Estimation: The Value of Imprecision

Many questions we have received have been about various aspects of estimation. Often this topic has been downplayed, because we tend to think of math as being all about precision; but it is essential in many applications, sometimes because there is nothing else to do, and other times because exactness wastes effort. I am starting a series of posts on this topic at various levels, starting here with why it is important, at an elementary level, to be able to estimate ordinary calculations.

The big picture

We’ll start with a very broad question, from 2002:

Why Estimate?

Why do we use estimation?

Doctor Ian took this one:

The _point_ of estimating things is that estimating will save you space, time, or effort, and in return for those savings, you agree to pay a price in accuracy. 

There are lots of different ways to estimate things: you can eliminate digits from a number (truncation or rounding), you can replace a decimal number with a fraction (which, technically, is the same thing), you can replace any number with a 'nearby' number that's easier to work with, you can use a graph to replace a list of individual values, you can use a simple function in place of a more complicated function, and so on.

In each case, you're making a trade-off: "It's not going to hurt much if my answer is off by a little bit, especially if I can get that answer ten times more quickly, or by storing a thousand times less information."

The _art_ of estimating things properly requires two different sets of skills.  The easier one is learning the various techniques of estimation: rounding, graphing, and so on.  The harder one is learning to recognize how much accuracy you can afford to throw away in a given situation, and how much benefit you'll get from doing it.

This is estimation in a nutshell: giving up some accuracy in order to gain some sort of efficiency; and deciding how much of the former is a worthwhile price for the latter. Estimating well is not just a matter of technique, but of understanding the application. It is a judgment call (and therefore hard to grade objectively, another reason teachers may avoid it or teach it ineffectively).

Many readers will not recognize what Doctor Ian means by using a simpler function, because this is not seen at lower levels. But, for example, complicated formulas (or even formulas that can’t be worked out at all) are often replaced with approximate polynomial formulas, which can be easily worked with. This is one of the reasons students learn so much about polynomials: not because real-life problems typically have polynomial solutions, but because pretending that they are polynomials makes it possible to solve them approximately. And that is good enough for practical purposes.

Checking by estimating

A month earlier, we had a specific question about estimation, which served as an example of why we do it:

Why Estimate?

Estimate to the nearest thousand a solution to Dave's subtraction problem. The numbers are:

   6,107
  -2,980
   -----
   4,127

I just don't know how to estimate.

I noticed that the problem could have been written without showing a result for the subtraction. Clearly, the problem is meant to teach something more than how to estimate. I answered:

Assuming that 4,127 is "Dave's" answer to the problem, it illustrates why it can be very important to estimate - it's wrong, it's an easy mistake to make, and it would have a huge effect on whatever he uses the number for, from spending money to building a house. And estimating makes it easy to see the error!

So, the reason  we might estimate the result here, even though someone has already claimed to have an exact answer, is to check it. In fact, it is quite easy with a calculator to think you have an exact answer, when it is really very wrong. Estimation is an essential companion to calculators: calculators bypass our own thinking, while estimation forces us to think. We’ll have more to say on this below.

Now I went through the process of estimating as instructed, leaving the actual work to Stephanie:

To estimate to the nearest thousand, the natural thing to do is to round the numbers you are adding to the nearest thousand. Which of these numbers is closest to 6,107?

    0   1000 2000 3000 4000 5000 6000 7000 8000 9000

Which of those numbers is closest to 2,980?

Now subtract those much simpler numbers, and the result is your answer. Sometimes you can get a better estimate by thinking about how much you increased or decreased each number when you rounded them, and how that will affect the result; but in this case the difference of the rounded numbers is good enough.

Notice that I didn’t teach a routine method for rounding to the nearest thousand, but emphasized the meaning of what we are doing. (I later gave links to pages with more detail on the routine process.) Carrying out the work, we see that the “thousand” nearest to 6107 is 6000 (because it is less than the halfway point between 6000 and 7000); and the “thousand” nearest to 2980 is 3000 (because it is beyond the halfway point between 2000 and 3000). We were told to estimate by rounding to the nearest thousand, so we replace the original numbers with the estimates, and do the subtraction: 6000 – 3000 = 3000. That is the answer they want. Since the given answer, 4127, is quite far from this (in fact, farther than the largest possible error introduced by rounding), it must be wrong.

