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Difficulty: HardPercentiles and Quartiles

A data set consists of 6060 distinct test scores arranged in ascending order. The 70th70\text{th} percentile of this data set is equal to the 42nd42\text{nd} score. If 1515 new test scores, all strictly lower than the lowest score in the original data set, are added to form a new data set of 7575 scores, what is the percentile rank of the score that was the 70th70\text{th} percentile of the original data set?

Answer: 76

Answer

76
The target score originally occupied position 42 in an ordered set of 60 values. When 15 values strictly smaller than all original values are introduced, they occupy the first 15 positions in the new ordered set. Consequently, the target score shifts to position 42 + 15 = 57 in the new set of 75 values. The percentile rank corresponds to the percentage of values at or below this position, calculated as (57 / 75) * 100% = 76%.

Step-by-Step Solution

1
Determine the rank position of the original 70th percentile score
The target score is at position 42 in the original sorted set of 60 scores.
The stem specifies that the 70th percentile corresponds to the 42nd score.
2
Calculate the target score's position in the expanded data set
The target score is now at position 57 in the new sorted set of 75 scores.
Since 15 scores smaller than the original minimum are prepended, every original score's rank position increases by 15. Thus, position 42 becomes position 42 + 15 = 57.
3
Compute the percentile rank of position 57 out of 75
76
The percentile rank is the percentage of values less than or equal to this score in the new set: (57 / 75) * 100% = 76%.

Key Concept

Percentile Rank and Rank Position Shifts
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Topics

Percentiles and Quartiles

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