LogicalDUAL-USE

Large-Number Innumeracy

What it is

Presenting a million, a billion or a trillion — or any figure beyond everyday experience — without a scale, a per-person figure or a comparison, so the audience reacts to the word rather than the size and can be steered to read it as enormous or as trivial.

How it works

Beyond a few thousand, numbers stop being quantities and become adjectives. Paulos's 1988 test: a million seconds is about eleven and a half days, a billion seconds is nearly thirty-two years, a trillion is around 31,700 years, and most educated adults cannot place them within an order of magnitude. Kahneman and colleagues found that people will pay about the same to save 2,000 birds as 200,000, and Slovic showed that concern for victims does not scale with their number and can even fall. The persuader works both directions. With no scale attached, “40 billion dollars” reads as vast; with the scale attached — a federal budget of some six trillion — it is under one percent, and a spending advocate leads with the fraction while a deficit hawk leads with the raw figure, each omitting the other. Worked: in March 2020 an MSNBC segment reported that a candidate's 500 million dollars of advertising could have given every American a million dollars; the correct figure is about 1.53 dollars each. Surveys have long found Americans placing foreign aid at a quarter of the federal budget when it is about one percent. The lever is that a number without a comparison cannot be checked against experience, and a number that cannot be checked is read however it is framed.

Real-world examples

  • •On 5 March 2020 an MSNBC anchor and a guest agreed on air that Michael Bloomberg's roughly 500 million dollars in campaign advertising, divided among 327 million Americans, would have given each of them a million dollars; the actual figure is about a dollar and a half, and the error passed two people and a producer before it was noticed.
  • •Kaiser Family Foundation polling in 2015 found that Americans estimated foreign aid at an average of about a quarter of the federal budget; the actual share was around one percent, and support for cutting it fell sharply when respondents were told the real figure.
  • •Deficit hawks quote the national debt in raw trillions without reference to national income; advocates of spending programs describe them as “a fraction of a percent of the budget” without the dollar figure per household. Each frame depends on the audience having no feel for the scale, and each party uses whichever frame suits the program under discussion.
  • •The 1980s reports of the Pentagon paying 435 dollars for a hammer and 640 for a toilet seat were vivid because the numbers were human-sized; the billions in procurement overruns in the same period, which mattered far more, were not, because they were not.

Ethical guidelines

Where the line is

Quoting a large figure is legitimate when it comes with a scale the audience can check — per person, per day, share of the total, a comparison — and when the same form is used for both sides of a comparison; it becomes manipulation when the scale is withheld or switched so that the audience reads the number as enormous or negligible according to the speaker's need.

  • ●Attach a scale to every large number: per person, per household, per day, or as a share of the relevant total, and give the comparison your opponent would give as well as your own.
  • ●Do not switch between raw figures and shares according to which makes your case; use both forms for both sides of an argument.
  • ●Check the arithmetic on any per-person conversion before publishing it; the errors are usually of a thousand-fold and always embarrassing.
  • ●When the point is the number of people affected, give the number and a comparison that can be felt — a stadium, a city — rather than relying on the word “thousands”.

How to defend against it

  • ►Divide by something. Any figure in millions, billions or trillions should be turned into a per-person, per-day or percent-of-total figure before you react to it.
  • ►Ask for the denominator the speaker is not giving: the budget for a spending figure, the population for a headcount, the total for a share.
  • ►Keep three anchors in memory — a million seconds is twelve days, a billion is thirty-two years, the US federal budget is around six trillion dollars — and test claims against them.
  • ►When a number is offered as “only a fraction of a percent”, ask what it is in dollars per household; when it is offered as “billions”, ask what it is as a share. Both answers together are the size.

From the Defense Playbook

Ask for the Denominatorseconds

Whenever you are shown a count ("4,000 complaints", "12 deaths", "9 out of 10 dentists"), ask "out of how many?", because a numerator on its own cannot tell you whether something is common, rare, rising, or falling.

Every playbook entry states how strong its evidence is and when not to use it. Browse the full playbook.

References

  1. Paulos, J. A. (1988). Innumeracy: Mathematical Illiteracy and Its Consequences. Hill and Wang
    The inability of educated adults to place large numbers, and the seconds-to-years anchors.
  2. Kahneman, D., Ritov, I., & Schkade, D. (1999). Economic preferences or attitude expressions? An investigation of the psychology of dollar responses to public issues. Journal of Risk and Uncertainty, 19(1-3), 203-235
    Scope insensitivity: willingness to pay that barely changes with the number of lives or animals at stake.
  3. Slovic, P. (2007). “If I look at the mass I will never act”: Psychic numbing and genocide. Judgment and Decision Making, 2(2), 79-95
    Concern that fails to scale with, and can decline with, the number of people affected.
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