# Why is rounding half away from zero the only method taught?

Rounding to the nearest even digit is very practical in a lot of areas (e.g. statistics, accounting), but is never taught anywhere from elementary school to college. Even in R, the go-to statistics language, the round function uses this method. There are a few other methods with their own appropriate situations; see http://en.wikipedia.org/wiki/Rounding#Tie-breaking for an overview.

Is there a reason this is not taught? What level/area of math would it be wise to introduce this concept for a minimum of exposure to the idea?

• As for R: Is there any situation where the rounding bias matters and using the round function is not a very bad idea? Mar 17 '14 at 7:15

To properly understand and appreciate the different rounding rules, one really needs to have some grounding in statistics and probability theory. It is a little hard to describe the full theory in detail (the least being that I don't have a entirely coherent formulation of the theory off the top of my head), but here are some examples:

• Naively if we idealistically think of data as living in $\mathbb{R}$ and the rounded counterparts as living in $\mathbb{Z}$, then tie-breaking rules do not matter, as for all reasonable probability measures, we would be thinking about events of probability zero.
• However, tie breaking rules do matter, and this is due to that our data really lives in $\mathbb{Z}$ and we are rounding to $\lambda\mathbb{Z}$ for some positive integer $\lambda$. The event that the data falls exactly halfway between the rounding points is now of positive probability, if $\lambda$ is even. (If $\lambda$ is odd, there's no halfway point.)
• But there are other subtleties in rounding rules: for physical science and statistical analysis, we want the rounding rules to be stable under addition of small random noise, so that we don't introduce rounding artifacts. In statistics the "round to the nearest even integer" rule for tie-breaking is useful, under the assumptions that
• we intend to sum/average our data
• the ones-digit of the data is uniformly distributed, at least in terms of evens and odds, and that the distribution is independent of the digits after the decimal point, which
• we assume again to be uniformly distributed.
• There are times however our data is multiplicative. In this case rounding from the half-way point is no longer appropriate: it is more appropriate to round $0.3$ down and $0.4$ up. The exact dividing line is $\frac{1}{10} \sqrt{10}$ which conveniently happens to be an irrational number, so in finite precision computations the problem with the tie-breaking rules are side-stepped somewhat.

The main point being that the correct rounding rules depend at least on

• A priori assumptions on your data: where they come from, what noises are inherent, what are reasonable a priori probability distributions of the digits...
• What you intend to do with those data: are they to be summed? are they to be multiplied? are they to be binned (thank to Sue VanHattum for pointing out this case)? Some more complicated functions?

And a proper discussion of the strengths and weaknesses of different rounding rules will necessarily involve quite some bit of statistics and mathematical modeling.

This is, however, not to say that there are not portions of the theory that can be taught to elementary schoolers. But one has to be very careful about the presentation:

• You do not want it to devolve into a discussion of "different conventions". As this just makes two different things that the poor kid will have to memorise, instead of just one.
• The issue of tie-breaking rules never occur when dealing with a single datum. Are you sure the kids can appreciate that rounding errors can accumulate when one works with many data points and that this is undesirable?

Let me give one more try about the theory:

One way to think about rounding (or approximations in general) is that we have some space of objects $X$ which are sometimes hard to deal with, but within it we have a subspace of objects $Y$ which are easier to handle. We also have a family of maps $\mathcal{F} = \{F_1, F_2, \ldots\}$ which are maps from some finite Cartesian power of $X$ to $X$ itself.

The ideal rounding rule is a map $\mathcal{R}:X \to Y$ (which induces also $\mathcal{R}_k: X^k \to Y^k$ by acting on components) such that

1. It is idempotent: $\mathcal{R}\circ \mathcal{R} = \mathcal{R}$.
2. It "commutes" with the family $\mathcal{F}$: for every $i$, suppose that $F_i: X^{k_i} \to X$, then $$\mathcal{R} \circ F_i \circ \mathcal{R}_{k_i} = \mathcal{R} \circ F_i$$

This would mean that we can approximate computations (actions of the family $\mathcal{F}$) on $X$ by computations in $Y$ with no ill effects, up to some well-defined finite error.

