When it comes to digits, this scintillation counter provides an embarrassment of riches, so which numbers should I ditches? Turns out a lot of them don’t matter – go figure! In science, we only should report numbers which are significant. Basically, you don’t want to write down that something you measured with a ruler is 1.372 mm (the lines are 1 mm apart, so you know that the 3 is an eyeball guess and the rest is just eyeball over-confidence!). And this “sig fig” cut-off extends to when you do calculations, so that you don’t divide something you’ve measured as 1 mm by 3 and then report 1.3333333333, making it seem like your measurement was way more precise. And and the sig fig cut-off extends to when you report error so that you don’t report something as 1.3 +/- .125 (error here’s that error has too many sig figs) or 1.3 +/- 5. (error here is that measured value has too many sig figs). When it comes to error you just get one digit! I want to review all of these different situations, but let’s start by getting on the same page about what significant figures actually are. 

SIGNIFICANT FIGURES are digits (whole numbers) that actually tell us info that we KNOW and you can think about it as what you can learn from a partly-filled theater row! (obviously this is a pre-COVID example!)

Imagine that a row of theater seats gets filled from the right and then you bring in a detective to try to see what you can learn from the audience members’ final positions. The 1st person sits down “blindly” anywhere so seats to her left don’t tell us anything (theater could pull back a curtain & reveal an infinite number of empty seats to her left, but it wouldn’t matter). 

BUT, now when choosing where to sit, people have to make a conscious decision about how close they want to be to person to their left. So the number of empty seats between them DOES tell us something (smelliness?)

Seats to right of last person to sit down may or may not hold info. If those seats really are available, but no one sat there, that has meaning. BUT if the seats are “reserved” or cordoned off, there aren’t really any more seats available so we’re “lying” if we say there are. And if someone sits there later, it’s your job as usher to kick them out 

What does this have to do with numbers and measurements? Person-filled chairs are like nonzeros & empty chairs are like 0s. Being “significant” is like being a (non-cordoned-off) chair that someone consciously did or didn’t sit in. If the chair was available and no one sat there that tells us something – those 0s *are* significant.* BUT if that chair was cordoned off the fact that it’s empty doesn’t tell us anything – those 0s are NOT significant.

When you make a measurement, the measuring device decides where the row ends. Even when measuring the exact same thing, you’ll get different numbers of sig figs depending on the tool. Say you’re measuring something that’s *exactly* 1m long. Some instruments have really long rows so you can get lots of sig figs (like 1.00000cm). But other ones have short rows so you can only get a few (1.000m) What’s it to you?

Once you start doing calculations with those measurements, those rows can appear to get longer but it’s really just people sitting in cordoned-off seats. (Think of dividing 1 by 3….) It’s your job as a good scientist to act as an usher and kick those uninvited folks out. 

So we need to know how many figures are significant. How do you know how many sig figs you have? Any NONZERO # IS significant & you start counting at the 1st one you meet. 0s may or may not be significant… 

If there are nonzeros on each side, the 0 must reside ✅ 

If the 0’s in the lead, erasing can proceed ❌ 

But if the 0’s at the end, well, then that depends…❓

If there’s a decimal preceding, that end 0 does have meaning ✅ 

If there’s a decimal at the rear, that 0 you hold dear✅If there’s no decimal but you had to round, confusion might abound. 

In such a situation, turn to scientific notation 

In non-rhyme…

🔹 0’s before 1st nonzero (LEADING 0s) are NOT significant ❌

🔹 0’s in between nonzeros ARE significant ✅

🔹 0s after last nonzero (TRAILING 0s) are NOT significant ❌ UNLESS there’s a decimal point ✅

For example, 1, 01, & 001 all have 1 sig fig, but 1.0 has 2 & 1.00 has 3. If I add 0s at the front, they’re just “fluff” ❌ BUT if they’re at end AND there’s a decimal point, they’re actually telling you something ✅ (on the other hand, 1, 10, & 100 all have only 1 sig fig bc there’s NO decimal point)

⚠️ EXCEPTION ALERT!!! EXACT NUMBERS have unlimited sig figs. What do I mean by “exact numbers”? 

🔹#s that are “defined” (like 1 yard = 3 feet)

🔹”counted” #s (like “6 people”)

📝 A note on SCIENTIFIC NOTATION: Scientific notation is where you use things like 2 X 10³ to write 2000 and 2 X 10⁶ to write 2 million. The “10^something” part only adds NONSIGNIFICANT 0s (those are all leading or trailing zeros)❌ So you only count sig figs from “unique part” (e.g 9.0×10⁴, 9.0×10¹⁵, & 8.4×10¹⁵ all have 2 sig figs BUT 9×10⁴ only has 1)

Once again, this is easiest explained with examples, so see you in the pics! 📸

Now that we know what sig figs are and how to count them, let’s take a look at how to work with them, starting with making measurements, then some simple arithmetic stuff and then getting into reporting error.

