"Significant Figures," "Precise and "Imprecise" Information

5 Apr 2006 in Information Quality, Glossary

If you can understand significant figures, you can understand the difference between precise and imprecise information.

The accepted use of significant figures can be found in various on line sources such as MathWorld or the Reference.com, where the following logical rule is specified:

Ignoring any decimal point in a number, start at the left end of it and move right; when a non-zero digit is found, that digit and all digits to its right are significant.

Reference.com offers a number of examples and a few exceptions to the rule (and is, unsurprisingly, less mathematical than MathWorld). They also suggest that numbers with multiple trailing zeros be expressed in scientific notation if they are supposed to be meaningful. This makes sense for scientific documents and technical publications, but it violates the style manuals of most (if not all) newspapers. This creates a lot of confusion in newspaper accounts: readers have a hard time discerning precise from imprecise information.

To give a simple example, the number 50,000 reported in a newspaper could mean any one of the following ranges of values (as well as many others):

50,000, following the definition above	49,999.5 to 50,000.4
5.0 x 10^4 rounded to the nearest 0.1 x 10⁴	49,000 to 51,000
5 x 10^4 rounded to the nearest 1 x 10⁴	40,000 to 60,000

If the exponent is increased even less precision is implied. For example:

0.5 x 10^5 rounded to the nearest 0.1 x 10⁵

0 to 100,000

And if that's not enough, sometimes numbers have precision that is based on a logarithmic base other than 10, such as the natural logarithm (e=2.718...)

Here's a good three-part rule of thumb to follow when reading numbers that reflect measurements or estimates of something:

All numbers are uncertain. It's the only thing that's certain about them.
A clear statement of how many digits are significant is an essential first step in figuring out what a number means. The statement tells you how much precision there is in the measurement or estimate. The amount of precision that matters depends on what you are going to do with the number.
Any number that does not include a clear statement of how many digits are significant is inherently imprecise. It's perfectly reasonable to admit that you don't know what such a number means. In fact, it's the only reasonable thing to do.

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Comments on "Significant Figures," "Precise and "Imprecise" Information

From Scott Ferson on 20 April 2006, 17:15

The numerical ranges you give for the intervals implied by the sigdigs are not quite correct. The intervals you should give are below.

[50000.] = [ 49999.5, 50000.5]

[5.0e4] = [ 49500, 50500]

[50000] = [ 45000, 55000]

If the values were either higher or lower than these ranges, the rounding you specify would have yielded a number other that 50,000, or 5.0e4, or 5e4. (Note that convention does not support 50000.4 in the first example because the value could be anything up to 50000.49999999... which is mathematically equivalent to 50000.5.)

The fourth example is completely incorrect and you should probably omit it. In fact, the sigdigs for [0.5e5] represent the interval [ 45000, 55000] which is the same as for the third example. This is not surprising since the value and the rounding you specify are identical to those in the third example.

In fact, you cannot get worse than one significant digit. (This is one of the problems with using sigdigs to represent precision of a measurement, because it is the case that uncertainty might be worse than this. But to represent this situation, you have to go to explicit intervals, rather than sigdigs.) The sentence in your blog after the fourth example about logarithmic bases is nonsense I think. Perhaps I misunderstand what you're getting at, but precision is not fundamentally related to the radix or base of the number system. Expressing numbers on a log scale doesn't alter their precision. Now, it's true that interpreting some convention based on sigdigs would be influenced by the radix, but you don't really see hexadecimal or binary or even log scale numbers reported in newspapers, so I think you should omit this sentence too.

The characterization in Mathworld of "accuracy" is strange. Accuracy is certainly NOT "given by the sigdigs to the right of the decimal point". This may relate to some weird feature of Mathematica. I think Wikipedia (http://en.wikipedia.org/wiki/Accuracy_and_precision) is a far better reference on this matter. They give a standard discussion.

Your three-part rule seems reasonable, except for two minor things. You should add "measured" as the second word in rule 1, because some numbers ARE perfectly precise such as those that are defined or mathematical creations. The point in the third rule goes a bit too far. If the number is expressed numerically (that is, with digits), then the sigdigs expresses its accompanying precision -- by the convention that you describe the blog. A particular individual may not know "what a number means" but it is not reasonable to pretend that it doesn't have a well defined meaning and an evident precision. Actually, it's far more common for a number to have vacuous meaning not because its precision is poorly defined but because its UNITS are. That is a real and pervasive problem in reporting. Also problematic is the failure to specify reference classes for risks and probabilities, but that's maybe beyond your scope.

Regards,
Scott Ferson

From Scott Ferson on 20 April 2006, 17:15

Gee, it'd have been helpful not to have stripped away all the linebreaks in the previous post!