The mean of 100 observations was calculated as 40. Later it was found that one of the observations had been misread as 83 instead of 53. What is the correct mean of the 100 observations?

Introduction / Context:
This is another average correction question, this time with a smaller error. An observation that should have been 53 was misread as 83. We must adjust the total sum accordingly and then recalculate the mean for the 100 observations.

Given Data / Assumptions:

• Number of observations = 100.
• Reported mean based on a misread value = 40.
• One value used was 83 but the correct value is 53.
• We must compute the correct mean of all 100 observations.

Concept / Approach:
The reported mean of 40 comes from an incorrect total. We first calculate this incorrect total by multiplying the mean by the number of observations. Since the incorrect value 83 is 30 more than the correct value 53, the total must be reduced by 30. After adjusting the total, we divide by 100 again to obtain the corrected mean.

Step-by-Step Solution:
Step 1: Incorrect total sum = reported mean * number of observations = 40 * 100 = 4000. Step 2: Wrong value used = 83; correct value = 53. Step 3: The error in the total is 83 - 53 = 30 extra units. Step 4: Corrected total = 4000 - 30 = 3970. Step 5: Correct mean = corrected total / number of observations = 3970 / 100 = 39.7.

Verification / Alternative check:
We can also express the correction directly on the mean. Subtracting 30 from the total and then dividing by 100 lowers the mean by 30 / 100 = 0.3. So the corrected mean is 40 - 0.3 = 39.7, which agrees with the detailed computation and confirms the answer.

Why Other Options Are Wrong:
Option 39 would imply a reduction of 1 point in the mean, equivalent to subtracting 100 from the total, which is far more than the 30 units of actual error. Option 40.3 would increase the mean, which makes no sense when we are replacing an inflated value with a smaller one. Option 42.7 is even further away. Only 39.7 correctly reflects the net reduction of 30 in the total.

Common Pitfalls:
Students sometimes add instead of subtracting the difference between the wrong and correct values or divide the difference by the wrong number of observations. Always identify whether the wrong value is higher or lower than the correct one so you know whether to add or subtract the difference from the total before recomputing the mean.

Final Answer:
The correct mean of the 100 observations is 39.7.