If the median of the data 30, 8, 7, 3, 17, 15, 21, 24, 29, 23 is x and the…
2024
If the median of the data 30, 8, 7, 3, 17, 15, 21, 24, 29, 23 is x and the median of the data obtained by replacing 3 by 33 and 8 by 18 in the above data is y, then what is the difference between y and x?
- A.
1
- B.
2
- C.
3
- D.
4
Attempted by 1 students.
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Correct answer: C
The median of a data set is the middle value when all observations are arranged in ascending order. For an even number of observations, the median is the average of the two middlemost terms — the (n/2)-th and (n/2+1)-th terms after sorting.
Arrange the original 10 values in ascending order: 3, 7, 8, 15, 17, 21, 23, 24, 29, 30.
With n = 10 (even), the median x is the average of the 5th and 6th terms: x = (17 + 21) / 2 = 19.
Replace 3 with 33 and 8 with 18 in the original data, then re-sort: 7, 15, 17, 18, 21, 23, 24, 29, 30, 33.
The median y of this new data is again the average of the 5th and 6th terms: y = (21 + 23) / 2 = 22.
Therefore, y − x = 22 − 19 = 3.
Cross-check: in the original sorted list, exactly 6 values are less than or equal to 21 (the 6th term), confirming its rank; in the new sorted list, exactly 6 values are less than or equal to 23 (the new 6th term), confirming its rank too. Both medians are therefore correctly identified.
Hence, the difference between y and x is 3.