Question

The mean of 100 items was found to be 30. If at the time of calculation, two numbers were wrongly taken as 32 and 12 instead of 23 and 11, find the correct mean.

Answer 1

Hint: When we calculate the mean of ‘n’ numbers, we basically add all the ‘n’ numbers and divide it by ‘n’ to get our result. In our problem, only two numbers have been wrongly written, which means that the sum of the rest 98 numbers is same as before, we will assume this as a variable and calculate this value, in order to find the mean with correct numbers.

Complete step by step solution:
Let us first assign some terms that we are going to use later in our problem.
Let the sum of 98 items that were correctly written be ‘x’.
Let the initial mean with two wrong items be ‘m’. And,
Let the final mean with all the correct items be ‘M’.

Now, according to our first statement, the mean of 100 items was 30 when the two wrong numbers are 32 and 12. Mathematically, this could be written as:
$\Rightarrow \dfrac{x+32+12}{100}=m$
Where, ‘m’ is equal to 30. Putting this in the above equation, we get:
$\begin{align}
  & \Rightarrow x+44=3000 \\
 & \therefore x=2956 \\
\end{align}$
Thus, the sum of the other 98 terms which were correctly written comes out to be 2956.
Now, according to the second statement, the correct terms are 23 and 11 respectively. Thus, the new mean can be calculated as follows:
$\Rightarrow M=\dfrac{x+23+11}{100}$
Putting the value of ‘x’ calculated above as 2956, we get:
$\begin{align}
  & \Rightarrow M=\dfrac{2956+23+11}{100} \\
 & \Rightarrow M=\dfrac{2990}{100} \\
 & \therefore M=29.90 \\
\end{align}$
Thus, the new mean with all the correct terms comes out to be 29.90 .

Note: While calculating the mean or average of a given number of values, we should always take care of the number of items actually present. In our above problem, we assumed the sum of 98 items as ‘x’. So, even after that there were still 100 items present, that is, we should not calculate the mean by diving the total sum by 3, as there are three entities only. These are the little things that one should take care of.