Question

The mean of 20 observations is 31. In this data, one observation was taken by mistake as 52 instead of 25. Find the correct mean.

Answer 1

Hint: We have given the mean of 20 observations as 31 so multiplying 31 by 20 will give the summation of 20 observations. Now, it is given that instead of 25, one observation has been mistakenly written as 52 so we are subtracting 52 from the multiplication of 31 by 20 to get the summation of 19 observations. Then adding 25 to the result of this subtraction followed by the division with 20 will give you the correct mean.

Complete step-by-step answer:
The mean of 20 observations is given as 31.
We know that mean of 20 observations is equal to the summation of all the 20 observations divided by 20.
 $ \text{Mean}=\dfrac{\text{Sum of 20 observations}}{20} $
Substituting the value of mean as 31 in the above equation we get,
 $ 31=\dfrac{\text{Sum of 20 observations}}{20} $
On cross multiplying the above equation we get,
 $ 31\times 20=\text{Sum of 20 observations} $
Multiplying 31 and 20 on the left hand side of the above equation we get,
 $ 620=\text{Sum of 20 observations} $
Now, it is given that by mistake instead of 25, one observation has been written as 52. Now, 31 mean is for the summation of the observations having 52 so subtracting 52 from 620 we will get the summation of 19 observations which have not 52.
 $ \begin{align}
  & 620-52=\text{Sum of 19 observations} \\
 & \Rightarrow \text{568}=\text{Sum of 19 observations} \\
\end{align} $
We are going to add 568 to 25 to get the sum of actual 20 observations.
 $ \begin{align}
  & 568+25=\text{Sum of 20 observations} \\
 & \Rightarrow 59\text{3}=\text{Sum of 20 observations} \\
\end{align} $
Now, to find the actual mean of 20 observations we are going to divide the sum of 20 observations by 20.
Mean of 20 observations $ =\dfrac{593}{20}=29.65 $
Hence, the actual mean of 20 observations is equal to 29.65.

Note:The above question demands the knowledge of ``mean” if you know what'' mean ``is, then the whole question is done. The point where you go wrong in the question is instead of subtracting 52 you might have subtracted 25 from the sum of 20 observations. It has been observed that students make such mistakes in the exam because the language “one observation was taken by mistake as 52 instead of 25” might confuse you in the pressure of the exam so carefully read the question and prevent yourself from making this mistake.