Question

If mean deviation about the mean of a particular data consisting of 10 observations is 7, then what will be the value of mean deviation when each data is multiplied by 5.
(a) 35
(b) 45
(c) 55
(d) 65

Answer 1

Hint: We know the formula for mean deviation is equal to \[\sqrt{\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}}{n}-{{\left( \dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n} \right)}^{2}}}\] . In this formula, ${{x}_{i}}$ represents the value of observations. We have given a number of observations as 10 and their mean deviation as 7. We are asked to find the mean deviation when each observation is multiplied by 5. So, we are going to multiply each observation by 5 and then will see the change that we are getting in the mean deviation.

Complete step by step answer:
We have given the mean deviation for 10 observations as 7.
We know the formula for mean deviation which is shown below:
\[\sqrt{\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}}{n}-{{\left( \dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n} \right)}^{2}}}\]
Now, for 10 observations mean deviation value is 7 so substituting n as 10 and equating the above expression to 7 we get,
\[\sqrt{\dfrac{\sum\limits_{i=1}^{10}{{{x}_{i}}^{2}}}{10}-{{\left( \dfrac{\sum\limits_{i=1}^{10}{{{x}_{i}}}}{10} \right)}^{2}}}=7\]
Now, we have multiplied each observation by 7 then the left hand side of the above expression will look like:
\[\begin{align}
  & \sqrt{\dfrac{\sum\limits_{i=1}^{10}{{{\left( 5{{x}_{i}} \right)}^{2}}}}{10}-{{\left( \dfrac{\sum\limits_{i=1}^{10}{5{{x}_{i}}}}{10} \right)}^{2}}} \\
 & \Rightarrow \sqrt{\dfrac{\sum\limits_{i=1}^{10}{25{{\left( {{x}_{i}} \right)}^{2}}}}{10}-{{\left( \dfrac{\sum\limits_{i=1}^{10}{5{{x}_{i}}}}{10} \right)}^{2}}} \\
\end{align}\]
As you can see that we can take 25 as common from both the expressions and we get,
\[\sqrt{25\dfrac{\sum\limits_{i=1}^{10}{{{\left( {{x}_{i}} \right)}^{2}}}}{10}-25{{\left( \dfrac{\sum\limits_{i=1}^{10}{{{x}_{i}}}}{10} \right)}^{2}}}\]
Taking 25 out from the square root we get,
\[\begin{align}
  & \sqrt{25}\sqrt{\dfrac{\sum\limits_{i=1}^{10}{{{\left( {{x}_{i}} \right)}^{2}}}}{10}-{{\left( \dfrac{\sum\limits_{i=1}^{10}{{{x}_{i}}}}{10} \right)}^{2}}} \\
 & \Rightarrow 5\sqrt{\dfrac{\sum\limits_{i=1}^{10}{{{\left( {{x}_{i}} \right)}^{2}}}}{10}-{{\left( \dfrac{\sum\limits_{i=1}^{10}{{{x}_{i}}}}{10} \right)}^{2}}} \\
\end{align}\]
In the above expression, we know the value of \[\sqrt{\dfrac{\sum\limits_{i=1}^{10}{{{x}_{i}}^{2}}}{10}-{{\left( \dfrac{\sum\limits_{i=1}^{10}{{{x}_{i}}}}{10} \right)}^{2}}}=7\] so substituting this value in above expression we get,
$\begin{align}
  & 5\left( 7 \right) \\
 & =35 \\
\end{align}$
From the above, we found that after multiplying each data by 5 our new mean deviation has become 35.

So, the correct answer is “Option A”.

Note: The possible mistake that could happen in this problem is that while writing the formula of mean deviation, you might commit the following mistake:
The formula for mean deviation is equal to:
\[\sqrt{\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}}{n}-\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}}\]
This is the wrong formula for mean deviation. In this formula, we have not taken the square of the mean. Generally, we forget to put the powers on the expression so make sure you won’t repeat this mistake.