Measures of dispersion for ordinal data

[Bob Schacht-3]

For ordinal data, "dispersion" has an intuitive sense: "clustering" around
a "mode" provides a model of "central tendency"; a "uniform" distribution
implies an absence of any such clustering, and a platykurtic or bimodal
distribution forms the ideological opposite of "central tendency." These
measures are all well understood for ratio data, but are less well known
(at least to me) for ordinal data. What quantitative measures are available
for these concepts, given ordinal but not ratio data? Well, the Mode is
easily understood (i.e., most frequent category). But what about the rest?

dispersion
clustering
central tendency
Uniform distribution (Kolmogorov-Smirnoff test with equal probabilities for
every category?)
Bimodal distribution

Better yet, is there a standard set of statistics for describing the
dispersion of ordinal data?

Thanks,
Bob Schacht

Robert M. Schacht, Ph.D. <[hidden email]>
Pacific Basin Rehabilitation Research & Training Center
1268 Young Street, Suite #204
Research Center, University of Hawaii
Honolulu, HI 96814

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[valferes]

Bob,

 

You can find measures of symmetry and kurtosis based in quantiles in the site:

 

 

http://www.fpce.uc.pt/niips/spss_prc/desc_dist/sc_quantis/sc_quantis.htm

 

 

(Note: the instructions are in Portuguese, but the syntaxes will work!).

 

Valentim

 

 

http://www.fpce.uc.pt/niips/spss_prc/index.htm

 

---

From: SPSSX(r) Discussion on behalf of Bob Schacht
Sent: Thu 2009-03-12 00:17
To: [hidden email]
Subject: Measures of dispersion for ordinal data

For ordinal data, "dispersion" has an intuitive sense: "clustering" around
a "mode" provides a model of "central tendency"; a "uniform" distribution
implies an absence of any such clustering, and a platykurtic or bimodal
distribution forms the ideological opposite of "central tendency." These
measures are all well understood for ratio data, but are less well known
(at least to me) for ordinal data. What quantitative measures are available
for these concepts, given ordinal but not ratio data? Well, the Mode is
easily understood (i.e., most frequent category). But what about the rest?

dispersion
clustering
central tendency
Uniform distribution (Kolmogorov-Smirnoff test with equal probabilities for
every category?)
Bimodal distribution

Better yet, is there a standard set of statistics for describing the
dispersion of ordinal data?

Thanks,
Bob Schacht

Robert M. Schacht, Ph.D. <[hidden email]>
Pacific Basin Rehabilitation Research & Training Center
1268 Young Street, Suite #204
Research Center, University of Hawaii
Honolulu, HI 96814

=====================
To manage your subscription to SPSSX-L, send a message to
[hidden email] (not to SPSSX-L), with no body text except the
command. To leave the list, send the command
SIGNOFF SPSSX-L
For a list of commands to manage subscriptions, send the command
INFO REFCARD

[Bob Schacht-3]

At 05:26 PM 3/11/2009, Valentim R. Alferes wrote:

Bob,

You can find measures of symmetry and kurtosis based in quantiles in the site:
http://www.fpce.uc.pt/niips/spss_prc/desc_dist/sc_quantis/sc_quantis.htm
(Note: the instructions are in Portuguese, but the syntaxes will work!).
Valentim

Thank you. The proposed measure of kurtosis is Moors Coefficient, which is defined for Octiles.
In English, there is a review of this coefficient by Kim & White (2003)
http://econ.ucsd.edu/~mbacci/white/pub_files/hwcv-092.pdf and the Moors Coefficient itself was published as
Moors, J. J. A. (1988), "A Quantile Alternative for Kurtosis," The Statistician, 37, 25-32. Moors has gone on to participate in a more general discussion, "Characterizing systems of distributions by quantile measures," with 4 co-authors, in 1996, in

Statistica Neerlandica,

Volume 50 Issue 3, Pages 417 - 430, which suggests a number of useful ways to proceed.

Thanks,
Bob