Question

Choose the correct option from the provided options to the following question:
The mean of $200$ items was $50$. Later on, it was discovered that two items were misread as $92$ and $8$ instead of $192$ and $88$. The correct mean is
A. $50.9$
B. $50$
C. $51$
D. $51.5$

Answer 1

Find the sum of the given items by the formula and name it as the wrong sum. Then subtract the two wrong terms from the sum and add the right terms to it to get the correct sum. Find the mean of the correct terms by using the formula.

Complete step-by-step solution:
Given,
The mean of $200$ items = $$$$50
Wrong terms $= 92,8$
Sum of wrong terms $= 92 + 8$
Sum of wrong terms $= 100$
Right terms $= 192,88$
Sum of right terms $= 192 + 88$
Sum of right terms $= 280$
Now,
Sum of $200$ items $= S$
Number of items $= f$ (frequency)
Mean = $$$$\dfrac{S}{f}
$\Rightarrow S = f \times$ mean
$\Rightarrow S = 200 \times 50$
Multiplying the right-hand side, we get;
$\Rightarrow S = 10,000$
Wrong Sum $= S = 10,000$
Subtracting the sum of wrong terms and adding the sum of the right terms, we get the correct sum.
Correct Sum $= 10000 + 280 - 100$
$\Rightarrow$ Correct Sum $= 10,180$
We have the correct sum and the frequency. Now we find the mean;
Mean = $$$$\dfrac{S}{f}
$\Rightarrow$ Mean $= \dfrac{{10180}}{{200}}$
Dividing the right-hand side with the denominator, we get;
$\Rightarrow$ Mean$= 50.9$
Hence, the mean $= 50.9$

$\therefore$ The correct option is A.

Note: We have to mind that, the mean is the mathematical average of a set of two or more numbers. The arithmetic mean and the geometric mean are two types of mean that can be calculated. Summing the numbers in a set and dividing by the total number gives you the arithmetic mean. The geometric mean is more complicated and involves multiplication of the numbers taking the nth root. The mean helps to assess the performance of an investment or company over a period of time, and many other uses.