Sample Quantiles in Statistical Packages. Hyndman, R. J. & Fan, Y. 50(4):361–365.
Sample Quantiles in Statistical Packages [link]Paper  doi  abstract   bibtex   
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 be used because it has most of the desirable properties of a quantile estimator and can be defined independently of the underlying distribution.
@article{hyndmanSampleQuantilesStatistical1996,
  title = {Sample Quantiles in Statistical Packages},
  author = {Hyndman, Rob J. and Fan, Yanan},
  date = {1996-11},
  journaltitle = {The American Statistician},
  volume = {50},
  pages = {361--365},
  issn = {1537-2731},
  doi = {10.1080/00031305.1996.10473566},
  url = {https://doi.org/10.1080/00031305.1996.10473566},
  abstract = {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 be used because it has most of the desirable properties of a quantile estimator and can be defined independently of the underlying distribution.},
  keywords = {*imported-from-citeulike-INRMM,~INRMM-MiD:c-12889338,~to-add-doi-URL,algorithms,comparison,definition,mathematical-reasoning,mathematics,robust-modelling,statistics},
  number = {4}
}
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