Average marks in Mathematics of Section A comprising 30 students is 65 and that of Section B comprising 35 students is 70. What are the average marks (approximately) of both the sections if it was detected later that an entry of 47 marks was wrongly made as 74 ?
67.28
Calculating Corrected Average Marks for Two Sections
Let's find the corrected average marks for both sections combined after a data entry error was found. We are given the average marks and the number of students for two sections, Section A and Section B.
Understanding the Initial Average Marks of Sections A and B
We have information for two groups of students:
- Section A has 30 students. Their average marks in Mathematics is 65.
- Section B has 35 students. Their average marks in Mathematics is 70.
To find the total marks for each section, we multiply the number of students by the average marks:
Total marks for Section A (\(T_A\)):
\(T_A = \text{Number of students in A} \times \text{Average marks in A}\)
\(T_A = 30 \times 65\)
\(T_A = 1950\)
Total marks for Section B (\(T_B\)):
\(T_B = \text{Number of students in B} \times \text{Average marks in B}\)
\(T_B = 35 \times 70\)
\(T_B = 2450\)
Combining Marks and Students Before Error Correction
Next, let's find the total number of students in both sections combined and the combined total marks based on the initial information (before correcting the error).
Total number of students (\(N\)):
\(N = \text{Number of students in A} + \text{Number of students in B}\)
\(N = 30 + 35\)
\(N = 65\)
Combined total marks recorded (\(T_{recorded}\)):
\(T_{recorded} = T_A + T_B\)
\(T_{recorded} = 1950 + 2450\)
\(T_{recorded} = 4400\)
Analyzing and Correcting the Data Entry Mistake
It was discovered that an entry of 47 marks was wrongly recorded as 74 marks. This error means the recorded total marks are higher than the actual total marks.
The difference between the wrong entry and the correct entry tells us how much the total sum was inflated:
\(\text{Error Difference} = \text{Wrong Entry} - \text{Correct Entry}\)
\(\text{Error Difference} = 74 - 47\)
\(\text{Error Difference} = 27\)
So, the recorded total marks (\(4400\)) are 27 marks higher than the actual total marks. To get the correct total marks, we must subtract this difference from the recorded total.
Corrected total marks (\(T_{corrected}\)):
\(T_{corrected} = T_{recorded} - \text{Error Difference}\)
\(T_{corrected} = 4400 - 27\)
\(T_{corrected} = 4373\)
Calculating the Corrected Combined Average
Now that we have the corrected total marks for both sections combined and the total number of students, we can calculate the corrected combined average marks.
Corrected Combined Average (\(\bar{x}_{combined, corrected}\)):
\(\bar{x}_{combined, corrected} = \frac{\text{Corrected Total Marks}}{\text{Total Number of Students}}\)
\(\bar{x}_{combined, corrected} = \frac{4373}{65}\)
Performing the division:
\(\frac{4373}{65} \approx 67.2769\)
The question asks for the approximate average marks. Rounding the result to two decimal places, we get 67.28.
Therefore, the corrected average marks of both sections combined is approximately 67.28.
Revision Table: Corrected Average Calculation Concepts
| Concept | Calculation | Value |
|---|---|---|
| Total Students (Both Sections) | \(30 + 35\) | 65 |
| Total Marks Section A (Recorded) | \(30 \times 65\) | 1950 |
| Total Marks Section B (Recorded) | \(35 \times 70\) | 2450 |
| Combined Total Marks (Recorded) | \(1950 + 2450\) | 4400 |
| Error in Entry | \(74 - 47\) | 27 |
| Corrected Combined Total Marks | \(4400 - 27\) | 4373 |
| Corrected Combined Average | \(\frac{4373}{65}\) | \(\approx 67.28\) |
Additional Information: Understanding Averages and Data Errors
An average, or arithmetic mean, is a fundamental concept in statistics used to represent a typical value in a dataset. It's calculated as the sum of all values divided by the number of values.
\(\text{Average} = \frac{\text{Sum of values}}{\text{Number of values}}\)
When dealing with combined groups, the overall average is the total sum of all values across all groups divided by the total number of items in all groups.
Data entry errors are common and can skew statistical calculations like the average. If a value is entered incorrectly, the sum of values changes, which in turn affects the average.
To correct an average affected by a single incorrect entry:
- First, determine the total sum based on the incorrect data.
- Calculate the difference between the wrong value and the correct value.
- Adjust the total sum: subtract the difference if the wrong value was too high, or add it if the wrong value was too low.
- Recalculate the average using the corrected total sum and the correct count of items.
In this problem, the total number of students remained constant (65), but the total sum of marks needed adjustment due to the error. We successfully corrected the total sum and then computed the new average.
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