Calculating the Correct Mean After Data Correction

This problem involves finding the accurate average (mean) marks of students when some initial data entries were incorrect. We are given the initial mean, the total number of students, and the specific values that were wrongly used and the values that should have been used instead.

Understanding the Initial Calculation

The initial calculation used 100 students with a mean score of 45. The formula for the mean is:

$ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} $

From this, we can find the initial sum of marks:

$ \text{Sum}_{\text{initial}} = \text{Mean}_{\text{initial}} \times \text{Number of Students} $

Substituting the given values:

$ \text{Sum}_{\text{initial}} = 45 \times 100 = 4500 $

Identifying the Errors and Corrections

It was found that two marks were recorded incorrectly:

• Incorrect values: 23 and 27
• Correct values: 32 and 72

Let's calculate the sum of the incorrect values and the sum of the correct values:

• Sum of incorrect values = $23 + 27 = 50$
• Sum of correct values = $32 + 72 = 104$

The difference between the sum of the correct values and the sum of the incorrect values tells us how much the total sum should change:

$ \text{Difference} = \text{Sum}_{\text{correct}} - \text{Sum}_{\text{incorrect}} $

$ \text{Difference} = 104 - 50 = 54 $

This means the initial sum of marks was underestimated by 54.

Calculating the Correct Sum and Mean

To find the correct sum of marks, we adjust the initial sum by adding the difference:

$ \text{Sum}_{\text{correct}} = \text{Sum}_{\text{initial}} + \text{Difference} $

$ \text{Sum}_{\text{correct}} = 4500 + 54 = 4554 $

Now, we can calculate the correct mean using the correct sum and the total number of students (which remains 100):

$ \text{Mean}_{\text{correct}} = \frac{\text{Sum}_{\text{correct}}}{\text{Number of Students}} $

$ \text{Mean}_{\text{correct}} = \frac{4554}{100} = 45.54 $

Conclusion

The correct mean marks of the 100 students, after correcting the two wrongly entered values, is 45.54.