25 Oct 2005 - 04:07

significant digits, part 1

I've wondered out loud recently, somewhere on KTM, about significant figures and what they are about. Significant figures fell through a crack in my education. They were taught -- not in a lot of detail -- when I was in chemistry in high school. I barely paid attention, the explanation was confusing, and I suspect that instead of buckling down and learning about them, I spent my time actively hoping that the topic of significant figures would go away soon. For a while there, I guess I was really in touch with the way a lot of people feel about math.

But math never goes away, and significant figures basically did.

Significant figures were definitely taught in a procedural way. We learned (or didn't, in my case) the rules for the number of significant figures that go into sums and products, and told to use these rules when doing science labs. I didn't get the point. But I'm not going to blame my teachers for the fact that I didn't get the idea, because I didn't even learn the procedures. I was a real bum in high school.

Significant figures weren't part of my educational recovery later on, either, being a Science Lab Thing and not a math thing.

So recently I went looking up some reading online on significant figures. Oddly enough, though I haven't thought about significant figures in decades, I have found that I have more than enough mental 'hooks' to hang the concepts on, due to my later adventures with probability and statistics. So I'm going to do a little miniseries here on significant figures (my favorite way to learn is always to teach).

Besides, anything I get wrong here will be quickly corrected by our very savvy bevy of engineers ("very savvy bevy" being my lead-in to tomorrow's discussion of alliteration). I hope they will bear with me. I'm going to take it fairly slowly.


significant figures explained, part 1

Every measurement we make has an uncertainty associated with it. If we have a ruler with tick marks in whole inches (and assume we are unable to eyeball values in between whole inches), then objects we measure with it will have an associated measurement error of up to 1/2", even if we use the ruler correctly.

The clearest way to represent this uncertainty is to say that an object we have measured to be 6", with this particular ruler, is actually 6±1/2". But scientists are always doing mathematical operations using quantities they've measured, and it gets to be a drag doing mathematical operations using those ±1/2 parts of numbers.

The "significant figures" way to represent a measurement is a sort of shorthand -- they use 6, and lose the ±1/2 part, but they understand (and all the other scientists understand) that the lack of any extra digits to the right of the 6 implies that you only know the answer to that degree of accuracy.

Every number a scientist reports therefore has two parts -- a "measurement" part, which is the part you see; and an "uncertainty" part, the "±" part, that you infer from the way the measurement part is shown.

For example, if a scientist reports "this thing is 6 inches long", he means that he might be off by as much as half an inch; in other words, the thing is really between 5.5 and 6.5 inches long.

But if the scientist reports "this thing is 6.0 inches long", with the decimal point and extra zero, he's telling you that his measurement was a lot more accurate than that. If there were half an inch of uncertainty in that measurement, he would have left off the decimal place and the extra zero. So he's actually telling you that he measured the thing with a ruler that had tick marks every tenth of an inch; and the actual measurement was between 5.95 and 6.05 inches -- that is, the number he's reporting is really 6±.05 inches.

Here's another example: suppose a scientist reports that he has determined, through radiocarbon dating, that the age of a sample is 35,500 years. In the scientist's shorthand, only the first three digits are meaningful; the two zeros are placeholders, indicating that the measurement is only accurate to the nearest hundred. That is, "35500 years" actually means "35500±50 years".

However, if he reports that the age of the sample is 35500.0 years, he is telling you that its age is 35500±.5 years; i.e., that his measurement is accurate to a half of a year (but no carbon dating method is that accurate, so you know he is lying).

So the rules are:

• The left-most nonzero digit (<-) is the most significant digit.

For example, in 123,000, the 1 is the most significant digit.

• If the number does not have a decimal point, the right-most nonzero digit (->) is the least significant digit.

In 123,000 the 3 is the least significant digit.

• If the number does have a decimal point, the right-most digit (->) is the least significant digit, even if it's a zero.

So, for example, in 1.3, the 3 is the least significant digit; in 1.30, the zero is the least significant digit. 1.3 actually stands for 1.3±.05, and 1.30 stands for 1.23±.005. The measurement part of the number is the same, but the uncertainty part is not.

• Every digit between the least and most significant digits should be counted as a significant digit. So 123,000 has three significant digits, and 1.230 has four.

one more tidbit

I now understand why Saxon 8/7 is continually giving problems like the one I described in ConceptualSaxon; I thought they were trying to teach the notion of quantization error, which is a bit abstruse for your average 7th grader. They are actually trying to get the kids ready to really understand the notion of significant digits, which they might get in science as early as 8th grade.

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I have printed this out!

I am putting it inside my ktm study book!

-- CatherineJohnson - 25 Oct 2005