Question

The mean of 100 observations is 50. If one of the observations which was 50 is replaced by 150 the resulting mean will be
(A) 51
(B) 52
(C) 51.5
(D) 53

Answer 1

Hint: We solve this question by first considering the formula for the mean, $Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$. Then we substitute the values of number of observations and mean in it and find the sum of observations. Then we subtract the value 50 and add 150 to it as 50 is replaced by 150. Then we use the formula for the mean again and find the resulting mean by substituting the value of new sum of observations and number of observations.

Complete step by step answer:
We are given that mean of 100 observations is 50.
We are also given that an observation which was 50 is replaced by 150.
We need to find the mean of new observations.
Now let us consider the formula for mean.
$Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$
Using this formula, and substituting the value of mean and number of observations we get,
\[\begin{align}
  & \Rightarrow 50=\dfrac{Sum\ of\ observations}{100} \\
 & \Rightarrow Sum\ of\ observations=50\times 100 \\
 & \Rightarrow Sum\ of\ observations=5000 \\
\end{align}\]
We are given that an observation 50 is replaced by 150.
First let us subtract the value 50 from the sum of observations. Then we get,
\[\begin{align}
  & \Rightarrow Sum\ of\ observations=5000-50 \\
 & \Rightarrow Sum\ of\ observations=4950 \\
\end{align}\]
As the observation 50 is subtracted, the number of observations is 99.
Now, let us add the observation 150 to the above sum.
\[\begin{align}
  & \Rightarrow Sum\ of\ observations=4950+150 \\
 & \Rightarrow Sum\ of\ observations=5100 \\
\end{align}\]
As the observation 150 is added, the number of observations is 100 again.
So, we get the sum of observations equal to 5100.
Now let us consider the formula for the mean.
$Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$
Now, substituting the new sum of observations and total number of observations in it we get the new mean as,
$\begin{align}
  & \Rightarrow New\ Mean=\dfrac{5100}{100} \\
 & \Rightarrow New\ Mean=51 \\
\end{align}$
Hence the value of resulting mean is 51.

So, the correct answer is “Option A”.

Note: There is a possibility of one making a mistake while solving this problem, after finding the new sum of observations one might divide it with 99 instead of with 100 and solve it as,
$\begin{align}
  & \Rightarrow New\ Mean=\dfrac{5100}{99} \\
 & \Rightarrow New\ Mean=51.5151 \\
\end{align}$
Then they might mark the answer as Option C. But it is wrong because the observation 50 is replaced by 150 so the number of observations will be 100.