Arrange the following statistical measures in increasing order A. 37th…

Concept

Percentiles, deciles and quartiles are all quantiles — cut points that divide an ordered data set into equal-sized parts — so they can all be placed on one common scale.

On that scale the kth percentile Pk is the value below which k% of the observations lie. A quartile divides the data into 4 equal parts, so Qk = P25k; a decile divides it into 10 equal parts, so Dk = P10k; and the median is the central cut point, P50.

A quantile never decreases as its percentile rank rises, so comparing measures of position reduces to comparing their percentile ranks — no numerical data set is required.

Application

Express each of the four measures as a percentile rank:

1. 37th percentile — already stated on the percentile scale, so its rank is P37, i.e. 37.

2. First quartile — Q1 = P25 × 1 = P25, i.e. 25.

3. Median — it is the second quartile, Q2 = P25 × 2 = P50, i.e. 50.

4. Third decile — D3 = P10 × 3 = P30, i.e. 30.

Sorting the four ranks from smallest to largest gives 25 < 30 < 37 < 50. Reading the measures off in that order gives first quartile, then third decile, then 37th percentile, then median — the sequence B, D, A, C.

Cross-check

The ranks in the middle column increase down the table, confirming the order. The two boundary checks agree with intuition: the first quartile, cutting off only the lowest 25% of the data, must sit lowest, and the median, the halfway cut, must sit highest of the four. The common trap is to read "third decile" as a larger position than "37th percentile" because 3 sounds smaller than 37; converting both to the same scale (30 against 37) removes that confusion.