Abstract
The arithmetic mean computed over ordinal data has to be interpreted with care. Instead it is often recommended to compute the sample median, or more generally sample quantiles. The virtue of raw quantiles is that they are not affected by an arbitrary rescaling of the data. Unfortunately, they are not very informative when an ordinal variable falls into only few categories. The informational content is larger for so-called grouped quantiles interpolating around one possible value. In this note we show, however, that they turn out to be not invariant with respect to strictly monotonous transformations as long as the possible outcomes are not equidistant. This motivates to suggest the new centered interpolated quantiles. They are designed to be invariant with respect to transformations that preserve the ranking of the possible values.
Similar content being viewed by others
References
Abberger K (1997) Quantile smoothing in financial time series. Stat Pap 38:125–148
Burke CJ (1953) Additive scales and statistics. Psychol Bull 60:73–75
Carifio J, Perla RJ (2007) Ten common misunderstandings, misconceptions, persistent myths and urban legends about likert scales and likert response formats and their antidotes. J Soc Sci 3:106–116
Hassler U (2018) Sample quantiles for ordinal data, Social Science Research Network Working Paper, https://ssrn.com/abstract=3237945
Hays WL (1963) Statistics for psychologists. Holt, Rinehart & Winston, New York
Hyndman RJ, Fan Y (1996) Sample quantiles in statistical packages. Am Stat 50:361–365
Jamieson S (2004) Likert scales: how to (ab)use them. Med Educ 38:1217–1218
Likert R (1932) A technique for the measurement of attitudes. Arch Psychol 22:5–55
Norman G (2010) Likert scales, levels of measurements and the “laws” of statistics. Adv Health Sci Educ 15:625–632
Senders VL (1953) A comment on Burke’s additive scales and statistics. Psychol Bull 60:423–424
Zhang Q, Yang W, Hu S (2014) On Bahadur representation for sample quantiles under \(\alpha \)-mixing sequence. Stat Pap 55:285–299
Author information
Authors and Affiliations
Corresponding author
Additional information
Uwe Hassler thanks two anonymous referees, Mehdi Hosseinkouchack and Marc-Oliver Pohle for helpful comments.
Rights and permissions
About this article
Cite this article
Hassler, U. Note on sample quantiles for ordinal data. Stat Papers 61, 2383–2391 (2020). https://doi.org/10.1007/s00362-018-1054-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-018-1054-5