The average of numbers 6, 4, 3, 8, 10, 11, 10, 5 and x is 7.2. The median of numbers 11, 10, y, 8, 6, 10, x and 4 is 8.5. What is the value of y?

Appeared in: CGHS SSC - 2018

Explanation

The problem requires a two-step calculation using fundamental statistical concepts.
First, the value of 'x' is determined using the formula for the average (mean): Mean = (Sum of observations) / (Number of observations). The sum is 7.2 * 9 = 64.8. Subtracting the sum of known numbers (57) gives x = 7.8.
Second, the value of 'y' is found using the definition of a median for an even-sized dataset. The numbers are {4, 6, 7.8, 8, 10, 10, 11, y}.
For 8 numbers, the median (8.5) is the average of the 4th and 5th terms. So, (4th term + 5th term) = 17.
After sorting the known numbers, the 4th term is identified as 8. Therefore, the 5th term must be 17 - 8 = 9. This means y = 9.

Option A: The value 10 is incorrect. If y were 10, the sorted list would be {4, 6, 7.8, 8, 10, 10, 10, 11}. The median would be (8 + 10) / 2 = 9, which is not 8.5.
Option C: The value 9.5 is incorrect. If y were 9.5, the median would be (8 + 9.5) / 2 = 8.75, not 8.5.
Option D: The value 8 is incorrect. If y were 8, the sorted list would be {4, 6, 7.8, 8, 8, 10, 10, 11}. The median would be (8 + 8) / 2 = 8, which is not 8.5.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Calculation of Mean and Median as background academic context rather than a clinical decision trigger.
This question assesses foundational mathematical and logical reasoning skills, which are part of the non-nursing syllabus for nursing entrance exams.
While not directly a clinical skill, the ability to perform accurate, multi-step calculations is crucial in nursing for tasks like medication dosage calculation, IV drip rate monitoring, and interpreting patient data.
What if the number of observations in the second set was odd (e.g., 9 numbers)? The median would simply be the single middle term (the 5th term), and the calculation would change accordingly.

How to Approach the Question

Identify that this is a two-part problem. You cannot find 'y' without first finding 'x'.
Step 1: Focus on the first sentence. Use the definition of average (mean) to write an equation and solve for 'x'.
Step 2: Substitute the calculated value of 'x' into the second set of numbers.
Recall the rule for finding the median of a dataset with an even number of terms: it's the average of the two middle terms after sorting.
Set up an equation using the given median (8.5) and solve for the unknown value 'y'.
Always double-check your answer by placing your calculated 'y' value back into the set and re-calculating the median to ensure it matches the value given in the problem.

Concept Tested & Keywords

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Practise the full CGHS SSC - 2018

Attempt every question from this paper in a timed mock, then review the full solution for each one.