Significant Digits and Useless Digits

A pet peeve of mine is measuring things to far too many decimal places. For example, notice that the thickness of these trash bags is 0.0009 inches (0.9 mil) but is 22.8 microns in metric. There are two mistakes:

While the conversion factor is correct, there’s no way that the thickness is known within only 0.1 microns, or 100 nanometers. That’s significantly that a typical cell nucleus.
Less importantly, if they rounded correctly, it should be 22.9 microns, not 22.8.

My favorite example that I’ve personally witnessed — that I wish I had a picture of — is measuring student’s perceptions of a professor’s teaching effectiveness is 13 decimal places.

This webcomic from xkcd illustrates the point both cleverly and perfectly.

Source: https://xkcd.com/2170/

Too many significant digits: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my short series on what happens when too many significant digits are reported.

Part 1: An analysis of the report of this Nike app:

Part 2: Links to resources that discuss how many significant digits should be reported.

Too many significant digits (Part 2)

In yesterday’s post, I had a little fun with this claim that the Nike app could measure distance to the nearest trillionth of a mile. The more likely scenario is that the app just reported all of the digits of a double-precision floating point number, whether or not they were significant.

In real life, I’d expect that the first three decimal places are accurate, at most. According to Wikipedia, the official length of a marathon is 42.195 kilometers, but any particular marathon may be off by as many as 42 meters (0.1% of the total distance) to account for slight measurement errors when figuring out a course of that length.

A history about the Jones-Oerth Counter, the devise used to measure the distance of road running courses, can be found at the USA Track and Field website. And my friends who are serious runner swear that the Jones-Oerth Counter is much more accurate than GPS.

green line
The same story often appears in students’ homework. For example:

If a living room is 17 feet long and 14 feet wide, how long is the diagonal distance across the room?

Using the Pythagorean theorem, students will find that the answer is $\sqrt{485}$ feet. Then they’ll plug into a calculator and write down the answer on their homework: 22.02271555 feet.

This answer, of course, is ridiculous because a standard ruler cannot possibly measure a distance that precisely. The answer follows from the false premise that the numbers 17 and 14 are somehow exact, without absolutely no measurement error. My guess is that at most two decimal places are significant (i.e., the numbers 17 and 14 can be measured accurately to within one-hundredth of a foot, or about one-eighth of an inch).

My experience is that no many students are comfortable with the concept of significant digits (or significant figures), even though this is a standard topic in introductory courses in chemistry and physics. An excellent write-up of the issues can be found here: http://www.angelfire.com/oh/cmulliss/

Other resources:

http://mathworld.wolfram.com/SignificantDigits.html

http://en.wikipedia.org/wiki/Significant_figures

http://en.wikipedia.org/wiki/Significance_arithmetic

Too many significant digits (Part 1)

The following appeared on my Facebook feed a while back:

Just look at that: the Nike app claimed to measure the length of my friend’s run with twelve decimal places of accuracy.

Let’s have some fun with this. Just suppose that the app was able to measure distance to the nearest trillionth of a mile. One trillionth of a mile is…

5.28 billionths of a foot,

or about 63.4 billionths of an inch,

or about 161 billionths of a centimeter,

or about 1.61 billionths of a meter,

or about 1.61 nanometers.

By way of comparison, the fingernails on the average adult grow about 3 millimeters a month. A world-class runner could run 6.25 miles in about 30 minutes; in those 30 minutes, his/her fingernails would grow about 2 microns, or about 2000 nanometers. (Of course, they’ll grow longer for less athletic runners covering the same distance at a slower speed.)

So if the Nike app can measure my distance to the nearest trillionth of a mile, it would have absolutely no difficulty measuring how much my fingernails grew while running.

Or, it could be that the Nike app really isn’t measuring the distance all that precisely. Probably the app used double-precision arithmetic, and whoever programmed the app didn’t tell it to truncate after a reasonable number of digits.