Quantile
A value that divides a distribution into equal-sized groups — quartiles, deciles, and percentiles are the common cases.
The classic item gives you a small ordered data set and asks for a specific percentile, so the whole problem is the position formula: the location of the yth percentile is L_y = (n + 1)(y/100), where n is the number of observations. The “tell” is a non-integer answer — say L = 3.75 — which means you linearly interpolate between the 3rd and 4th ordered values, adding 0.75 of the gap to the 3rd value. The trap is grabbing the value at position 3.75 instead of interpolating, or forgetting the +1 (it’s n+1, not n). Order the data smallest-to-largest first; an unsorted list is the setup for a wrong answer.
Distinguish the quantile (a position-based cut point) from the median, which is just the 50th percentile (the second quartile), and from probability, a 0-to-1 likelihood measure rather than a data value. Students conflate the quartile number with the quartile value — Q1 is the observation at the 25th percentile, not “25.” Memory hook: q is for “quarter-style cuts,” any equal slice you choose.
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