Question

The mean of 25 observations is 32. It was later discovered that 38 and 29 are misread as 25 and 42. Find the correct mean.
A) 36
B) 32
C) 34
D) 33

Answer 1

Hint: Mean is the average values of data, which can be found by dividing the sum of all the observations with the number of observations. In this question, a mean of 25 observations has been given, out of which two numbers are wrongly calculated. We need to determine the correct mean of the data.

Complete step-by-step solution
Given the mean of the 25 observations is 32.
Let the 25 observations be \[{x_1},{x_2},{x_3}...........{x_{25}}\]
Two of the 25 observation are given, hence assume:
\[{x_{24}} = 25\]
\[{x_{25}} = 42\]
Therefore,

\[
  \Rightarrow \dfrac{{{x_1} + {x_2} + ......... + 25 + 42}}{{25}} = 32 \\
 \Rightarrow {x_1} + {x_2} + ......... + 47 = 800 \\
 \Rightarrow {x_1} + {x_2} + ........ + {x_{23}} = 733 - - - - (i) \\
 \]
Hence the sum of the first 23 observations is 733.
Now it is said that two of the data which were used to find the mean were miss read, and the real data are given as
\[{x_{24}} = 38\]
\[{x_{25}} = 29\]
Now let us replace the misread data to find the mean

\[\dfrac{{{x_1} + {x_2} + ......... + 38 + 29}}{{25}}\]

The value of the sum of\[{x_1} + {x_2} + ........ + {x_{23}} = 733\], so find the new mean
\[
\Rightarrow \dfrac{{{x_1} + {x_2} + ........ + {x_{23}} + 38 + 29}}{{25}} \\
\Rightarrow \dfrac{{733 + 38 + 29}}{{25}} \\
 \Rightarrow \dfrac{{800}}{{25}} \\
   = 32 \\
 \]
Hence the correct mean is\[ = 32\]

Thus the correct answer is option (B).

Note: Students should note that if any of the data of the observations are changed, then the final value of the mean will also change. Also, in the case when the number of observations is changed, it will also affect the mean.