The average of 9 observations was initially calculated as 35. Later, it was discovered that one of the observations was 81 but had been misread and recorded as 18. What is the correct average of the 9 observations after this error is corrected?

Introduction / Context:
This problem illustrates how a single incorrect value can affect the average of a data set. You are given the original average, then told that one observation was misread. By adjusting the total sum to correct the mistake, you can compute the new, accurate average.

Given Data / Assumptions:
• Number of observations = 9.• Initially calculated average = 35.• One observation was actually 81.• It was misread and used in the calculation as 18.• All other observations were recorded correctly.

Concept / Approach:
The average is defined as total sum divided by the number of observations. When a value has been misread, the originally used total is incorrect. To correct the average, we first compute the initial total based on the wrong average, then adjust the total by removing the misread value and adding the correct one. Finally, we divide the corrected total by the same number of observations.

Step-by-Step Solution:
Step 1: Let S be the incorrect total sum used originally.Step 2: Since average = 35 for 9 observations, S = 9 * 35 = 315.Step 3: One value 81 was misread as 18, so 18 contributed to S instead of 81.Step 4: Correct total sum = S - misread value + correct value = 315 - 18 + 81.Step 5: Compute 315 - 18 = 297, then 297 + 81 = 378.Step 6: Correct average = corrected total sum / number of observations = 378 / 9.Step 7: Compute 378 / 9 = 42.

Verification / Alternative check:
We can reason that the difference between the correct observation and the misread observation is 81 - 18 = 63. So the correct total must be 63 more than the incorrect one. Hence the average must increase by 63 / 9 = 7. The original average was 35, so the corrected average is 35 + 7 = 42, which agrees with the detailed calculation.

Why Other Options Are Wrong:
28: This would mean the average decreased, which is impossible because the corrected value is greater than the misread one.35: This is the original incorrect average before adjustment.45: This would require an increase larger than 7 per observation, which is not supported by the difference of 63 in the total.

Common Pitfalls:
Some learners forget to adjust the total by both subtracting the misread value and adding the correct one. Others mistakenly divide the difference directly by the new average or number of observations without adjusting the original total. Always recompute the total carefully and then divide by the number of data points.

Final Answer:
The correct average of the observations is 42.