In reality finding such a rounding function $\mathcal{R}$ can be very hard. Consider just the rounding of real numbers ($X = \mathbb{R}$) to integers ($Y = \mathbb{Z}$). No tie-breaker rule for the half-way rounding commutes with simple addition. So we often, in applications, relax the "commutation" rule by asking that it be satisfied for "generic" data, so that only very unlikely outliers will cause the commutation rule to break down. But to understand the notion of generic and to get a quantitative understanding of how badly the commutation rule can break requires some knowledge in probability and statistics.

• Implicit in my answer is the sentiment that "rounding should not be taught in isolation, but it should be taught in view of the fact that it is easier to add 2 to 3 than to add 2.14589 to 2.98027." Rounding is really about trade-offs: simpler representation of data and ease of manipulation against loss of precision (and also possibly accuracy, which is why there's all those different tie-breaking rules). Mar 17 '14 at 9:55
• Interesting. If you round to the nearest even, won't even numbers show up more in your data, and be an 'artifact of rounding'? Isn't this a problem? Mar 17 '14 at 16:07
• @SueVanHattum: that's the whole point. There is no one correct rounding method. Each method has its strength and weaknesses. My answer is about how the assessment of the strengths and weaknesses, and by extension the choice of the correct rounding method for a particular application often requires some mathematical sophistication. (And yes, you interpret my notation correctly.) Mar 17 '14 at 16:16
• @jpd527: I am no researcher in mathematical pedagogy, but my position is that rounding (and the methods for rounding) should be tied to their applications. As long as you have a good application in mind that the students can understand, you can use it to discuss the pros and cons of different rounding methods. The well-known biases that can creep in when using one of the common rounding rules can probably be converted to classroom exercises for students to discuss. But "simply introducing" something for students to learn by rote, is, imho, not the best way. Mar 18 '14 at 15:29
• (What I have in mind: get yourself a roulette wheel but label each space with a multiple of 1/4, use this to generate a bunch of random numbers. Show how that the 5-up-4-down rounding method will systematically overestimate the sum. Explain to students why this can be a problem for, e.g. banks. Show that rounding to evens solve the problem. Then do some binning on the numbers generated, show that there's a bias toward even numbers using the rounding to evens method (you will see a rather large excess now), and explain that the other method is better.) Mar 18 '14 at 15:36

The reason is that rounding half up is by far the simplest rounding rule, because you only have to look at the next relevant digit.

For example, everyone agrees that you should round $12.5134$ up to $13$, since $12.5134$ is closer to $13$ than to $12$. The issue only arises when you are rounding exactly $12.5$, with no further digits. In this case, it's easier to just remember the rule "always round up 5's" than to worry about the distinction between these two cases.

Of course, it would be possible to discuss further rounding rules at some point, as well as the advantages and disadvantages of each rule. But the reality is that it just isn't important enough to be worth a spot in the curriculum. The few professionals who have to deal with such things (accountants, some computer scientists, etc.) are aware of the distinctions, as are those of us who appreciate mathematical arcana, but there's just not a pressing need to inform the average math student.

• This invokes memorys of endless discussions with my elementary-school teacher who insisted that 12.5 were closer to 13 than to 12. Mar 17 '14 at 6:52
• Also, to add to this: In contrast to the general concept of rounding, individual rounding methods are easy to learn, so it’s no problem to learn other rounding methods when required. It may be even better, since at that point you can easily motivate the respective method. Mar 17 '14 at 7:05

Mathematics is logical. I do not see how always rounding up when you are halfway between two numbers is a logical thing to do. If a number is exactly halfway between two numbers, then rounding to either would make sense.

If you are looking at data, and want no change in the distribution, it seems to me that you would want to alternate between rounding up and rounding down.

As a math educator, I think it's very important that math always make sense to students. Each step they take should have reasons that make sense to them.

In response to "[At] what level/area of math would it be wise to introduce this concept for a minimum of exposure to the idea?" It's clear from the complex answer provided by Willie Wong that this sort of analysis cannot come before college (or advanced high school) courses. Until then, students could discuss the purposes of rounding, and work together on what would be a sensible way to do it.

