## RE: Significant digits? Take II

Subject: RE: Significant digits? Take II
From: "Halter, Meg" <HalterMC -at- navair -dot- navy -dot- mil>
To: "TECHWR-L" <techwr-l -at- lists -dot- raycomm -dot- com>
Date: Tue, 14 Nov 2000 13:35:12 -0800

Geoff writes in one message

> What I meant was that _removing_ the digit would make a difference in the
> meaning, not _changing_ the number. So for example, 0.020 has only two
> significant figures (because removing the final zero makes no difference
> in the meaning); in contrast, 0.002 has three significant figures (because
> removing any of the zeros to the right of the decimal would dramatically
> change the meaning).
>
And in another message he writes

Though it's true that zeros _often_ don't count as significant figures, you
have to look beyond that simple rule. If the number is 0.00123, and no other
numbers are involved that have a different pattern (i.e., all numbers
resemble 0.00XYZ), then there's no question that there are only three
significant figures: X, Y, and Z. But let's say you have two or more numbers
with different patterns: if the author presents 0.00123 and 0.04123, the
picture changes dramatically: now the first number actually has four
significant figures, to match the number used in the second number. Using
the logic expressed for scientific notation, the numbers become 0.123x10^-2,
and 4.123x10^-2, and it's clear that both have four significant figures, not
three.

Wellllll I disagree. Two concepts "significant figures (or digits)" and
accuracy are being mixed.

The number 0.020 does have 2 significant digits, but not for the reason
stated. Written as 0.020 (or as 2.0x10^-2), it means that we know that the
zero is significant. Hmmm that's a bit circular. In other words, the actual
value is between 0.0195 and 0.0205, so it's accurate to within +/-0.0005. If
we write the number as 0.02 or 2.x10-2, then the number has 1 significant
figure and we know that the actual value is between 0.015 and 0.025, or it's
accurate to within +/-0.005. So, though the magnitudes are the same,
accuracy with which we claim to know the magnitude is very different.

The number of sig figs does not depend on the context with other numbers. I
think you are really referring to accuracy here. Both 0.04123 and 0.00123
are accurate to within +/- 0.000005. However, 0.04123 has 4 sig figs and
.00123 has 3 sig figs.

According to _Applied Numerical Analysis_, "Sometimes the accuracy is
expressed as the number of digits that are correct; in other cases, the
number of correct digits after the decimal point is used." So there's a bit
more confusion thrown in.

(My trusty Schaum's Outline on Statistics by Spiegel and _Applied Numerical
Analysis_ by Gerald and Wheatley were helpful here.)

Them's my thoughts.

-- Meg

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