Quantiles, percentiles and quartiles

The p-th quantile is a value with about a proportion p of the data below it; software differs in how it interpolates, and Statistics Quest uses Hyndman–Fan type 7 by default.

Q(p) = x_{(j)} + g\,\bigl(x_{(j+1)} - x_{(j)}\bigr), \quad h = (n-1)p + 1,\ j = \lfloor h \rfloor,\ g = h - j

Type 7 (default; R, NumPy, Excel PERCENTILE.INC / QUARTILE.INC).

$p$: Probability between 0 and 1 (the 90th percentile is $p = 0.9$ ).
$x_{(j)}$: The $j$ -th smallest value.
$h$: Position in the sorted data; $j$ is its whole part, $g$ its fractional part.

h = (n+1)p

Type 6 (Excel PERCENTILE.EXC / QUARTILE.EXC, Minitab): same interpolation, different position.

$h$: Position; values beyond the ends are clamped to the minimum or maximum.

\text{IQR} = Q_3 - Q_1 = Q(0.75) - Q(0.25)

Interquartile range.

$Q_1,\ Q_3$: First and third quartiles.

When to use

Describing position ("the 90th percentile of delivery time is 41 minutes"), spread (IQR), pay bands, risk quantiles (VaR-style).
Comparing distributions with different shapes; quantiles do not assume symmetry.
The engine offers all nine Hyndman–Fan types because software disagrees; state the type when the exact number matters.

How to interpret

$Q_1$ , the median and $Q_3$ split the sorted data into four parts with about 25% each. The IQR is the spread of the middle half and is not affected by the most extreme quarter on each side. A percentile rank is the reverse question: what percentage of values lie below a given value?

Common mistakes

Expecting identical quartiles from every program: with small samples type 6 and type 7 can differ noticeably. Neither is "wrong".
Computing quantiles of unsorted data.
Confusing a percentile (a value in data units) with a percentage.

Example

Data 7, 15, 36, 39, 40, 41 ( $n = 6$ ). Type 7: $Q_1 = 20.25$ , median 37.5, $Q_3 = 39.75$ , IQR = 19.5, 90th percentile 40.5. Type 6 (Excel .EXC): $Q_1 = 13$ , $Q_3 = 40.25$ , 90th percentile 41. Same data, different conventions.