Significant figures for inherently bound values

[jambaugh]

TL;DR

I assert that e.g. p=0.99991234 has 4 significant figures where p is a probability since it is by definition bound between 0 and 1. Who is the authority on the meaning of "significant figures"?

I have been insisting to my statistics students that for probabilities, the rule is the number of significant figures is the number of digits past the leading zeros or leading nines. For example to give 4 significant figures for a probability: 0.000001234 and 0.99999991234 are the correct number of decimal places. That way the complementary probability can also be given to the same significant figures ( 0.999998766 and 0.00000008766 respectively).

More generally if you have a value that is inherently bound between two exact values should not the significant digits start where the decimal representation of that value differs from the decimal representation of its bounds?

Does anyone know of an authoritative source on this question?

 

[willyengland]

I'm a layman in this field. In my opinion, there's a difference between expressing a probability as 0.1234 and as 0.000001234. The latter is more precise, isn't it?

 

[fresh_42]

Significance is an evaluation - always. Significant to demonstrate what? Your chances of winning the lottery are significantly zero, but the price for a ticket is usually insignificantly low. That's why it works. There cannot be an "authority" as it depends on what you are talking about.

 

[FactChecker]

What if you know, at a 6-sigma significance level, that the value is between 0.99991233 and 0.99991235? Wouldn't all the digits 0.9999123 be within the 6-sigma significance range? Why does a limit of 1 even matter?

 

[jambaugh]

@willyengland Precision is how far the expressed (rounded) value is from the intended value. Hence pi to 4 significant figures is 3.142 and pi/1000 to 4 significant figures is 0.003142.

For values with units, the number of significant figures shouldn't depend on the units. Thus in grams the mass of the electron is not zero to four significant figures; it is 9.109x10^-28 grams. Significant figures apply typically in a multiplicative setting.

@FactChecker My specific interest in this question is in grading my exams where due to the student's over-rounding their answers, I can't distinguish correct methods from common calculation errors for those cases when the probabilities are close to the definitional boundaries of 0 and 1.

I believe the "significance" in "significant figures" is not the same as the "significance = 1-alpha" used in hypothesis testing. If that wasn't your intent I'm not clear why you brought up sigmas.

I think the key principle with significant figures is in how the value is used multiplicatively. A value good to n sig. figs. should, when multiplied by an exact value still lead to a value good to n sig. figs. This however may contradict my assertion about probabilities near 1... or may not.

To my mind if two values at to an exact constant e.g. p+q = 1 or alpha + beta = 180deg then such complementary conversion between the values should yield the same number of "significant figures".

I may just have to instruct my students directly without using the phrase "significant figures" per se.

 

[FactChecker]

@FactChecker My specific interest in this question is in grading my exams where due to the student's over-rounding their answers, I can't distinguish correct methods from common calculation errors for those cases when the probabilities are close to the definitional boundaries of 0 and 1.

I believe the "significance" in "significant figures" is not the same as the "significance = 1-alpha" used in hypothesis testing. If that wasn't your intent I'm not clear why you brought up sigmas.

I mentioned the statistical significance because you said it was for a statistics class.
In any case, the significance of digits, whether statistical or calculation/roundoff, can go beyond your claim that the true value must be 0.9999999999... so only 0.9999 are significant in 0.99991234

 

[FactChecker]

These days it is easy to get a lot of significant digits in a calculation. Here is a Perl example that does a simple division calculation to 90 digits.

use Math::BigFloat; Math::BigFloat->accuracy(90);
#$pi = Math::BigFloat->new('3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342');
$n = new Math::BigFloat '1.0';
$d = new Math::BigFloat '3.0';
$x = $n/$d;
print "$x\n";

When run, it prints 0.333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333