Question

The mean of 100 observations is 40. It is found that an observation 53 was misread as 83, the correct mean is
A. 39.7
B. 39
C. 40
D. 39.5

Answer 1

Hint: We know that the mean of n observations is given by dividing the sum of the observations by n .using this we can calculate the sum of observations and we need to remove the wrong observation by subtracting it from the sum and adding the correct observation and find the new mean by dividing the new sum of observation by 100.

Step by step solution :
We are given that the number of observations is 100
And the mean is 40
We know that the mean of n observations is given by the formula
$ \Rightarrow mean = \dfrac{{{\text{sum of the observations}}}}{n}$
Since here mean is 40 and n is 100
We get the sum of observations to be
$
   \Rightarrow 40 = \dfrac{{{\text{sum of the observations}}}}{{100}} \\
   \Rightarrow 40\times 100 = {\text{sum of the observations}} \\
   \Rightarrow 4000 = {\text{sum of the observations}} \\
$
Now we are said that a number 53 was misread as 83
So first let's remove the incorrect observation from the sum of the observation
$ \Rightarrow 4000 - 83 = 3917$
And now let's add the correct observation
$ \Rightarrow 3917 + 53 = 3970$
So now lets find the new mean with the new sum of observation
$ \Rightarrow mean = \dfrac{{3970}}{{100}} = 39.7$
Hence the new mean is 39.7

Therefore the correct answer is option A.

Note :
Mean, in mathematics, a quantity that has a value intermediate between those of the extreme members of some set. Several kinds of mean exist, and the method of calculating a mean depends upon the relationship known or assumed to govern the other members.