Sample quantiles in statistical packages

 

There are a large number of different definitions used for sample quantiles in statistical computer packages. Often within the same package one definition will be used to compute a quantile explicitly while other definitions may be used when producing a boxplot, a probability plot or a QQ-plot. We compare the most commonly implemented sample quantile definitions by writing them in a common notation and investigating their motivation and some of their properties. We argue that there is a need to adopt a standard definition for sample quantiles so that the same answers are produced by different packages and within each package. We conclude by recommending that the median-unbiased estimator is used since it has most of the desirable properties of a quantile estimator and can be defined independently of the underlying distribution.

Keywords: sample quantiles, percentiles, quartiles, statistical computer packages.

R code: The quantile() function in R from version 2.0.0 onwards implements all the methods in this paper.

Errata:

• Table 1, p361. P2 should have lower bound equal to \lfloor np\rfloor.
• p362, right column. The upper-tail frequency for Definition 3 should be \text{Freq}(X_k \ge \hat{Q}_3(1 - p)) = \begin{cases} \lceil pn + \frac{1}{2}\rceil + 1 & \text{if } g = 0 \text{ and } \lfloor (1 - p)n\rfloor \text{ even}, \\ \lceil pn + \frac{1}{2}\rceil & \text{otherwise.} \end{cases}
• p363, left column. The upper-tail frequency for Definitions 4–9 should be \text{Freq}(X_k \ge \hat{Q}_i(1 - p)) = \lfloor pn + \beta + p(1 - \alpha - \beta)\rfloor, because m must be evaluated at 1 - p. The printed expression is correct only for Definitions 4 and 5, where m does not depend on p.
• p363, left column. P2 is satisfied if and only if \alpha \ge 0 and \beta \le 1.
• p363, left column. P3 is satisfied for all p if and only if \alpha = \beta.
• p364, Table 3. Definitions 6–9 also satisfy P3.
• p364, right column. \hat{Q}_5(p), \hat{Q}_6(p), \hat{Q}_8(p) and \hat{Q}_9(p) satisfy all six properties, and \hat{Q}_7(p) satisfies five (all except P5).

The frequency expressions on pp.362–363 assume distinct observations. With tied observations, P3 can fail for every definition; for example, x = (1, 2, 2) with p = 0.5.

Thanks to Eric Langford and Alan Dorfman for pointing out the errors concerning P2 (8 May 2007), to Ati Ghoreyshi for pointing out the inconsistency between Table 3 and the text on p364 (27 April 2022), and to Takanori Kawabata for pointing out the errors concerning Definition 3 and P3 (3 October 2026).

For further discussion, see Sample quantiles 20 years later on Hyndsight.

Cite

@article{HF96,
  author  = {Rob J Hyndman and Yanan Fan},
  doi     = {10.1080/00031305.1996.10473566},
  journal = {The American Statistician},
  number  = {4},
  pages   = {361--365},
  title   = {Sample quantiles in statistical packages},
  volume  = {50},
  year    = {1996}
}