I, too, used to be amused when Mr. Spock would correct people with computer-like precision on various numbers.
MR CHEKHOV: We should be there in 22 minutes, sehr.
MR SPOCK: I should say 22.47589 minutes, Ensign. I do wish you would endeavor to be more precise.
MR CHEKHOV: (Mutters something in Russian under his breath)
Now, when I watch those episodes I’m just annoyed. Because Mr Spock is ignoring the concept of significant figures.
The idea of significant figures is that when you’re doing experimental work, you’re taking measurements – and measurements always have a limited precision. The fact that your measurements – the inputs to any calculation or analysis that you do – have limited precision, means that the results of your calculations likewise have limited precision. Significant figures (or significant digits, or just “sigfigs” for short) are a method of tracking measurement precision, in a way that allows you to propagate your precision limits throughout your calculation.
Before getting to the rules for sigfigs, it’s helpful to show why they matter. Suppose that you’re measuring the radius of a circle, in order to compute its area. You take a ruler, and eyeball it, and end up with the circle’s radius as about 6.2 centimeters. Now you go to compute the area: π=3.141592653589793… So what’s the area of the circle? If you do it the straightforward way, you’ll end up with a result of 120.76282160399165 cm2.
The problem is, your original measurement of the radius was far too crude to produce a result of that precision. The real area of the circle could easily be as high as 128, or as low as 113, assuming typical measurement errors. So claiming that your measurements produced an area calculated to 17 digits of precision is just ridiculous.
Back in a Previous Portion of My Career, I dealt with something similar in the environmental sciences biz. We were building a database of chemical analyses of groundwater and soil. You would think you would just get back a breakdown from the lab — X parts per million of acetone, Y parts per billion of PCBs, etc.
But there were some other important figures that had to be included in the analyses: Detection Limit and Quantitation Limit. The QL was the limit at which the lab instruments could detect the amount of something vs. its presence. So if the QL was 5 ppm and you got back a result of 10 ppm, you could be pretty sure there were 10 ppm. If you got back a result of 3 ppm, all you knew is that there was something there, but it could be 5, 4, 3, or 1 ppm.
The Detection Limit was also important. That was the threshold at which the analyte could actually be detected. Let’s say in the same example the DL for the test and instrument was 2 ppm. If the result came back as 0, it didn’t mean that there was none of the contaminant there. It just meant that there was too little to detect — there could still be 1 ppm of it.
Note that both the DL and QL are not, themselves, magically absolute figures, but figures within a certain confidence factor. They could also vary between different runs of the test (we’re talking such minuscule numbers that a slight variation could be significant), so, for example, we’d always send a blank sample along with a test batch (e.g., a distilled water sample along with the groundwater sample) to see how the instrument dealt with that (usually some level of “noise” in the instrument or preparation process would lead to some amount of X showing up, even if nothing was there). The DLs could also be affected by both the instrument used and by the test method being used on them.
Wikipedia has a good analogy, using “LOD” for DL and “LOQ” for QL.
Suppose you are at an airport with lots of noise from jets taking off. If the person next to you speaks softly, you will probably not hear them. Their voice is less than the LOD. If they speak a bit louder, you may hear them but it is not possible to be certain of what they are saying and there is still a good chance you may not hear them. Their voice is >LOD but <LOQ. If they speak even louder, then you can understand them and take action on what they are saying and there is little chance you will not hear them. Their voice is then >LOD and >LOQ. Likewise, their voice may stay at the same loudness, but the noise from jets may be reduced allowing their voice to become >LOD. Detection limits are dependent on both the signal intensity (voice) and the noise (jet noise).
In short, it’s good to be precise, but it’s important to remember that there are limits to precision, and that people who show off or claim precision greater than what is significant, or possible, are trying to sell you something, even if it’s just their ostensible expertise.
(via DOF)
A pingback via Twitter (a tweetback?): http://activeanswers.blogspot.com/2009/02/significant-figures-and-scientific.html
What bugged me was when somebody would mention a time increment that would pass while they were mentioning it. “Less than two seconds to impact!” Perhaps they were timing it from when they were going to be finished speaking, but that seems unrealistic.
I also notice when time doesn’t flow correctly in countdowns. A timer shows 20 seconds. After ten seconds, it shows 17 seconds.
It’s odd that I can suspend disbelief for things like a guy having spider-powers, but not for fairly insignificant things like this, isn’t it?
I agree on both points. It’s a danger for writing on TV (movies, too, I suppose, but they pay closer attention to that kind of continuity).