What Numbers Are; And What They’re Not. *Or* Why (When) Is 12 Times 12 Not 144

In 1963 and 1964, I was in 3^rd Grade, and Mrs. Waugh taught me my 12-times tables. And I learned that 12 times 12 is 144. So, what am I going on about? Let me throw out three different questions:

How many is a dozen dozen eggs? (A gross)
If a single egg weighs 12 ounces, how much does a dozen eggs weigh?
And a more focused question for us, if the regulation says do not exceed 1.2, is a measurement result of 1.24 failing?

Well, Gene, you’re saying to yourself, 144 in the first two cases, and yes. Duh…

Not so fast.

Yep. A dozen dozen eggs is exactly 144 eggs. We have exactly 12 eggs in the dozen and exactly 12 dozen eggs.

Let’s look at the single egg. There’s two points to be made here: 1) We weighed this with a scale that measures to the nearest ounce and 2) we only weighed one egg. So, what we know, in a little more detail, is that our 12-ounce egg is somewhere between 11 and 13 ounces (big egg, right?). 12 eggs is somewhere between 132 and 156 ounces. Not 144 ounces.

Before we unpack the egg story, let’s explore an analogy. We all know how to tell time. And we all know that there are different meanings depending on how you say what time it is. Or what time you’ll do something:

• 10:15 is the same time as “quarter after ten”. But we don’t think about it in the same way. Is quarter after ten somewhere between 10:00 and 10:30, while 10:15 is between 10:10 and 10:20?
• “Come by the house at 10:15” feels different than “we’re serving brunch at 10:15”
• 10:15 sounds “sloppy” and 10:16 sounds “tight”. 10:15 has some slop with it (between 10:10 and 10:20) but 10:16 just sounds way tighter.
• Of course, if you’re cooking to a clock, and it’s supposed to be done in 7 minutes, and you look at your watch at 10:08, then 10:15 means 10:15, sharp.

Our point here is that what the number means (to you, to me) is somewhat contextual, based on the conversation, and based on the way the number is presented. Perhaps this provides a little background for a more detailed, more geeky conversation about measurements and calculations.

What happened with our eggs? We accounted for measurement uncertainty. That’s a big-word way of saying we understand the difference between “exactly 12” and a “fuzzy, measured 12”. I turned around the value we have (12), to assess what the value is not (11 or 13). When I think about it that way, I don’t want to claim that my dozen eggs weighs 144 ounces, and by implication, not 143 or 145 ounces.

Now, let’s think about what 140 means. In this context, it means “not 130” and “not 150”, while 144 implies “not 143 and “not 145”. If we look at my dozen eggs above, “not 130 or 150” seems like a lot better expression of the true value and possible range (132-156) than not “143 or 145”.

Okay Gene, I follow you this far, but how do I reflect this uncertainty in my calculations and expression of values? There is a formal statistical tool. We use words like “error”, “standard deviation”, and “confidence interval”. For every value you use in a calculation, you need to have (derive or assign or measure) an uncertainty, and then propagate that uncertainty across the calculation. Excel can do that for you (I think), but do you really want to do this? Hint: NO. There’s also a less formal, shortcut tool called “significant figures (sigfigs)”. Most of us learned this, and how to use it, but never really explored the reason why it works or how it reflects the idea of propagated uncertainty.

There are simple rules of sigfigs, different for multiplication and addition. For multiplication (or division), you simply count the number of sigfigs in your starting value(s) and use the smaller number in the result. 37/59 becomes 0.63; 37/59.04 is likewise 0.63.

37/59 = 0.63 ( between  36/60 = 0.600 and 38/58 = 0.655 )

Interestingly

37/59.04 = 0.63 ( between 36/59.05 = 0.610 and 38/59.03 = 0.644 )

Understanding why we use the lowest number of significant figures is left as an exercise for the reader.

For addition (or subtraction), it has to do with the relative magnitudes. Here’s an example: 0.34 + 0.0684 is 0.41.

0.34 + 0.0684 = 0.41 ( between 0.33 + 0.0683 = 0.3983 and 0.35 + 0.0685 = 0.4185

Even though one value had 2 significant figures, and the other 3, the assigned number of significant figures is associated with the “hundredths” value of 0.34. Said more formally, the final result is rounded to the same number of decimal places as the number with the least decimal places. In our case, 0.34 has the fewest decimal places and is used to define the significant figure in the final sum.

Is this a perfect solution? No. Sigfigs is a shortcut and an approximation. If you look a little closer, you note that the relative magnitude of the implied uncertainty changes with the size of the value; 84 (not 83 or 85) has a lot less uncertainty than 27 (not 26 or 28), despite having the same numbers of significant figures). And if you’re just looking at 470, do you know if it’s 460-480 or 469-471? And for that matter, the approach is a bit stilted. What if your reality is 465-475?

What’s the takeaway on this? Let’s try to be cognizant of the significance of the values used in a calculation. Don’t just put in 0.7024 because that’s what the calculator says. Think about what you’re implying with your number, and what your reader/user can infer from the number you present.

Does it matter? It matters to us. We want it to matter to you. When you see a permit or regulation that tells you that a limit is 30.47 pounds per hour, be cognizant that whoever wrote that likely didn’t understand what we just explained.

Hey Gene, you missed a question. That one above about 1.24 being passing if the limit is 1.2. There is a reference¹ that all EPA specifications are to be considered to be to (at least) two significant figures. That means that we round our result to the same two significant figures. If the limit is 1.2, that means 1.2, and not 1.20, and we are to round our result to the same level. 1.24 rounds to 1.2 and is passing. (However, if the limit in your permit is 1.20 and not 1.2, then 1.24 is not passing.)

A frequent question is “should we round and then calculate or calculate and then round?” Best Practice is to do all the calculations using unrounded values, and then round all values for presentation only at the final stage. An unavoidable consequence of this is that there may be rounding errors in the final presentation table. That is, it may not be possible to exactly reproduce the values in a table if all of them have been rounded for presentation. You might want to note that in a footnote.

In Summary…

Thanks for hanging with us all the way to the end. Talking about numbers/values – what they are, what they’re not – can get twisty and uncomfortable sometimes. We hope it’s a bit clearer, and that the concept of uncertainty, because that’s the heart of this, rings a bit truer. If you want to explore this, or any other data interpretation or use, feel free to contact either of us:

• Gene Youngerman, PhD Chemist and unrepentant science geek, 40 years experience doing stack testing and complex measurement programs. gyoungerman@all4inc.com, 512.649.2571
• Aditya Shivkumar, MS Environmental Engineer and data enthusiast, 9 years experience in data handling, statistical analysis, and environmental compliance. ashivkumar@all4inc.com, 281‑201-1239

¹https://www.epa.gov/sites/default/files/2020-08/documents/tid-024.pdf