Question

If total sum of square of the observations is 40, mean of the observations is 2 and sample variance is also 6 then total number of observations is:
(a) 15
(b) 35
(c) 25
(d) 4

Answer 1

Hint: We have given the sum of squares of the observations as 40 and mean of the observations is also given and is equal to 2 and we have given the variance as 6. We know the relation between variance, mean and mean square which is equal to: \[Variance=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}}{n}-{{\left( \dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n} \right)}^{2}}\] and in this formula, we have considered the number of observations as “n”.

Complete step by step answer:
In the above problem, we have given the sum of the square of the observations as 40. Let us assume the number of observations to be “n”.
So, we can write mean of the squared observations as:
\[\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}}{n}\]
In the above, the expression written in the numerator is the sum of the square of n observations which is given in the above problem as 40 so substituting that in the above we get,
$\dfrac{40}{n}$
Now, we have given the mean of “n” observations also which is equal to 2 so we can write the mean of “n” observations as:
\[\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\]
This expression written above is the mean of “n” observations so we can equate the above mean expression to 2.
\[\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}=2\]
We know that the variance is equal to:
\[Variance=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}}{n}-{{\left( \dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n} \right)}^{2}}\]
Substituting the values of the expressions written on the right hand side of the above equation we get,
\[\begin{align}
  & 6=\dfrac{40}{n}-{{\left( 2 \right)}^{2}} \\
 & \Rightarrow 6=\dfrac{40}{n}-4 \\
\end{align}\]
Adding 4 on both the sides we get,
$10=\dfrac{40}{n}$
On cross multiplying the above equation we get,
$\begin{align}
  & 10n=40 \\
 & \Rightarrow n=\dfrac{40}{10}=4 \\
\end{align}$
From the above solution, we got the value of n as 4 and we have assumed “n” as the total number of observations.

So, the correct answer is “Option D”.

Note: The mistake that could happen in the above problem is that, while substituting the value of mean in the variance formula you would have substituted in the following way:
We have written the mean of n observations as:
\[\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\]
So, we have substituted the value of mean which is 2 in the above expression as:
\[\dfrac{2}{n}\]
This is the wrong way to write the mean of all the observations. In this way, we have written the sum of observations as 2 but mean is the whole expression given above. In this way, we have written the sum of observations as 2 but mean is the whole expression given above which is the division of sum of observations to number of observations.