Question

Mean of $100$ observations is $45$. It was later found that two observations $19$ and $31$ were incorrectly recorded as $91$ and $13$. The correct mean is:
A) $44.0$
B) $44.46$
C) $45.00$
D) $45.54$

Answer 1

Mean of observations is the given of the question. There is some mistake in the record of that observation. We have to find the correct mean of the re-corrected record. At first, we will find the total value of 100 observations. Then, we will remove the incorrect observation. Next, we will add the correct observation. And finally, dividing by 100, we will get the mean value of the 100 observations.

Complete step-by-step solution:
It is given that; the mean of $100$ observations are $45$. It was later found that two observations $19$ and $31$ were incorrectly recorded as $91$ and $13$.
We have to find the correct mean.
As, the mean of $100$ observations are $45$.
The total value of $100$ observations are $= 45 \times 100$
Simplifying we get,
The total value of $100$ observations are $= 4500$
Among these observations, $19$ and $31$ were incorrectly recorded as $91$ and $13$.
At we will remove the incorrect observations those are $91$ and $31$.
After removing, the incorrect observations, we get the value as $= 4500 - 91 - 13$
Simplifying we get,
The value is $= 4396$
Now, we will add the correct observations that is $19$ and $31$.
After adding, the correct observations, we get the value as $= 4396 + 19 + 31$
Simplifying we get,
The value is $= 4446$
Now, the mean of the correct 100 observations is $= \dfrac{{4446}}{{100}}$
Simplifying we get,
The mean of the correct 100 observations is $= 44.46$

$\therefore$ The correct option is B

Note: The mean error is an informal term that usually refers to the average of all the errors in a set. An error in this context is an uncertainty in a measurement, or the difference between the measured value and true/correct value. The more formal term for error is measurement error, also called observational error.