relative vs. absolute

[James A. Landau <JJJRLandau@netscape.com>]

On Wed, 3 Jun 2009 11:52:13 at the ever-popular Zulu minus 0400
Mark Mandel thnidu at GMAIL.COM wrote:

<quote>
"Relative" here is being used for an ordinal scale, which is more
meaningful than a nominal one but less than an interval or ratio
scale. There are four classes of value: nominal, ordinal, interval,
and ratio, in ascending order of rigor. (My mnemonic for them is
"noir".)

Nominal: Each value has a name, but there is no order or comparison
between them. E.g., for "nationality" you might have values like
France, UK, Trinidad and Tobago, Japan, USA, Philippines. I'd be
unwilling to call such a classification a "scale" at all.

Ordinal: The values are ordered, but there is no arithmetic comparison
between them. For example, cancer is "staged" in increasing order of
development and seriousness as I, II, III, IV, with substages like
IIIa and IIIb, but it's meaningless to ask whether Stage III is twice
as far from Stage I as it is from Stage II. The Mohs scale of hardness
is a classic example of an ordinal scale.

Interval: The values are ordered and can be subtracted, but not added,
multiplied or divided. The interval between Sept. 12, 2000 and Sept.
12, 2009 is 9 years, but "Sept. 12, 2000 + Sept. 12, 2009" is
meaningless. Similarly,   Jan. 1, 2000 isn't twice as (anything) as
Jan. 1, 1000,   100 degrees Fahrenheit isn't twice as hot as 50, and
40 degrees west longitude isn't four times as west as 10 degrees west,
except as measured from their scales' arbitrary zeros.

Ratio: The values can be treated as numbers: meaningfully subtracted,
added, multiplied or divided. $6 is twice as much as $3,   300 degrees
Kelvin is three times as hot as 100, and a person of 60 is twice as
old as a person of 30.   40 degrees north latitude is twice as far
north as 20 degrees north (and the equator, unlike the Greenwich
meridian, is not arbitrary).

Google gives about 18k hits for /+nominal +ordinal +integer +ratio/.
This one (http://tinyurl.com/qskmrh) is helpful:

Data mining: practical machine learning tools and techniques with Java
implementations
By Ian H. Witten, Eibe Frank   Edition: 2
Published by Morgan Kaufmann, 2000   ISBN 1558605525
</quote>

Quite a good answer, but two comments:

1.      "100 degrees Fahrenheit isn't twice as hot as 50" is correct, but 100 degrees Rankine *is* twice as hot as 50 degrees Rankine.  That’s because the Rankine scale uses Fahrenheit-sized degrees but its zero is set at absolute zero, thereby making it a Ratio rather than an Interval scale.  Similarly, as you point out, degrees Kelvin (degrees Celsius with zero at absolute zero) is a ratio scale.  See below my signoff.

2.      There is a fifth kind of scale, namely logarithmic.  Decibels are a logarithmic scale.  53 decibels is twice as loud (or twice as strong) as 50 decibels, and 47 decibels is half as loud/strong as 50 decibels.  E.g. electrical types speak of a “3-decibel drop” when they talk about splitting a current into two, as when one wire comes in and two go out.  Other logarithmic scales you might be familiar with include the Richter scale for earthquakes, the pH scale for acid/base strength in chemistry, Malthus (who said food production can be measured on a ratio scale and population on a logarithmic), and in music octaves.

It is possible to extend this list beyond logarithmic, but with the exception of power towers (e.g. one, googol, googolplex…) I can’t recall ever having seen a power tower scale in use, and even in advanced mathematics power towers are quite rare.

   James A. Landau
   Test Engineer
   Northrop-Grumman Information Technology
   8025 Black Horse Pike, Suite 300
   West Atlantic City  NJ  08232  USA

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