Question

If the mean of 8 observations is x and one of the observations y is removed, then what is the new mean?

Answer 1

Hint: We have given the mean of 8 observations as x. We know the formula of mean of n observations as $Mean=\dfrac{\text{Sum of n observations}}{n}$. Substitute the value of n as 8 in this formula then rearrange this expression to find the value of “Sum of 8 observations”. Now, one of the observations y is removed then the number of observations becomes 7 and we have to subtract y from the sum of 8 observations. In the above question, we are asked to find the new mean after removal of y which is the sum of 7 observations divided by 7.

Complete step by step answer:
It is given in the above problem that the mean of 8 observations is x.
We know the formula of the mean of 8 observations which is the division of sum of n observations by total number of observations (i.e. n).
$Mean=\dfrac{\text{Sum of n observations}}{n}$
Now, in the above problem we have given 8 observations so substituting the value of n as 8 and mean as x in the above formula of mean we get,
$x=\dfrac{\text{Sum of 8 observations}}{8}$
Cross multiplying the above equation we get,
$8x=\text{Sum of 8 observations}$
Now, the question is saying that we have removed one observation (i.e. y) from the set of 8 observations. As one observation out of 8 has been removed so we are left with 7 observations and we have to subtract y from $8x$ to get the sum of 7 observations.
Subtracting y from $8x$ we get the sum of 7 observations as:
$8x-y=\text{Sum of 7 observations}$
We are asked in the above question to find the new mean of the observations. Now, in the formula of mean we are substituting the value of n as 7 and sum of 7 observations as $8x-y$.
$Mean=\dfrac{8x-y}{7}$

Hence, we got the new mean as $\dfrac{8x-y}{7}$.

Note: The mistake that could happen in this problem is to write 8 in place of 7 in calculating the new mean of 7 observations because you have just solved for the mean of 8 observations so subconsciously your mind will put the value of 8 in place of 7.
You can also check if the value of the new mean is correct or not by multiplying the new mean by 7 to get the sum of 7 observations.
The new mean that we got above is equal to:
$\dfrac{8x-y}{7}$
Multiplying this new mean by 7 we get,
$\begin{align}
  & \dfrac{\left( 8x-y \right)}{7}\left( 7 \right) \\
 & =8x-y \\
\end{align}$
Now, adding the eliminated observation y to it to get the sum of 8 observations.
$\begin{align}
  & 8x-y+y=\text{Sum of 8 observations} \\
 & \Rightarrow \text{8x}=\text{Sum of 8 observations} \\
\end{align}$
To get the original mean of 8 observations, we have to divide the above equation by 8 on both the sides.
$\begin{align}
  & \dfrac{\text{8x}}{8}=\dfrac{\text{Sum of 8 observations}}{8} \\
 & \Rightarrow x=\dfrac{\text{Sum of 8 observations}}{8} \\
\end{align}$
Hence, we are getting the mean of 8 observations as x which is given in the above question. This means that the value of the new mean which we have got in the above solution is correct.