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From: jeff@rtech.UUCP (Jeff Lichtman)
Newsgroups: net.med
Subject: Re: The Perils of Nutrasweet: digits of precision
Message-ID: <587@rtech.UUCP>
Date: Thu, 8-Aug-85 02:54:18 EDT
Article-I.D.: rtech.587
Posted: Thu Aug  8 02:54:18 1985
Date-Received: Wed, 14-Aug-85 02:39:21 EDT
References: <771@burl.UUCP> <394@petrus.UUCP> <182@steinmetz.UUCP>
Organization: Relational Technology, Alameda CA
Lines: 46

> In article <1634@orca.UUCP> andrew@orca.UUCP (Andrew Klossner) writes:
> >>> I am positive that more than .007% ...
> >>
> >>	3-digit precision based solely on anecdotal evidence?
> >
> >Both the quoted numbers (.007% and 10%) have only one significant
> >digit.  Leading and trailing zeroes are not significant.
> 
> I agree. 10% has only one significant  digit,  but  .007%  has  three.  The
> decimal  point  makes the two leading zeros significant (i.e.: the original
> article claims accuracy to three decimal places).
> 
> The Polymath (aka: Jerry Hollombe)

The usual method of writing numbers (e.g. 10, .007) carries no information
about accuracy.  .007 could be accurate to one, two or three decimal places.
The only way to express accuracy is to put the number in scientific notation,
or to explicitly specify the accuracy of the number.

In scientific notation, 7 * 10 ** -3 has one digit of accuracy, 7.0 * 10 ** -3
has two digits of accuracy, and 7.00 * 10 ** -3 has three digits of accuracy.
This method of notation makes incidental zeros impossible.  If you saw
1000000, how would you tell which of the zeros were part of the measurement,
and which were padding?  1.00 * 10 ** 6 leaves no question that there are
three significant digits.  This and the fact that scientific notation allows
huge numbers to be written compactly are the two main advantages of this
method of expressing numbers.

However, one should remember that the accuracy of real measurements
doesn't often conform to the decimal system of expressing numbers.  That is,
real measurements are not usually accurate within some integral number of
digits.  The number of significant digits in a measurement usually only
gives a rough idea of the accuracy of the measurement; a better method is
to give some measure of accuracy along with the measurement, such as standard
deviation or range of 95% confidence (e.g. 7.00 * 10 ** -3 +/- 2.5 * 10 ** -4
with 95% confidence).  Figuring this out can take a bit of calculus if the
number in question is calculated from several variables, each with its own
accuracy: one must factor in not only the accuracy of each of the variables,
but also the sensitivity of the derived measurement to changes in each of
the raw measurements.
-- 
Jeff Lichtman at rtech (Relational Technology, Inc.)
aka Swazoo Koolak

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