The mean of 100 observations was originally calculated as 40. Later it was found that one observation was misread as 48 instead of 84. What is the correct mean of the 100 observations after this correction?

Introduction / Context:
This problem deals with correction of a mean when an error is detected in one of the observations. Such questions commonly appear in statistics sections of aptitude tests. They test whether you understand that the mean is based on the total sum of observations and that any change in the data must be reflected by adjusting the total before recomputing the mean.

Given Data / Assumptions:

There are 100 observations in total.
The originally calculated mean of these 100 observations is 40.
One value was wrongly taken as 48 instead of the correct value 84.
We must find the corrected mean after fixing this single error.
All other observations are assumed to be correct.

Concept / Approach:
The mean is equal to the total sum divided by the number of observations. When one observation is misread, the original total is incorrectly low or high. To correct the mean, we first compute the original total from the original mean. Then we adjust this total by removing the wrong value and adding the correct value. Finally we divide the corrected total by the same number of observations, which is still 100 in this case.

Step-by-Step Solution:
Step 1: Compute the original total of all observations. Original mean = 40, number of observations = 100. Original total = mean * number of observations = 40 * 100 = 4000. Step 2: Adjust the total for the misread observation. The wrong value used was 48; the correct value should be 84. Corrected total = original total - wrong value + correct value. Corrected total = 4000 - 48 + 84. Compute 4000 - 48 = 3952, then 3952 + 84 = 4036. Step 3: Compute the new mean using the corrected total. Corrected mean = corrected total / number of observations = 4036 / 100 = 40.36.

Verification / Alternative Check:
The difference between the correct value and the wrong value is 84 - 48 = 36. Since there are 100 observations, the mean should change by 36 / 100 = 0.36. Adding 0.36 to the original mean 40 gives 40.36, which matches our detailed calculation. This quick method confirms that the corrected mean 40.36 is reliable and consistent.

Why Other Options Are Wrong:
Option 41.24 would correspond to an adjustment of 1.24 in the mean, which would mean a total correction of 124, not 36, so it is not correct.
Option 41.92 represents an even larger change in the total and does not match the actual difference between 84 and 48.
Option 42.05 would require a difference in total of 205, which is completely inconsistent with the true correction required.

Common Pitfalls:
A typical mistake is to divide the difference 84 - 48 by the wrong number of observations or to add and subtract in the wrong order. Some students also forget that only one value was corrected and incorrectly change the count of observations. Others recompute the mean without first recalculating or adjusting the total, which leads to inaccurate answers.

Final Answer:
The correct mean of the 100 observations after correction is 40.36.