In my post on March 12 about rounding, I discussed how in the context of estimation it is often better not to “round to the nearest”. I just hinted at that here. You may want to try using some of those ideas; there are several things to be learned from this example.

The correct answer is 3127, not 4127; “Dave” made an error in the thousands digit. Note that if he was right there, but wrong in, say, the tens digit, estimation would not catch the error. We would just compare, say, 3217 to 3000 and say it looks good.

So estimates are a good way to become suspicious of an answer, but they don’t catch every error, and can never prove it is correct.

Do calculators make mental math unnecessary?

Let’s move on to a very different question, from 2004:

Why Is Mental Math Important?

How can I convince a 14 year old girl who is in 8th grade the importance of mental math?  I tried to point out some real life situations but her teacher told her it is not really important since there are calculators and computers.  I also pointed out that if she does not learn such things it could make her unsuccessful further down the road in life.

I think that things like adding 2 digit numbers together using techniques such as adding the tens place first and then adding the ones place and then adding the results are critical skills.  Similar skills include doing subtraction mentally when using 2 digit numbers and being able to mentally estimate multiplying 2 digit large numbers.

John is asking about the importance of two different things: “mental math” (getting an accurate answer for simple problems without paper or calculator), and “estimation” (getting approximate answers easily). Two of us answered. Doctor Link started with a pedagogical reason:

Mental math, irrespective of its applications, is an excellent way to stimulate one's mind.  Not only does it generally stimulate your mind, but it also helps one get a better "number sense."  In other words, one becomes more familiar with how numbers interact--this is very important, because as you know, math is something that builds on itself.  If you don't have a good grasp on how numbers interact, then more complicated math will seem like more of a challenge.

Working only with a calculator insulates a student from the numbers, making her just a copyist rather than a participant. Then he moved on to a social reason:

I wasn't going to mention it, but since you brought it up I will.  If you can't add and subtract without the help of a calculator, it can certainly reflect poorly on you.  Even if you really are a very intelligent person otherwise, this inability may give people a different idea.  For better or for worse, our society puts a lot of stock into appearances.  If one doesn't appear to be educated or "smart" at first glance, society will often just label that person as such.

Being able to do basic math suggests intelligence, and in some settings that will be important.

There is also a simple, practical reason : habitual use of a calculator can be a hindrance; sometimes it is actually quicker to do the mental math:

Also, the fact is that calculators are most useful in a setting that requires either a) things that are essentially impossible for humans to compute at any reasonable speed (like evaluating the cosine of a number) or b) for calculations with rather large numbers.  For two digit addition, not using a calculator should actually be faster, given that you have learned the techniques well.

In working with students face to face, it is far too common to find them picking up a calculator to multiply -1 times 3, or 34 divided by 34! This makes it clear that they are completely bypassing their minds, turning to the calculator when it is absolutely not needed. As Doctor Link said, this reflects poorly on the student, and it also means they don’t gain the experience of thinking. I often tell such students, “Your calculator doesn’t need the practice — you do!”

There are a myriad of other reasons, from being able to dazzle one's friends and future coworkers to giving one confidence about their abilities.  Ultimately though, I am very surprised to hear that your daughter's teacher fueled this.  When I was growing up (which wasn't too long ago), students constantly questioned the usefulness of mental math.  My teachers always maintained that it was very important to be able to do these calculations ourselves.  After all, thinking is what makes us human--not dependence on machines to do our thinking for us!

Finally, he touched on a specific practical reason for at least estimating:

My brother also reminded me of one other important thing.  What if you are getting your change, and the cashier has either accidentally or purposely given you less change then they should have.  Without the ability to do mental math, you may overlook this--so there's your real life scenario for you.

I picked up from here, emphasizing estimation:

In the modern world of computers and calculators, mental estimation skills are MORE important, rather than less!  That's because, although calculators and cash registers make it unnecessary to do exact calculations in many instances, it can be easy to trust them too much. People can make mistakes in using them, entering the wrong amount or doing the wrong thing with it, and if we just blindly accept the answer it gives without checking, we'll lose out a lot.  One major kind of error is putting the decimal in the wrong place (or failing to enter it at all), another is to miss a key so that what should have been 1234 is taken as 124, off by a factor of ten!  Or how about hitting "percent" instead of "divide"!  These kinds of mistakes are easily caught by estimating the correct answer.  Estimation won't catch errors of pennies, but it will catch dollars, which are more important anyway.  You can probably find other examples where this will be important in her life, but money is a good motivator!

