Significant figures (sig figs) - in measurements, calculations, etc.

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 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! 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 because 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 9.0x10⁴, 9.0x10¹⁵, & 8.4x10¹⁵ all have 2 sig figs BUT 9x10⁴ only has 1)

finished in comments

[thebumblingbiochemist]

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).


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 because 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