Percentile rank

A percentile rank runs from a value to a percent. On this site it counts the values at or below $x$ and divides by $n$, the number of observations, then multiplies by 100. The answer is a percent with no units, and it describes where a value sits inside one particular group rather than anything about the value itself.

Suppose 22 of the 25 students in a class scored at or below Maya's 84. Her percentile rank is $\frac{22}{25} \times 100 = 88$. Ties come along for the ride. In the five scores 70, 70, 70, 80, 80 each of the three 70s has three values at or below it, so all three have percentile rank $\frac{3}{5} \times 100 = 60$, and both 80s land at 100.

"I am at the 88th percentile, so I beat 88 percent of the class" is close, and not what the number says. The count is at or below, which includes Maya herself and anyone tied with her, so the honest reading is that 88 percent of the class scored 84 or lower. In a class of 25 the gap is one student. Where scores are heavily tied it is not small: the three students at 70 beat nobody and still carry a rank of 60.

Two consequences follow from counting at or below. The largest value always has a percentile rank of 100, and no value ever has a rank of 0, because every value counts itself: the lowest of 25 scores comes out at $\frac{1}{25} \times 100 = 4$. Other sources count only the values strictly below, or those below plus half the ties, and they give different numbers on the same data, so say which count you used.

A percentile rank is meaningful only against a stated reference group. Move that same 84 into a stronger class and the rank drops, though nothing about the score changed. On a smooth distribution the counting stops and the rank becomes an area under the curve, which is how a z-score converts into a percentile once the distribution is close enough to normal.