If you divide 1 by 2 you get 0.5. If you divide 1 by 4 you get 0.25 & if you divide 1 by 8 you get 0.125. An answer of 0.125 might sound more knowledgeable than an answer of 0.5, but do you really know more? Of course not! The more digits you include, the more it *seems* like you know. BUT the amount you REALLY know is limited by amount you know about the thing you know the least about & this is limited by your measuring tool.

Every measuring tool has limits. When I weighed some salt on a “kitchen scale” it told me it weighs 1.00g (3 sig figs). Factoring in rounding, it could “actually” weigh anywhere from 0.95-1.04g, but this scale couldn’t tell me because it’s limited to hundredths place. If I instead weigh it on an analytical balance, it gave me value of 0.9999 (4 sig figs) (which could actually be 0.9995-1.0004g).

This # of sig figs relates to our past discussion of the difference between accuracy & precision in the context of pin the tail on the donkey. https://bit.ly/accuracyprecisionerror 

In that case, we were dealing w/multiple “measurements” that we were averaging. If you take multiple measurements of the same thing, PRECISION tells you how close those measurements are to *each other* & ACCURACY tells you how close the average of those measurements is to the REAL value 

But often when we’re measuring something in a lab, we only get “1 shot,” to measure. BUT we can still provide an indication of how precise the value is – in this case a measure of how confident we are in our measuring tool. This is where SIGNIFICANT FIGURES (sig figs) come in – each # you can *confidently* measure (even 0s) IS SIGNIFICANT (tells you “real info”)), then you get 1 significant “estimate” as your last sig fig.

So when you pin the tail on the meniscus, how many sig figs do you report? If you’re measuring liquid w/a graduated cylinder (like a tall skinny measuring cup) & bottom of meniscus (the “smile” you measure from) is between 2 lines (aka gradations – hence “graduated” cylinder 💡), you’re sure it’s above the one, so you write down that # confidently, but then you have to “eyeball” how close it is to line above.

You only get to write down 1 sig fig here (so if it’s between 100 & 200 you can write 150 but not 155 because your eyes aren’t that good without lines to help out & that trailing 0 isn’t significant, it’s just a placeholder.

How close together lines are will determine how many decimal places & sig figs we can get. The closer the lines, the more precise you can get (assuming you’re reading the readout correctly!). Digital displays take out the “eyeballing,” BUT they too have to round to their limit.

Another quick word of caution. The analytical balance gives me more sig figs than the lab scale, so it’s more precise, BUT is it more accurate? Not necessarily! What if I forgot to tare it (0 out weight of the weigh boat)? I could get a really precise value ✅ but it would give me an INACCURATE weight of the solid ❌ (though it might give me an accurate weight of solid + weigh boat) The balance has an extra aid to help w/accuracy – the “cage” around it prevents “distractions” from altering readout. 

It’s easier to tell if something is precise than if it’s accurate bc to know if something’s accurate you need to know the real value & that’s what you’re trying to find in the 1st place… A good sign is if multiple different measuring tools give you the same answer. Since I’m got the “same” value from both scales, I can feel pretty sure the measurement’s accurate (assuming I remembered to tare!)

So, that whole example might have seemed like a bit of a tangent ramble (which I am known to do…) but it demonstrates an important point – the original # of sig figs (which you now know how to count) comes from the # you get when you MEASURE something. 

But, if you then perform CALCULATIONS w/those measured #, you need to make sure you’re not adding “fake” sig figs. Going back to our theater analogy, fake sig figs are like people sitting in reserved seats. And, as a careful scientist, it’s your job as usher to kick them out. So you have to decide when and where to round

Speaking of ROUNDING…

  • round at the end, or else where you round earlier on will affect your final answer 
  • conventionally, if something is 5 or more, round up (e.g 1.4 rounds to 1 & 1.5 rounds to 2)
  • add 0s as “placeholders” for nonsignificant numbers you had to remove (e.g. 17 200 has 3 sig figs rounded to 2 sig figs: 17 000 or 1 sig fig: 20 000)
  • ⚠️ Things can get confusing if you have to round to a 0 (e.g. if you round 19 900 to 2 sig figs, you have to round to 20 000 which looks like it only has 1 sig fig (eek!) to avoid ambiguity, you can instead write it in scientific notation, 2.0 X 10⁴ )

So how do you know where to round? It depends on what kind of calculation you did

ADDITION ➕ & SUBTRACTION ➖: answer should have same # of DECIMAL PLACES as starting value w/least DECIMAL PLACES (you can remember this by thinking back to a measuring stick and how the last decimal place is the smallest increment you can report)

🔹 e.g. 10.37-1.2 = 9.2 (NOT 9.17)

MULTIPLICATION ✖️ & DIVISION ➗: answer should have same # of SIG FIGS as starting value w/least SIG FIGS (here the decimal places don’t represent any ruler lines or anything so they’re not important, it’s the numbers themselves that count)

🔹 e.g. 20 x 13.4 = 300 (NOT 268)

so remember: 

When you add or subtract, the decimal *points* to your fact

But multiply or divide, then you need to consider both sides 

Try it out for yourself in pics! 

