relative vs. absolute
Mark Mandel
thnidu at GMAIL.COM
Fri Jun 5 18:32:59 UTC 2009
On Fri, Jun 5, 2009 at 1:25 PM, James A. Landau
<JJJRLandau at netscape.com> <JJJRLandau at netscape.com> wrote:
>
> Quite a good answer, but two comments:
Thanks.
> 1. Â Â Â "100 degrees Fahrenheit isn't twice as hot as 50" is correct, but 100 degrees Rankine *is* twice as hot as 50 degrees Rankine. Â Thatâ€™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. Â Similarly, as you point out, degrees Kelvin (degrees Celsius with zero at absolute zero) is a ratio scale. Â See below my signoff.
Good point. That's part of why I specified Fahrenheit, but the
Fahrenheit and Kelvin scales differ on two dimensions (size of
interval and zero point), which damages the comparison.
> 2. Â Â Â There is a fifth kind of scale, namely logarithmic. Â Decibels are a logarithmic scale. Â 53 decibels is twice as loud (or twice as strong) as 50 decibels, and 47 decibels is half as loud/strong as 50 decibels. Â E.g. electrical types speak of a â€œ3-decibel dropâ€ when they talk about splitting a current into two, as when one wire comes in and two go out. Â 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.
Nice. I knew that, of course, but I never connected them with the
four-part classification... which is itself an ordinal scale :-). I
would draw a distinction, though, with the musical example:
Acoustically (in terms of waveform: frequency) the musical scale --
the octave and its divisions -- is a logarithmic scale, but auditorily
(in terms of perception: pitch) it is an interval scale: from middle C
to G is a fifth, just the same as from C' to G' (in the octave above
middle C) and from C, to G, (in the octave below). Similarly for
decibels with amplitude vs. loudness.
> It is possible to extend this list beyond logarithmic, but with the exception of power towers (e.g. one, googol, googolplexâ€¦) I canâ€™t recall ever having seen a power tower scale in use, and even in advanced mathematics power towers are quite rare.
Stopping to think about it -- a bad idea! -- the log and power tower
types of scale seem to veer off in a different direction than the
nominal - ordinal - interval - ratio sequence. Each type in N-O-I-R
allows more kinds of operation than the ones before it, but a log
scale is just a different way of looking at a ratio scale, more
convenient to us because it compresses the larger orders of magnitude.
What operations are possible on a log scale that aren't possible on a
ratio scale?
In fact, can you do anything with a log scale *except* compare levels
and intervals? All these examples have a reference value, an arbitrary
zero point like the Greenwich meridian for longitude:
- pure water at some temperature for pH
- "one micrometre on a seismograph recorded using a Wood-Anderson
torsion seismometer 100 kilometres (62 mi) from the earthquake
epicenter" for the Richter scale (WP)
- middle C for pitch
- somehow determined threshold of perception for decibels
I'm not counting Malthus here, as he didn't actually propose a scale,
and zero food production very quickly means zero population.
--
Mark Mandel
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