• Sorry to be a DV, but clearly, it's math. Others have stated the need for these rules. Mar 17 '14 at 13:18
• No problem, Joe. Mar 17 '14 at 16:11
• I hope both of you, and the third downvoter, will let me know whether my edits help my case. Mar 17 '14 at 16:11
• Math has plenty of conventions that are set for sensible, but not rigorous reasons, and students would benefit from acknowledging this and discussing the reasons earlier in the curriculum than we do. (I think of the posts that go around social media designed to trick people about order of operations, and the many comments from people who aren't prepared for the existence of variations in the convention.) Even if students cant't do the rigorous analysis until later, I like the suggestion that students discuss the purpose of rounding and consider potential benefits to different schemes. Oct 14 '14 at 16:54
• I totally agree about the scorn people show for the conventions with those silly Facebook posts. I can explain to students why we have those (order of operations) conventions. I can't explain the round up convention adequately, and I'd rather my students use common sense, which they tend to want to leave at the door. Oct 15 '14 at 17:50

I mostly teach physics, not math, so I get a lot of experience in my lab classes where I see students manipulating numerical data. In particular, I often see them do numerical calculations that are long enough so that rounding errors really could accumulate. (I doubt that math teachers very often see their students do a really lengthy numerical calculation.)

The thing that stands out to me most from this experience is that most students at this level (freshman college STEM majors) show an almost complete inability to reason about how a calculation works. This extends to the most incredibly simple concepts, such as the idea that when you change the input to a calculation, the output will also change.

As an example, I will often see students write down intermediate steps in their lab notebooks, recording all the digits from their calculators. As an imaginary example, suppose that they've just immersed a cube in water and measured its volume to be 124 cubic centimeters. They next use this to calculate the length of the cube's edge, which is going to be a step in a longer calculation. They write it down as 4.98663095223864 cm. When I ask them why they're doing that, here are some typical answers I get:

• That's what my calculator gave me.

• In chemistry you're not supposed to round off. Is it different in physics?

Answers like these show that they're not reasoning at all about how the calculation works. They expect to be told an arbitrary rule to follow, which will ensure them a good grade.

For these reasons, I agree very much with Sue VanHattum. When something like this comes up, its value lies in the fact that we can use it to guide students toward thinking more deeply about what would make sense. Teaching the student a more statistically desirable procedure should not be the point, unless the student is ready to make sense of the procedure. Sometimes we can say things to our students that will nudge them into thinking:

• When I was a kid, my family got our first electronic calculator, and it had a 6-digit display. I see that you iPhone calculated that cube root with 14 digits of precision, and you wrote them all down in your lab notebook. If you got a newer phone and it displayed 100 digits, would you write all of those down?

• This other lab group measured 123 cm$^3$ for the volume of the cube. If we took the cube root of that number instead of your 124 cm$^3$, which of those digits in your result do you think would change?

As a practical matter, for people who are more numerate, I don't think it makes a lot of sense to follow any particular rule for rounding fives. In almost all real-world applications, the answer is that if rounding a 5 up or down is enough to make a difference in your life, then you're rounding too much; you should retain more digits of precision.

• When I was an undergraduate, I remember doing similar labs. What I wanted to do was manipulate intervals of numbers rather than numbers. So in this example, I would write something like " We measure the volume $V \in (123,125)$, so the side length $s \in (11.090536506409418, 11.180339887498949)$, etc. I was told to stop doing that, and learn conventions about significant figures. I never thought that the significant figures were very honest, or transparent. Just curious for your opinion on this idea. Oct 14 '14 at 13:34
• Significant figures are nearly equivalent to your method, provided that all the operations are multiplication and division. But for addition and subtraction it makes more sense to keep track of the number of decimal places. Your idea about ranges is reasonable, but the error bars on a measurement are usually more like a standard normal curve rather than a fixed range with zero probability of being outside the range. I would normally retain one or two extra sig figs, then at the end do propagation of errors, which is more efficient and statistically accurate than the range idea.
– user507
Oct 14 '14 at 23:58
• @StevenGubkin Your range convention is called interval arithmetic, and it does exist, it’s just extremely cumbersome in manual or slide-rule calculations (the origin of most of the conventions) and difficult to include in more complex numerical algorithms like PDE solvers. [cont.] Nov 27 '18 at 19:02
• [cont.] Even on a computer you’ll have a finite number of digits to store your boundaries, so doing this sort of thing properly requires (and is, indeed, the main reason for having) explicit round-to-{negative,positive}-infinity modes, which your calculator most likely didn’t have. Nov 27 '18 at 19:05