It's worth noting that estimation uses the same skills as exact mental arithmetic, and more.  It also builds number sense by making you aware of what matters most.  A teacher who says arithmetic skills are unimportant is depriving students of an essential life skill.

John wrote back that the social argument might be the most persuasive; but money can be, too.

Ballpark figures

Another question from 2004 about a very different kind of estimating (related to “Fermi problems”, which I will be discussing later in this series) suggests that number sense can take on bigger dimensions than the little problems we have been discussing:

The Importance of Number Sense and Estimating Answers

I don't understand how to solve problems using number sense.  For example, if an employee at the deli counter slices a 2 foot long salami, about how many slices will he get?  How can I figure that out? Why is it important?

Our previous estimation problems involved approximating given numbers. Here, we aren’t even given all the numbers we need! This takes us beyond math proper, into the real world, where our “number sense” has to fill in numbers, and give us confidence that we don’t need accurate numbers at all in order to find sensible answers. This sort of question often arises in real life, where you may need to decide if a project is feasible, without knowing all the facts.

Doctor Ian took this; again, I will only excerpt parts that are relevant to my topic, so you will want to read the whole answer.

One thing that makes this confusing is that there isn't enough information to come up with a definite answer.  That is, the number of slices he'll get will depend on how thick each slice is. ...

So where does number sense come into this?  Basically, you want to make a reasonable assumption about the width of a slice, and use that to get a ballpark figure for how many slices we might end up with.  I'd probably say that 1/10 of an inch is a reasonable width for a slice, and that would give me

   24/(1/10) = 24 * 10 = 240

slices. Maybe there are more, maybe there are fewer, but this is a _reasonable_ number.

So when we don’t have a number, we can make one up based on experience.

Why do you care?  Well, suppose someone works the problem and tells you that the answer is 3, or 150,000.  Would you know right away that these are ridiculous answers?  The way you'd check them is by making some reasonable assumptions and using those to come up with an answer of your own. 

So if someone says "About 500 slices", we could say: "Okay, that's about twice what we got, so he's making his slices half as thick.  That's still reasonable."  Or if someone says "About 100 slices", we could conclude that he's making his slices twice as thick.  Which is pretty thick, but not unrealistic.

So, as in our calculator problems, we can recognize a nonsense answer, or validate one that is not far off.

Again, why do you care?  Well, this is a made-up problem, but as you get out in the world, you're going to be hearing people throwing all kinds of numbers around, and you'll need to be able to get some sense of whether they're realistic.

Doctor Ian then discusses a typical claim one might hear repeated (which happens far more now, with social media and “fake news”!). On the surface, it doesn’t sound unreasonable — that’s why it might be passed on by so many people without being checked. Read the original for this long example.

The point is that people throw around all kinds of numbers in efforts to convince other people to pass laws, raise or lower taxes, donate money, change lifestyles, declare war, and make other major decisions. Quite frankly, a lot of these numbers are ridiculous.  But if you can't figure out for yourself which ones make sense and which ones don't, then you just have to believe what you're told, and hope for the best.  

And cynicism aside, much of what goes on in business and engineering involves making predictions--e.g., estimating how many burgers people will probably buy next week, or calculating how much electrical current will be running through a wire in a given situation.  People make mistakes (especially where computers or calculators are involved!) and sometimes they come up with ridiculous predictions.  Number sense--i.e., being able to quickly come up with ballpark estimates--can be crucial in detecting these mistakes early on, when they can still be corrected easily.

These can be very practical matters, and are good reasons for adults to make such estimates all the time. But too many don’t, because they haven’t been trained to do so.

So, how do you develop this mysterious number sense?  In a word: practice.  One good way to get this practice is to make a point of never using a calculator unless it's absolutely necessary.  When I was trying to convert "one person every 30 seconds" to an equivalent number of people per year, I could have used a calculator, but I deliberately chose not to, because I knew that whatever number I came up with would be approximate, instead of exact; and I knew from experience that I would be able to change the numbers around a little to end up with an easy calculation that would be "close enough for government work".

Be sure to read the original, where you will see some of the ways you can make a problem easier by adjusting the numbers, when you know you don’t need an exact number anyway. Doctor Ian’s choice not to use a calculator is much like choosing to park farther from the mall to give yourself exercise. You do what you need to do to keep both mind and body strong.