To find the median of the distribution of marks based on the given data, we first need to analyze the information provided about the candidates' performance in the examination. Let's break down the details step by step.
Understanding the Distribution of Candidates
We have a total of 100 candidates, and the performance can be categorized as follows:
- 21 candidates failed (below 40% marks)
- 6 candidates secured distinctions (75% marks and above)
- 43 candidates were placed in the third division (40% to below 50%)
- 18 candidates were in the second division (50% to below 60%)
- The remaining candidates must be in the first division (60% to below 75%)
Calculating the Number of Candidates in Each Division
From the total of 100 candidates, we can determine how many are in the first division:
- Failed: 21
- Third Division: 43
- Second Division: 18
- Distinctions: 6
Now, let's calculate the number of candidates in the first division:
First Division Candidates = Total Candidates - (Failed + Third Division + Second Division + Distinctions)
First Division Candidates = 100 - (21 + 43 + 18 + 6) = 100 - 88 = 12
Summary of Candidates by Division
Now we have the following distribution:
- Failed: 21 candidates
- Third Division: 43 candidates
- Second Division: 18 candidates
- First Division: 12 candidates
- Distinctions: 6 candidates
Finding the Median
The median is the middle value in a sorted list of numbers. Since we have 100 candidates, the median will be the average of the 50th and 51st candidates when arranged in order of their marks.
To find the positions of the 50th and 51st candidates:
- First 21 candidates (failed) have marks below 40%.
- Next 43 candidates (third division) have marks between 40% and 49.99% (positions 22 to 64).
- Next 18 candidates (second division) have marks between 50% and 59.99% (positions 65 to 82).
- Next 12 candidates (first division) have marks between 60% and 74.99% (positions 83 to 94).
- Last 6 candidates (distinctions) have marks 75% and above (positions 95 to 100).
From this distribution:
- The 50th candidate falls in the third division (marks between 40% and 49.99%).
- The 51st candidate also falls in the third division.
Conclusion on the Median Marks
Since both the 50th and 51st candidates are in the third division, we can conclude that the median marks are within the range of 40% to 49.99%. For a more precise value, we could assume a midpoint of this range, which is approximately 45%. Thus, the median of the distribution of marks is around 45%.
For practice, consider a similar scenario: If there were 120 candidates with 30 failing, 10 securing distinctions, 50 in the third division, and 20 in the second division, how would you calculate the median marks? Think through the distribution and apply the same logic!
