Mean of 100 observations is found to be 40. At the time of computation, two items were wrongly taken as $30$ and $27$ instead of $3$ and $72$. Find the correct meaning.

Verified
175.5k+ views
Hint: In this question, we have the number of observations and its mean given to us. So by using the formula of mean we will get the total sum of observations. And then we will calculate the correct mean by substituting the values in the formula.

Formula used:
Mean,
$\bar x = \dfrac{{\sum x }}{N}$
Here,
$\bar x$ , will be the mean
$N$ , will be the number of observations.
$\sum x$ , will be the sum of the observation.
Also, the correct mean will be calculated by using the formula as
$Correct{\text{ }}\bar x = \dfrac{{\sum x \left( {Wrong} \right) + \left( {Correct{\text{ value}}} \right) - \left( {Incorrect{\text{ value}}} \right)}}{N}$

First of all we will see the values given to us. So we have the values
$N = 100$
$\bar x = 40$
So by using the formula of mean, we have the equation as $\bar x = \dfrac{{\sum x }}{N}$
On substituting the values, we will get the equation as
$\Rightarrow 40 = \dfrac{{\sum x }}{{100}}$
And on solving the above equation by doing the cross multiplication, we will get
$\Rightarrow \sum {x\left( {Wrong} \right)} = 40 \times 100$
And on solving the multiplication, we get
$\Rightarrow \sum {x\left( {Wrong} \right)} = 4,000$
So we have the correct values which are equal to
$\Rightarrow 3 + 72$
And on solving it, we get
$\Rightarrow 75$
And also we have the incorrect values which are equal to
$\Rightarrow 30 + 27$
And on solving it, we get
$\Rightarrow 57$
So by using the formula for the correct mean, we have the formula as $Correct{\text{ }}\bar x = \dfrac{{\sum x \left( {Wrong} \right) + \left( {Correct{\text{ value}}} \right) - \left( {Incorrect{\text{ value}}} \right)}}{N}$
So on substituting the values in the above position, and equating it we will get
$\Rightarrow Correct{\text{ }}\bar x = \dfrac{{4,000 + 75 + 57}}{{100}}$
And on adding the numerator of the above part, we get
$\Rightarrow Correct{\text{ }}\bar x = \dfrac{{4,018}}{{100}}$
And on solving it again, we get
$\Rightarrow Correct{\text{ }}\bar x = 40.18$
Therefore, the correct value of the mean will be equal to $40.18$ .

Note: So here we have calculated the correct mean and we can see that not too high-level solutions are needed for such types of questions. So this method is suggested doing this type of question .