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In my high school chemistry class, my teacher insists that the "round to even" rule means rounding to the nearest even number whenever the next digit is 5, regardless of any digits that follow. For example, she teaches that 2.59 should be rounded to 2, claiming that rounding 2.59 to 3 would be "chain rounding" because it considers digits beyond the tenths place.

From my understanding, "round to even" should apply only when the number is exactly halfway between two options (like 2.5 or 3.5), and not when there are additional digits. When I explained this, my teacher said, "this is what we do in honors chemistry."

Is there a different "round to even" convention in chemistry, or is this just a misunderstanding of the "round to even" rule?

ryang
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Herbert The Bird
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  • https://en.wikipedia.org/wiki/Rounding#Rounding_half_to_even "also called convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, odd–even rounding, or bankers' rounding. [...] This is the default rounding mode used in IEEE 754 operations for results in binary floating-point formats". – PM 2Ring Aug 27 '24 at 04:28
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    Cross-posted to Chemistry SE. – Semiclassical Aug 27 '24 at 05:23
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    Your teacher would round $2.54$ to $2.5$ (by ignoring the last digit) and then round $2.5$ to $2$. That is "chain rounding." Worse, she is rounding $2.59$ down to $2.5$, which isn't even close to "round to nearest". – David K Aug 27 '24 at 05:44
  • Is there any benefit in using this rounding "half-to-even" in chemistry compared to rounding "half-up"? I've never seen this half-to-even rule used in practice and I could see the benefits in banking, but I wonder if there is any benefit in chemistry or if the teacher is misrepresenting the status quo. – AccidentalTaylorExpansion Aug 27 '24 at 13:27
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    Here is a sensible answer on the chem SE, that is in line with the comments and answers here on maths SE, which gives some hope that chemists have not collectively come up with a bizarre self-defeating convention: https://chemistry.stackexchange.com/questions/38227/how-to-round-significant-figures-correctly – tkf Aug 27 '24 at 16:38
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    @AccidentalTaylorExpansion Most floating-point arithmetic hardware & software uses IEEE 754, where "Round to nearest, ties to even" is the default for binary floating point. – PM 2Ring Aug 27 '24 at 17:03

2 Answers2

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It is certainly not the round to even that I know. $2.59$ is closer to $3$ than to $2$. This is clearly wrong. Following your teacher's convention, the numbers that you would round to $2$ would be all those in the interval $[1.5,2.6)$ and those that you would round to $3$ would be all those in the interval $[2.6,3.5)$. This shows a large bias for even numbers.

RLH
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Ross Millikan
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  1. Rounding half to even (also called Banker's rounding and, misleadingly, rounding to even) is an algorithm for rounding numbers to the nearest rounding digit such that this tie-breaking rule is applied:

    • if the original number is equidistant from its nearest rounded options and the rounding digit is even, then round towards zero (i.e., retain that even rounding digit and drop all subsequent digits)
    • if the original number is equidistant from its nearest rounded options and the rounding digit is odd, then round away from zero (i.e., increase that odd rounding digit by one and drop all subsequent digits).

    As opposed to tie-breaking by, for example, always rounding away from zero, Banker's rounding is an attempt to mitigate bias.

  2. In particular, for a number that's exactly midway between its two nearest integers, this algorithm rounds it to the nearest even integer.

  3. Applying rounding half to even: $$3.5\approx4\\\boldsymbol{2.59\approx3}\\2.5\approx2\\-2.5\approx-2\\-3.5\approx-4\\x\in(2.5,3.5)\implies x\approx3\\x\in[1.5,2.5]\implies x\approx2\\x\in(0.5,1.5)\implies x\approx1\\x\in[-0.5,0.5]\implies x\approx0\\x\in(-1.5,-0.5)\implies x\approx-1\\x\in[-2.5,-1.5]\implies x\approx-2.$$

  4. On the other hand, rounding to the nearest even integer: $$2.59\approx2.$$

P.S. In agreement with your teacher, though, two other apparent claims that Banker's rounding gives $24.8514\approx24.8$ and $0.2533\approx0.2$ (instead of $24.9$ and $0.3,$ respectively):

https://www.chemteam.info/SigFigs/Rounding.html

https://chemistry.stackexchange.com/questions/38227/how-to-round-significant-figures-correctly#comment122766_40104.

ryang
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