Calculating the Average Marks of Multiple Batches

The problem of finding the average marks of students from different batches, each with varying numbers of students and average marks, can be approached through a concept known as the weighted average. This method is particularly useful when dealing with datasets of different sizes but similar characteristics, such as student performance.

In this article, we will explore how to calculate the average marks of three batches of students with different numbers of students and average marks. We will use the example provided in the initial problem to demonstrate the process.

Problem Understanding

We are given data for three batches:

First batch: 40 students with an average of 65 marks. Second batch: 50 students with an average of 45 marks. Third batch: 70 students with an average of 60 marks.

The task is to find the overall average marks of all 160 students combined.

Step-by-Step Solution

To calculate the overall average marks, we can follow these steps:

Calculate the total marks for each batch. Sum the total marks of all batches. Divide the total marks by the total number of students.

Step 1: Calculate the Total Marks for Each Batch

Total marks for the first batch: (40 times 65 2600) Total marks for the second batch: (50 times 45 2250) Total marks for the third batch: (70 times 60 4200)

Step 2: Sum the Total Marks of All Batches

Total marks for all batches combined: (2600 2250 4200 9050)

Step 3: Calculate the Overall Average Marks

Total number of students: (40 50 70 160)

Overall average marks (frac{9050}{160} 56.5625)

Therefore, the average marks of all 160 students is approximately 56.56.

Alternative Approach: Using the Concept of Weighted Average

The approach of calculating a weighted average can be used to solve the same problem. The weighted average takes into account the number of students in each batch and their corresponding average scores.

Formulas:
[text{Weighted Average} frac{sum (xifi)}{sum fi}]Where (x) is the score and (f) is the number of students.

Conclusion

By using the weighted average method, we were able to accurately calculate the overall average marks of students from different batches. This method is particularly useful in scenarios where the groups have varying sizes or average scores. The process involves summing the total marks of all batches, dividing by the total number of students, and, in some cases, utilizing the weighted average formula for a more detailed analysis.