Question

The mean of 36 observations is 12. One observation 47 was misread as 74. Find the correct mean.

Answer 1

Hint: First consider the given data. As the wrong mean is given of 12 observations, we need to find the correct mean by replacing the incorrect observation with correct observation. First find the incorrect sum of 36 observations by multiplying the incorrect mean with the total number then subtract the incorrect value from it and add the correct value in it which will give us the correct sum, further divide the correct sum with the total number of observations to get the correct mean.

Complete step by step answer:

Consider the given data that is the mean of 36 observations is 12.
We have to find the correct mean of 12 Correct observation by replacing the value 74 by 47.
Here, given in the question that the incorrect observation is 74 and correct observation is 47
Now we will find the sum of 36 observations, which can be determined by multiplying the number of observations by the mean, that 36 and 12 respectively.
So, the sum of 36 observations will be equal to the product of 36 and 12.
Thus, we get,
Sum of 36 observation\[ = 36 \times 12 = 432\]
Now find the correct sum of all the observations, which can be found by subtracting incorrect observation from the sum of 36 observations and correct observation.
That is,
\[{\text{Correct}}\;{\text{sum}} = S{\text{um}}\;{\text{of}}\,{\text{36}}\;{\text{observations}} + {\text{correct}}\;{\text{observation}} - {\text{incorrect}}\;{\text{observation}}\]
Thus, correct sum will be:
\[
   \Rightarrow {\text{Correct}}\;{\text{sum}} = 432 + 47 - 74 \\
   \Rightarrow {\text{Correct}}\;{\text{sum}} = 405 \\
 \]
The correct mean can be found by using the actual or correct sum of all the 36 observations.
As the mean is the ratio of the sum of observations to the total number of observations, so we can find the correct mean by dividing the actual sum of 36 observations by 36.
Thus, we have,
\[
   \Rightarrow {\text{Correct}}\;{\text{Mean}} = \dfrac{{405}}{{36}} \\
   \Rightarrow {\text{Correct}}\;{\text{Mean}} = 11.25 \\
 \]
Hence, the correct mean is \[11.25\].

Note: As one value is misplaced which means the mean is also incorrect so we have converted the incorrect mean into the correct mean by finding the incorrect sum and then convert it into the correct sum and then find the correct mean. Mean is given by the total sum divided by the total number of observations. The number of observations that is the denominator will remain the same as when finding the correct sum, we have added one observation that is the correct value and simultaneously subtracted one observation that is the incorrect value so there is no change in the number of observations.