - $8^{2}$ = 2012 and number of observation is 20 - 1 = 19

Therefore, Correct S.D. = $\sqrt{\frac{2012 - 19*10.105^{2} }{19-1} }$ = 1.998

(ii)  If wrong item is replaced by 12;

Mean formula is given by, $Xbar$ = $\frac{\sum X}{n}$  

$\sum X$ = 10 * 20 = 200.

Now if wrong item of 8 is replaced by 12, then ;

Correct $\sum X$ = 200 - 8 + 12= 204  and number of observation is 20 - 1 +1 = 20

So, Correct Mean = 204÷ 20 = 10.2

Now, Standard deviation formula, S.D. = $\sqrt{\frac{\sum X^{2} - n*Xbar^{2} }{n-1} }$

$\sum X^{2}$ = $(2^{2}*19) + (20*100)$ = 2076

Now if wrong item of 8 is replaced by 12, then ;

Correct $\sum X^{2}$ = 2076 - $8^{2}$ + $12^{2}$= 2156 and no. of observation is 20 - 1 + 1 = 20  

Therefore, Correct S.D. = $\sqrt{\frac{2156 - 20*10.2^{2} }{20-1} }$ = 1.9894

Answer 2

Answer:

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