Question

The mean and standard deviation of five observations are 9 and 0 respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their standard deviation is?
A.0
B.4
C.2
D.1

Answer 1

Hint: We are given with the mean of five observations as 9 it means in first observation all the terms are the same because the deviation is also 0. For set 2 we will consider the last observation as a and using the formula of mean we will find the fifth observation so changed. Now to find the standard deviation we will use the formula of standard deviation using the terms of the second observation.

Formula used:
Mean \[\bar x = \dfrac{{\sum {{x_i}} }}{n}\]
Standard deviation \[S.D. = \sqrt {\dfrac{{{{\left( {{x_i} - \bar x} \right)}^2}}}{n}} \]

Complete step by step solution:
Given are the five observations. So set one is given by \[\left( {9,9,9,9,9} \right)\]
Now if the last observation is changed that is if it is considered as a then the set will be given by,
\[\left( {9,9,9,9,a} \right)\]
Now we will use the formula for finding the last term using the mean formula;
\[mean = \bar x = \dfrac{{\sum {{x_i}} }}{n}\]
Here n is the number of observations and that are 5.and the mean is 10. So let’s put the values;
\[10 = \dfrac{{9 + 9 + 9 + 9 + a}}{5}\]
On cross multiplying we get,
\[10 \times 5 = 36 + a\]
Taking the product,
\[50 - 36 = a\]
On subtracting we get,
\[a = 14\]
This is the observation after changing the terms.
Now to find the standard deviation we will use the formula and observations so obtained as \[\left( {9,9,9,9,14} \right)\]
\[S.D. = \sqrt {\dfrac{{{{\left( {{x_i} - \bar x} \right)}^2}}}{n}} \]
Now putting the values, we get;
\[S.D. = \sqrt {\dfrac{{{{\left( {9 - 10} \right)}^2} + {{\left( {9 - 10} \right)}^2} + {{\left( {9 - 10} \right)}^2} + {{\left( {9 - 10} \right)}^2} + {{\left( {14 - 10} \right)}^2}}}{5}} \]
On calculating we get,
\[S.D. = \sqrt {\dfrac{{{{\left( { - 1} \right)}^2} + {{\left( { - 1} \right)}^2} + {{\left( { - 1} \right)}^2} + {{\left( { - 1} \right)}^2} + {{\left( 4 \right)}^2}}}{5}} \]
Taking the square,
\[S.D. = \sqrt {\dfrac{{1 + 1 + 1 + 1 + 16}}{5}} \]
On adding we get,
\[S.D. = \sqrt {\dfrac{{20}}{5}} \]
On dividing,
\[S.D. = \sqrt 4 \]
Taking the square root,
\[S.D. = 2\]
This is the standard deviation obtained after the change in the observations.
So, the correct answer is “Option C”.

Note: Here note that in the first set the all observations are safe because the mean is already given but we cannot continue this for second because we have the previous observations with a new observation changed but the mean is also changed to 10. So, we took help to find the changed observation.