In the distribution of heights in a group of a thousand people, height p is at the 42nd percentile, height r is at the 48th percentile, height s is at the 54th percentile, and height t is at the 60th percentile. Quantity A: r-p Quantity B: t-s

Rationale

The two quantities are equal.

In the context of percentiles, the difference in height values between percentiles is consistent across intervals of the same width. Since both Quantity A (r - p) and Quantity B (t - s) represent the difference between heights at percentiles that are equally spaced apart (6 percentile points each), these quantities will yield the same numerical difference in height.

A (Quantity A is greater.) YOUR ANSWER

This choice suggests that the difference in height between the 48th percentile (r) and the 42nd percentile (p) is larger than the difference between the 60th percentile (t) and the 54th percentile (s). However, since both increments span the same range of height percentiles, this conclusion is incorrect.

B (Quantity B is greater.) YOUR ANSWER

This option implies that the height difference between the 60th percentile (t) and the 54th percentile (s) exceeds the difference between the 48th percentile (r) and the 42nd percentile (p). Like the previous choice, this fails to recognize that both differences correspond to equal percentile intervals, making this statement inaccurate.

C (The two quantities are equal.) CORRECT

This is the correct choice because both r - p and t - s represent height differences over intervals of 6 percentile points. Thus, their values are determined by the same relative spacing in the height distribution, ensuring they are indeed equal.

D (The relationship cannot be determined from the information given.) YOUR ANSWER

This choice suggests that there is insufficient information to compare the two quantities. However, the defined percentiles provide enough context to ascertain that both quantities are equal due to their equal interval nature, rendering this statement false.

Conclusion

In this scenario, since both Quantity A (r - p) and Quantity B (t - s) cover the same distance in the percentile distribution, they must necessarily be equal. Understanding percentile differences in height showcases how statistical measures can provide consistent comparisons, reaffirming the importance of recognizing equal intervals in such analyses.

In the distribution of heights in a group of a thousand people, height p is at the 42nd percentile, height r is at the 48th percentile, height s is at the 54th percentile, and height t is at the 60th percentile. Quantity A: r-p Quantity B: t-s

The two quantities are equal.

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