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On 10/13/13 3:40 PM, Simon Spero wrote:
I took my stats classes in the psychology department
at UNC- the Stevens typology was not always considered gospel
(esp. in regards to taking means of ordinal scales).
I assume you mean that some considered the mean of an ordinal scale
as an appropriate statistic (whereas the gospel would be that only
the mode and median are meaningful)?
The mean is often taken for ratings scales, which are in general
ordinal but if carefully designed can reasonably be considered as
approximately interval . Krantz et al discuss that briefly.
Tara
Simon // Not under Thurston :-)
On Oct 13, 2013 3:20 PM, "Tara Athan"
< taraathan@xxxxxxxxx>
wrote:
A classical
reference regarding concepts in metrology is Krantz et al
http://www.amazon.com/Foundations-Measurement-Volume-Representations-Mathematics/dp/0486453146
.
Since it is not downloadable, I'll attempt to give a summary
of the
points I think are relevant to the development of ontologies
of measurement.
The foundational concepts employed in this reference are
* a "relational structure", consisting of a set of
mathematical entities
(such as positive real numbers) and one or more relations
among them.
* A "numerical assignment" is a homomorphism that maps
physical entities
and measurement procedures into a relational structure in a
way that the
properties of the procedures are preserved.
From these foundations, it can be shown that measurement
theories
corresponding to particular relational structures may be
grouped into
families according to the "permissible transformations" that
may be
applied to a numerical assignment that result in another
numerical
assignment. The familiar Stevens scale types correspond to
certain families:
* Ratio scale: entities are 1-dimensional (e.g. real or
positive real
numbers), and permissible transformation are only
multiplication by a
positive scalar constant. Extensive measurements such as
length fall
into this family.
* Log-Interval scale: entities are again 1-dimensional, but
typically
only the positive real numbers, and permissible
transformations include
raising to a power in addition to multiplication by a positive
scalar
constant. Intensive measurements such as absolute temperature
and
density fall into this family. It's called log-interval
because taking
the log gives a theory in the "interval-scale" family, and
Stevens
recognized its existence in his original work, although it is
usually
not included in elementary treatments today.
and so on. For ordinal-scale measurements, permissible
transformations
include any strictly monotonic function.
However, there are plenty of measurement theories that do not
fit into
these families, notably, multi-dimensional theories.
Tara
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