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Fifteen candidates appeared in an examination. The marks of the candidates who passed in the examination are 9, 6, 7, 8, 8, 9, 6, 5, 4 and 7. What is the median of marks of all the fifteen candidates?

This question was previously asked in
CDS I 2020 Elementary Mathematics Previous Year Paper (02-Feb-2020)
The correct answer is

6

This problem requires us to find the median value of the marks obtained by all fifteen candidates in an examination. We are given the marks of only those candidates who passed.

Median Calculation Explained

The median represents the middle value in a dataset when the data points are arranged in ascending or descending order. If there is an odd number of data points (\(N\)), the median is the value at the middle position, which is calculated using the formula \(((N+1)/2)\). If there is an even number of data points (\(N\)), the median is the average of the two middle values located at positions \((N/2)\) and \(((N/2)+1)\).

Identifying All Candidate Marks

  • Total Candidates: 15
  • Marks of Passed Candidates (10): 9, 6, 7, 8, 8, 9, 6, 5, 4, 7
  • Number of Candidates who Passed: 10
  • Number of Candidates who Failed: Calculated as Total Candidates - Passed Candidates = \(15 - 10 = 5\).
  • Marks of Failed Candidates: The specific marks for the 5 failed candidates are not provided in the question. When dealing with such missing information for failed candidates while needing to find the median for the entire group, a common and reasonable assumption is that they scored 0 marks.
  • Combined Marks List: Based on this assumption, the complete list of marks for all 15 candidates is {9, 6, 7, 8, 8, 9, 6, 5, 4, 7, 0, 0, 0, 0, 0}.

Sorting All Candidate Marks

To determine the median, the first step is to arrange the combined list of all 15 marks in ascending order:

  • Sorted Marks: 0, 0, 0, 0, 0, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9

Determining the Median Mark

We have a total of \(N = 15\) data points (marks). Since \(N\) is an odd number, the median is the single value located exactly in the middle of the sorted dataset.

  • Median Position Calculation: The position of the median is found using the formula \(((N+1)/2)\).
  • Substituting \(N=15\): Position = \(((15+1)/2) = (16/2) = 8\). This means the median is the 8th value in our sorted list.
  • Finding the Median Value: We locate the 8th value in the sorted list of marks.

Let's identify the 8th value from the sorted list:

0, 0, 0, 0, 0, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9

The 8th value in this ordered list is 6.

Conclusion

Based on the analysis and the assumption for the marks of the failed candidates, the median of the marks of all fifteen candidates is 6.

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