The mean of 100 observations was calculated as 40. Later, it was found that one observation had been misread as 83 instead of the correct value 53. What is the correct mean of the 100 observations after this error is corrected?

Introduction / Context:
This question tests the ability to correct an average when one data value was recorded incorrectly. Such mistakes are common when working with large data sets, and the correct method is to adjust the total sum and then recompute the mean.

Given Data / Assumptions:
• Number of observations = 100.• Initially calculated mean = 40.• One observation was misread as 83.• The correct value for that observation is 53.• All other values are correctly recorded.

Concept / Approach:
The mean or average is total sum divided by the number of observations. If one observation is misread, the total sum is incorrect. To fix this, we subtract the incorrect value from the total and then add the correct value. The number of observations remains the same, so the new mean is computed using the corrected total divided by 100.

Step-by-Step Solution:
Step 1: Let S be the incorrect total based on the wrong mean.Step 2: Since the original mean is 40 for 100 observations, S = 100 * 40 = 4000.Step 3: The misread value is 83 but should have been 53.Step 4: Correct total = S - incorrect value + correct value = 4000 - 83 + 53.Step 5: Compute 4000 - 83 = 3917, then 3917 + 53 = 3970.Step 6: Correct mean = corrected total / number of observations = 3970 / 100.Step 7: 3970 / 100 = 39.7.

Verification / Alternative check:
The difference between the correct and incorrect values is 53 - 83 = -30. This means the correct total must be 30 less than the initially used total. Dividing this difference by 100 gives a change in mean of -30 / 100 = -0.3. The original mean was 40, so the corrected mean should be 40 - 0.3 = 39.7, which matches the computed result.

Why Other Options Are Wrong:
39: This would imply a change of -1 in the mean, which is too large for a difference of 30 spread over 100 observations.40.3: This would require adding to the total, but the correct value is smaller than the misread value.42.7: This is far from the original mean and does not reflect the direction of change caused by the correction.

Common Pitfalls:
Students may wrongly add the difference instead of subtracting it or may forget that only one observation changed. Another common error is to divide the difference by the wrong number of observations. Always recompute the total sum carefully and then divide by the exact count of data points.

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