Note on sample quantiles for ordinal data
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Note on sample quantiles for ordinal data Uwe Hassler1 Received: 11 January 2018 / Revised: 25 September 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
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 socalled 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. Keywords Likert response format · Interpolated quantiles · Invariance under strictly monotonous transformations Mathematics Subject Classification 62-07 · 62P10 · 62P15 · 62P20 · 62P25
1 Introduction Observations collected from questionnaires are often ordinal. Ordinal data arise e.g. from so-called Likert response format items in medical, psychometric or socio-political questionnaires that also play a major role in business or marketing, and are frequently encountered in everyday life. In such a context, the variable A measures the attitude or degree of agreement where we allow here for only four items for simplicity: A:
“strongly disagree” , “disagree” , “agree” , “strongly agree” .
Uwe Hassler thanks two anonymous referees, Mehdi Hosseinkouchack and Marc-Oliver Pohle for helpful comments.
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Uwe Hassler [email protected] Goethe University Frankfurt, Campus Westend, RuW Building, 60629 Frankfurt, Germany
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U. Hassler
It is sometimes argued that an equidistant variable, X ∈ {1, 2, 3, 4}, is the most natural choice to reflect A; but one may equally argue that the distance between “agree” and “disagree” is larger than between “agree” and “strongly agree”, and consequently choose e.g. Y ∈ {0, 3, 7, 10} in order to represent A. The Likert response format can be traced back to Likert (1932). More recently, there has been a vivid debate on what kind of statistical methods are appropriate for such Likert data, see Jamieson (2004), Carifio and Perla (2007) and Norman (2010). While “it does not take a lot of thought to recognize that Likert scales are ordinal” (Norman 2010, p. 627), the controversy is about robustness, i.e. the validity of statistical tests that have been designed for interval data. In this note we are not interested in the validity of statistical inference for ordinal data, but only in the appropriateness of descriptive methods. And we agree with the classical textbook view stressed by Jamieson (2004, p. 1217) that “The mean (and standard deviation) are inappropriate for ordin
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