Question

The mean of 40 observations was 160. It was detected on rechecking that the value of 165 is wrongly copied as 125 for computation of mean. Find the correct meaning.

Answer 1

Hint: We solve this question by first considering the formula for mean, $Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$. Then we substitute the given value of mean and number of observations and solve it to find the value of the sum of observations. Then we subtract the observation 125 and add 165 to the sum and find the value of the corrected sum of observations. Then we use the formula for mean again to find the correct mean.

Complete step by step answer:
We are given that the mean of 40 observations is 160.
We are also given that one of the observations of value 165 is wrongly noted as 125 and the mean is computed.
We need to find the correct meaning.
Now let us consider the formula for mean.
$Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$
Using this formula, we can write the mean of our given observations as,
$\Rightarrow Mean=\dfrac{Sum\ of\ observations}{40}$
As we are given that mean is 160, substituting it we get,
$\begin{align}
  & \Rightarrow 160=\dfrac{Sum\ of\ observations}{40} \\
 & \Rightarrow Sum\ of\ observations=160\times 40 \\
 & \Rightarrow Sum\ of\ observations=6400 \\
\end{align}$
Now we are given that an observation is wrongly noted as 125 whose correct value is 165. So, let's subtract the value 125 from the obtained sum. Then we get,
$\begin{align}
  & \Rightarrow Sum\ of\ 39\ observations=6400-125 \\
 & \Rightarrow Sum\ of\ 39\ observations=6275 \\
\end{align}$
Now let us add the value 165 to the above sum. Then we get,
$\begin{align}
  & \Rightarrow Sum\ of\ 40\ observations=6275+165 \\
 & \Rightarrow Sum\ of\ 40\ observations=6440 \\
\end{align}$
Now let us use the formula of mean discussed above. Then we get,
$\begin{align}
  & \Rightarrow Correct\ Mean=\dfrac{Sum\ of\ observations}{40} \\
 & \Rightarrow Correct\ Mean=\dfrac{6440}{40} \\
 & \therefore Correct\ Mean=161 \\
\end{align}$
So, the value of the corrected mean is 161.
Hence the answer is 161.

Note:
The common mistake one makes while solving this problem is one might take the total number of observations at the end as 41 thinking that we are adding 165 to the before sum. But it is wrong because we are adding 165 but subtracting 125. So, the number of observations remains 40.