The mean of 50 observations was calculated as 36. Later, it was discovered that one observation had been recorded as 23 instead of the correct value 48. What is the corrected mean of the 50 observations?

Introduction / Context:
This question tests your understanding of how a single incorrect observation affects the calculated mean and how to adjust the mean once the correct value is known. Instead of recomputing everything from scratch, we use the difference between the wrong value and the correct value to update the total and thus the mean efficiently.

Given Data / Assumptions:

• Number of observations = 50.
• Initially computed mean = 36.
• One observation was wrongly taken as 23.
• The correct value of that observation is 48.
• We need to find the corrected mean.

Concept / Approach:
The mean is defined as: mean = total sum / number of observations. We first compute the total sum based on the incorrect mean. Then, we adjust the total by adding the difference between the correct value and the wrong value. Finally, we divide the corrected total by the same number of observations (50) to obtain the corrected mean.

Step-by-Step Solution:
Step 1: Incorrect total sum based on mean 36: Incorrect total = 50 * 36 = 1800. Step 2: The wrong value recorded was 23; the correct value is 48. Difference = correct value - wrong value = 48 - 23 = 25. Step 3: Corrected total sum = incorrect total + difference. Corrected total = 1800 + 25 = 1825. Step 4: Corrected mean = corrected total / number of observations. Corrected mean = 1825 / 50. Step 5: Compute 1825 / 50 = 36.5.
Verification / Alternative check:
If the single value had originally been 48 instead of 23, the total would indeed increase by 25, so the mean would rise by 25 / 50 = 0.5. So the corrected mean is 36 + 0.5 = 36.5, which matches our detailed calculation.
Why Other Options Are Wrong:
Option A (35) and Option C (40) both differ significantly from 36.5 and do not correspond to adding the correct adjustment of 25 to the total sum. Option D (42) and Option E (37) similarly do not reflect the small 0.5 increase in the mean that results from the extra 25 marks spread over 50 observations.
Common Pitfalls:
Students might subtract instead of add the difference, leading to 1800 - 25, which is incorrect. Another frequent mistake is to divide the difference 25 by the wrong number of observations, such as 49 instead of 50.
Final Answer:
The corrected mean of the 50 observations is 36.5.