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Topic: levels of meas.
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Bob Hayden

Posts: 2,384
Registered: 12/6/04
levels of meas.
Posted: Aug 21, 1996 6:34 PM
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I blow hot and cold on the levels of measurement stuff discussed
recently on EdStat-L. On the one hand, you DO need to pay attention
to what kind of data you have in deciding what technique to use. I
think of the wonderful example in Brase and Brase where they ask the
students to do a chi-square test on a table in which the cells give
weights of nuts rather than counts (so you can get any level of
significance or nonsignificance you want by simply changing the units
of weight). On the other hand, there are other circles where "levels
of measurement" seems to have become an esoteric religion, quite
contrary to science in its outlook on the world.

With my own students, I like to distinguish between what I call
categorical and measurement data, which I present as two ends of a
continuum. Each technique presented comes with some clue as to the
type of data it is intended for. In the one-variable case, we have

for measurement data

means, medians, modes (maybe), variances, ranges, standard
deviations, quantiles, stem and leaf, histogram, boxplot, etc.

for categorical data

counts, percentages, barcharts, pie charts, etc.

The ideas are even more important when we look at relationships
between variables.

For relationships between two measurement variables

regression and correlation

For relationships between two categorical variables

chi-squared tests on contingency tables

For a measurement variable depending on a categorical variable

analysis of variance

While this is only a first approximation as to what to use when, most
textbooks DO NOT GIVE THIS INFORMATION! (For some ideas on why, see
A. Toom's article in the August MAA Focus.) But, while it is
important to give some guidelines, it is also important to understand
that there are many exceptions. Perhaps an extreme one is to use an
ordinary t-test with data placed in two categories and coded 0-1. In
most cases, you will get the same result as if you had applied the
binomial distribution, and there are good underlying reasons why this
is so. Yet we are applying a method from one end of the spectrum to
data from the other.

| | Robert W. Hayden
| | Department of Mathematics
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