Now, finally, let’s talk about reporting uncertainty or “error.” “Error” is when the measured values differ from the “real” value. It’s the “+/- something” in the value you report, and it can be reported a few different ways. A couple of the main ones are standard deviation, which tells us about underlying variation in the thing we’re measuring, and standard error, which tells us about how accurate our calculated values are. 

So, say for instance you you have a really competitive game of pin the tail on the donkey – judges and all. Contestants each get a certain number of tails to pin & the winner’s chosen based on their precision (how close together they pin the tails) & accuracy (how close to the desired location (x on donkey butt) the tails are. 

Determining how accurate they are requires measuring the distance between the x and the tail pin. And this is where the judges come in. Yes – judgeS, plural! Multiple judges all measuring how close the tail is to the x marking where it should be. Each takes a turn measuring the distance and then the game-master averages those distances and announces the official reported distance. We already discussed how many digits he should report (as many ruler ticks as are certain and then one guess for how far in between that and the next tick)). We’ll pretend they’re using a meter stick, so they get cm lines & one guess. 

And we discussed how the judge should deal with rounding if he averages the values the judges give him. Since he’ divided, he has to round this number to have the same number of sig figs as the starting value with the least number of sig figs. 

But how should he report how much agreement there was among the judges (how to report the variance or deviation)? If you want to learn more about how you actually calculate these and stuff, check out this post: https://bit.ly/standarddeviationerror 

and more on reporting them with examples in this post: https://bit.ly/errorreporting 

But here let’s just focus on what you do once you have that value your calculator spits out at you…

When it comes to reporting error, you only get a single digit. That’s it. 

Say you have an average of 2.5 and an error that your calculator says is 0.20816659994…. Where do we round here? The first place you see a digit – so the game-master can report the value as 2.5 +/- 0.2. He canNOT say 2.5 +/- 0.21. This gives the false impression that the judges were measuring with a ruler and not a meter stick. And, furthermore, there’s no possible way a judge could have reported a value of 2.71. The judges can only report to the tens place. So even if they were “sure” that it was 2.71 they’d have reported 2.7. So we don’t gain any information by adding that 1 in the deviation. 

What if you had a really bad judge so had a really big error, we’ll say “7.05053189483” (at least that’s what the calculator tells you). 

What to report here? We only get a single value for reporting error, so we have to round our error at the first digit BUT our SD has to be on the last place -> when you report a number you’re implicitly saying that you’re confident about all the numbers except the last, which is where there’s “room for error” (but note that the second to last doesn’t have to be absolutely agreed upon if you have something like 2.0 +/- 0.2 which encompasses both a 1 & a 2)

Imagine back to the judges squinting to estimate how far the tail is between those 2 lines. If they can’t agree on the tens place they certainly can’t agree on the 100s place. So you don’t need to tell us that. We just want to know where you start disagreeing – or at least where most of you start disagreeing. 

So, when you report uncertainty, you take the digit that you get and round the reported value to that digit. Even if your measuring device is “better than that” 

With the 1st example, we could use that single digit in the tens place and still encompass the values, but here there’s so much disagreement in the single’s digit that the error’s spread there – so we have to round our average value to match. 

We have to round our error at the first digit and we have to round our average to where we rounded the error. So the SD rounds to 7 but we can’t say 7.5 +/- 7. Our SD has to be on the last place. So we have to round the average to 8 and report 8 +/- 7. Eek! 

For the curious… The computer attached to the scintillation counter (a device used to measure radiation in racks of little sample vials) doesn’t save files and instead gives you an old-fashioned printout that seems to go on for miles! It’s a super sensitive machine, so it goes out a couple of digits passed the 0, which might be further than some of the other measurements I made, and when I do calculations on the data (once I type it in) I need to make sure that my spreadsheet doesn’t add on even more. Yup, there’s some potential for some significant sig fig mistakes, hence the motivation for this review in case others need it too!

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If you want to learn more about all sorts of things: #365DaysOfScience All (with topics listed) 👉 http://bit.ly/2OllAB0  ⠀

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