Question

If mean deviation about Mean of a particular data consisting of 10 observations is 7, then what will be the value of mean deviation when each is multiplied by 5?
A) 35
B) 45
C) 55
D) 65

Answer 1

Hint: We have been given the value of mean deviation as 7 for 10 observations. We will use the formula of mean deviation which is \[\dfrac{{\sum {sum\,of\,deviation} }}{{No.\,of\,observations}}\]. As we are not given the readings we will not calculate the deviation. We will assume deviations to be \[{x_1},{x_2}...\] and so on. We will substitute this in the formula which will get us \[\dfrac{{{x_1} + {x_2} + {x_{3......}} + {x_{10}}}}{{10}} = 7\]. We will get a sum of deviation = 70. Now we have been given that each deviation term will be multiplied by 5. So the terms will now change to\[5{x_1},5{x_2}...\]. We will substitute in the main formula to get \[mean\,deviation\,\dfrac{{5{x_1} + 5{x_2} + 5{x_{3......}} + 5{x_{10}}}}{{10}}\]. Taking 5 common, we will get the equation as mean deviation =\[\dfrac{{5({x_1} + {x_2} + {x_{3......}} + {x_{10}})}}{{10}}\]. We can now substitute this \[\dfrac{{{x_1} + {x_2} + {x_{3......}} + {x_{10}}}}{{10}} = 7\] to get value of mean deviation which will be mean deviation=\[5 \times 7\].

Complete step by step answer:
We have been given,
Mean Deviation=7
No. of observations=10
Let us start with the formula of mean deviation which is as follows,
Mean Deviation\[ = \dfrac{{\sum {sum\,of\,deviation} }}{{No.\,of\,observations}}\]
Where Deviation is the difference between the class mark of the corresponding interval and the mean of the group of data.
Deviation\[ = \left( {{x_i} - X} \right)\]
Now it is clear that to calculate Deviation, we require the class interval and corresponding frequencies for the set of data.
As we are not provided with any of the above, we will have to assume deviation as \[{x_1},{x_2}...\] .
Let us substitute all the values in the Mean Deviation formula.
Mean Deviation=\[\dfrac{{{x_1} + {x_2} + {x_{3......}} + {x_{10}}}}{{10}} = 7\] . . . . . . (1)
We have been given the question that each deviation is multiplied by 5.
So we will get the new deviations as \[5{x_1},5{x_2}...\] and so on.
Now let us calculate mean deviation for these new values of deviation.
We will substitute the values in the formula for mean deviation.
Mean Deviation\[ = \,\dfrac{{5{x_1} + 5{x_2} + 5{x_{3......}} + 5{x_{10}}}}{{10}}\]
As 5 is multiplied to each deviation from \[{x_1}\,to\,{x_{10}}\] we can write the above equation as
Mean Deviation\[ = \,\dfrac{{5({x_1} + {x_2} + {x_{3......}} + {x_{10}})}}{{10}}\]
Now we can see that when we take 5 common we get the term \[\dfrac{{{x_1} + {x_2} + {x_{3......}} + {x_{10}}}}{{10}}\], and on comparing to equation (1), we get
Mean Deviation\[ = \,\dfrac{{5({x_1} + {x_2} + {x_{3......}} + {x_{10}})}}{{10}}\]
\[\begin{gathered}
   = 5 \times 7 \\
   = 35 \\
\end{gathered} \]

Therefore we get the new value of mean deviation as 35.

Note: Do not try to find the sum of deviations, we will just assume it as we have been told each deviation is multiplied by the same number 5, we can take it common to simplify further.
Keep the solution simple and do not carry out unnecessary calculations.