Hack70.Use Numbers Carefully

Hack 70. Use Numbers Carefully

Our brains haven't evolved to think about numbers. Funny things happen to them as they go into our heads.

Although we can instantly appreciate how many items comprise small groups (small meaning four or fewer [Hack #35] ), reasoning about bigger numbers requires counting, and counting requires training. Some cultures get by with no specific numbers higher than 3, and even numerate cultures took a while to invent something as fundamental as zero.¹

So we don't have a natural faculty to deal with numbers explicitly; that's a cultural invention that's hitched onto natural faculties we do have. The difficulty we have when thinking about numbers is most apparent when you ask people to deal with very large numbers, with very small numbers, or with probabilities [Hack #71] .

This hack shows where some specific difficulties with numbers come from and gives you some tests you can try on yourself or your friends to demonstrate them.

The biases discussed here and, in some of the other hacks in this chapter, don't affect everyone all the time. Think of them as forces, like gravity or tides. All things being equal, they will tend to push and pull your judgments, especially if you aren't giving your full attention to what you are thinking about.

7.2.1. In Action

How big is:

9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

How about:

1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9

Since you've got both in front of you, you can easily see that they are equivalent and so must therefore equal the same number. But try this: ask someone the first version. Tell her to estimate, not to calculatehave her give her answer within 5 seconds. Now find another person and ask him to estimate the answer for the second version. Even if he sees the pattern and thinks to himself "ah, 9 factorial," unless he has the answer stored in his head, he will be influenced by the way the sum is presented.

Probably the second person you asked gave a smaller answer, and both people gave figures well below the real answer (which is a surprisingly large 362,880).

7.2.2. How It Works

When estimating numbers, most people start with a number that comes easily to mindan "anchor"and adjust up or down from that initial base. The initial number that comes to mind is really just your first guess, and there are two problems. First, people often fail to adjust sufficiently away from the first guess. Second, the guess can be easily influenced by circumstances. And the initial circumstance, in this case, is the number at the beginning of the sum.

In the previous calculations, anchors people tend to use are higher or lower depending on the first digit of the multiplication (which we read left to right). The anchors then unduly influence the estimate people make of the answer to the calculation. We start with a higher anchor for the first series than for the second. When psychologists carried out an experimental test of these two questions, the average estimate for the first series was 4200, compared to only 500 for the second.

Both estimates are well below the correct answer. Because the series as a whole is made up of small numbers, the anchor in both cases is relatively low, which biases the estimate most people make to far below the true answer.

In fact, you can give people an anchor that has nothing to do with the task you've set for them, and it still biases their reasoning. Try this experiment, which is discussed in Edward Russo and Paul Schoemaker's book Decision Traps.²

Find someonepreferably not a history majorand ask her for the last three digits of her phone number. Add 400 to this number then ask "Do you think Attila the Hun was defeated in Europe before or after X," where X is the year you got by the addition of 400 to the telephone number. Don't say whether she got it right (the correct answer is A.D. 451) and then ask "In what year would you guess Attila the Hun was defeated?" The answers you get will vary depending on the initial figure you gave, even though it is based on something completely irrelevant to the questionher own phone number!

When Russo and Schoemaker performed this experiment on a group of 500 Cornell University MBA students, they found that the number derived from the phone digits acted as a strong anchor, biasing the placing of the year of Attila the Hun's defeat. The difference between the highest and lowest anchors corresponded to a difference in the average estimate of more than 300 years.

7.2.3. In Real Life

You can see charities using this anchoring and adjustment hack when they send you their literature. Take a look at the "make a donation" section on the back of a typical leaflet. Usually this will ask you for something like "$50, $20, $10, $5, or an amount of your choice." The reason they suggest $50, $20, $10, then $5 rather than $5, $10, $20, then $50 is to create a higher anchor in your mind. Maybe there isn't ever much chance you'll give $50, but the "amount of your choice" will be higher because $50 is the first number they suggest.

Maybe anchoring explains why it is common to price things at a cent below a round number, such as at $9.99. Although it is only 1 cent different from $10, it feels (if you don't think about it much) closer to $9 because that's the anchor first established in your mind by the price tag.

Irrelevant anchoring and insufficient adjustment are just two examples of difficulties we have when thinking about numbers. ( [Hack #71] discusses extra difficulties we have when thinking about a particularly common kind of number: probabilities.)

The difficulty we have with numbers is one of the reasons people so often try to con you with them. I'm pretty sure in many debates many of us just listen to the numbers without thinking about them. Because numbers are hard, they lend an air of authority to an argument and can often be completely misleading or contradictory. For instance, "83% of statistics are completely fictitious" is a sentence that could sound convincing if you weren't paying attentionso watch out! It shows just how unintuitive this kind of reasoning is, that we still experience such biases despite most of us having done a decade or so of math classes, which have, as a major goal, to teach us to think carefully about numbers.

The lesson for communicating is that you shouldn't use numbers unless you have to. If you have to, then provide good illustrations, but beware that people's first response will be to judge by appearance rather than by the numbers. Most people won't have an automatic response to really think about the figures you give unless they are motivated, either by themselves or by you and the discussion you give of the figures.

7.2.4. End Notes

1. The MacTutor History of Mathematics Archive: a History of Zero (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html).

2. Russo, J. E., and Schoemaker, P. J. H. (1989). Decision Traps. New York: Doubleday.