Question

Mean of 50 observations was found to be 80.4. But later on, it was discovered that 96 was misread as 69 at one place. Find the correct mean.

Answer 1

Hint: We can make two cases here, one when the wrong and one when the right number is taken to calculate the mean. We can use the formula of mean, use the first case to find certain values and then substitute in the second case so as to find the value of the correct mean.
\[Mean = \dfrac{{Sum{\text{ }}of{\text{ }}observations}}{{Total{\text{ }}observations}}\]

Complete step-by-step answer:
Mean of observations is given by the sum of observations divided by the total number of observations.
One observation was misread, 96 as 69.
Let the 50 observations be $ {x_1},{x_2}....{x_{49}},{x_{50}} $
In the first case, when the number was misread, the mean is equal to sum of 49 observations with 69 as the $ {50^{th}} $ observation, its value is given to be 80.4.
\[
   \Rightarrow \dfrac{{{x_1} + {x_2} + .... + {x_{49}} + 69}}{{50}} = 80.4 \\
  \Rightarrow {x_1} + {x_2} + .... + {x_{49}} + 69 = 80.4 \times 50 \\
  \Rightarrow {x_1} + {x_2} + .... + {x_{49}} = \left( {80.4 \times 50} \right) - 69\_\_\_\_(1) \\
 \]
Now, in the second case, when the correct mean (M) is to be found, we will use the correct number i.e. 96 as the $ {50^{th}} $ observation.
\[ \Rightarrow \dfrac{{{x_1} + {x_2} + .... + {x_{49}} + 96}}{{50}} = M\]
We can substitute the value of the sum of 49 observations from (1):
  \[
  \Rightarrow \dfrac{{\left( {80.4 \times 50} \right) - 69 + 96}}{{50}} = M \\
  \Rightarrow M = \dfrac{{4020 + 27}}{{50}} \\
  \Rightarrow M = \dfrac{{4047}}{{50}} \\
  \Rightarrow M = 80.94 \\
 \]
Therefore, the correct mean if 96 would not have been misread as 69 would have been equal to 80.94

Note: We could have calculated the complete value in (1), but it would have been an extra work, so it’s better skipping the complete calculations when we know that the calculations will be easier when these get substituted further. We divide the sum of observations by total observations because the mean is used to evaluate the average of all the observations. The mean can also be referred to as arithmetic mean