Significant Figures 7/30/19

By Richard Bleil

Yes, gars and goyles, it’s the favorite subject of everybody taking science. Significant figures.

As I’m writing this, I’m debating on if I will actually give the rules or not, but, I probably will. It depends on if I can make the “750” word target I have for my daily posts.

But, while I debate this in my mind and make sentences unreasonably long just to hit the target number of words, let me start with explaining why significant figures, or “sig figs”, matter. See, it’s more than just a way to irritate students and rob them of points on exams and labs, it actually conveys information. And the information that it conveys is how precisely we know a number.

People often confuse accuracy and precision, using the terms interchangeably when, truth be told, they have different and distinct meanings. See, accuracy is how close the (typically) mean is to the “true” value (which is sometimes known, sometimes not), and precision is how close your measurements are to each other. For example, if a bathroom scale is fluctuating wildly, but completely randomly both in positive and negative directions, after repeatedly stepping on and off the scale might give a mean value that is very close to the correct weight (good accuracy) but no one number will be anywhere near any other (poor precision). On the other hand, if it is working well but somebody turned the adjustment knob so all of the readings are low, then the repeated readings will all be very close to each other (high precision) but completely wrong (poor accuracy). Obviously, scientists want to shoot for high precision and high accuracy, which is why they tend to design experiments that are very tedious to perform, and repeat them many times over. The care provides good precision, and repeated measurements should provide good accuracy (assuming the results will be high as often as they are low and by the same amount).

But the WAY in which scientists will report their findings is just as important as the numbers themselves. Sig figs are how scientists tell each other to what level the numbers are known. For example, suppose that you asked me how much money is in my bank account. There are several ways that I might answer.

If I told you that I had ninety bucks in my account, you would likely assume that I do not have exactly ninety dollars to the penny. This is an example of significant figures. If I say “a hundred bucks”, you’d assume it’s near one hundred, but, give or take a few. Since I said “ninety”, you would probably assume it’s closer to ninety than, say, eighty bucks, or even one hundred. The exact number might be a bit high, or a bit low, but since it would be just as easy to say “eighty”, or “one hundred”, you’d assume it’s closest to ninety. Eighty four is closer to eighty, so it’s probably more than eighty-four, and if it were ninety-five, that’s actually closer to one hundred. This means that you might assume that the value is within five dollars, above or below, of ninety.

If I said ninety-three, you still wouldn’t assume exactly ninety-three. There would be change involved. If, in fact, I had exactly ninety dollars, give or take one dollar (that is, give or take the value of change), I might say “exactly ninety dollars”. If I mean I have ninety dollars to the penny, I would say “ninety dollars to the penny”. In science, we would write this as $90 (for ninety dollars plus or minus five), or $90. (notice the decimal point, which means ninety dollars to the dollar) or $90.00 (meaning ninety dollars to the penny).

We’re actually kind of familiar with sig figs. When we hear that one corporation is purchasing another for, for example, $4.3 billion, we know it’s not exactly $4.3 billion to the dollar (which would be $4,300,000,000.00). We would safely assume it is $4.3 billion within 100 million dollars (not $4.4 billion, and not $4.2 billion). The sig figs tell us the value within $100 million, but we can assume it is within plus or minus $50 million. The interesting thing is that the corporations themselves may not even know the exact purchase value themselves. Often such large purchases include stocks, and the exact values of the stocks are not known until they’re sold. As such, they’ll use the best-guess estimate of the value of stocks to figure out how many shares to include, but this is only an estimate.

I’ve reached my target number of words (750, so it may be a little bit higher or lower), so I don’t have to put the rules, and chances are, you don’t have to know them anyway. But the next time somebody complains about significant figures, you’ll understand that it’s just another level of providing information.

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