LogicalDUAL-USE

Percentage of a Percentage

What it is

Describing a change in a rate that is itself a percentage either as a relative percentage change or in percentage points, whichever makes the movement sound larger or smaller — “a 25 percent jump” or “one point” for the same move from 4 to 5 percent.

How it works

When the quantity being discussed is already a percentage, a change in it has two honest descriptions that differ by an order of magnitude. Unemployment rising from 4 to 5 percent is a rise of one percentage point and a rise of 25 percent; a tax rate going from 35 to 39.6 is 4.6 points or 13 percent; an interest rate moving from 0.25 to 0.5 has “doubled” while changing by a quarter of a point. Both descriptions are arithmetically correct, and the lever is that ordinary speech uses “percent” for both, so an audience hearing “support fell 10 percent” cannot tell whether it went from 50 to 45 or from 50 to 40. Presenters choose the relative form to dramatize and the point form to minimize, and the choice can flip within one debate as each side describes the same change. The device is the relative-versus-absolute problem applied to rates, with a linguistic ambiguity on top. The honest practice is to state both — “from 4 to 5 percent, a one-point or 25 percent increase” — because the pair is short and the reader can then judge the size for themselves.

Real-world examples

  • •When the UK raised VAT from 17.5 to 20 percent in January 2011, supporters described a 2.5 percent rise and opponents a 14 percent rise; both were correct, and neither side volunteered the other figure.
  • •The 2012 debate over letting the top US income-tax rate return from 35 to 39.6 percent was conducted as “a 13 percent tax increase” by opponents and “a 4.6 point increase” by supporters — the same change, chosen by each side in the form that served it.
  • •The US Federal Reserve's December 2015 move from a 0-0.25 percent target range to 0.25-0.5 was headlined as rates “doubling”; the change was a quarter of a percentage point, the smallest step the Fed makes.
  • •Poll coverage that reports a candidate's support “falling by half” when a minor candidate goes from 6 to 3 percent, or “surging 50 percent” from 2 to 3, is arithmetically accurate and gives readers no sense that the movement is inside the poll's margin of error.

Ethical guidelines

Where the line is

Either form is legitimate when the other stands beside it or the before-and-after rates are given, since the reader can then judge the size; it becomes manipulation when the form is chosen for its sound, switched between one's own figures and the opponent's, or used so that a change in a rate is heard as the rate itself.

  • ●Give both figures — the points and the percentage — whenever the quantity is itself a rate; the pair takes six words.
  • ●Say “percentage points” when you mean points, and do not let a relative change in a rate be heard as a change in the rate itself.
  • ●Use the same form for movements that favor you and movements that do not.
  • ●For small rates, prefer the point form with the absolute numbers of people affected; “doubling” a tiny rate is a case for the relative-risk entry, not a headline.

How to defend against it

  • ►Ask for the before and after values of the rate itself: “from what to what?” Once you have them, both descriptions are yours to compute.
  • ►When you hear “percent” applied to something that is already a percentage, ask whether points or a ratio is meant; the ambiguity is usually where the persuasion lives.
  • ►Convert relative changes in rates into people: a one-point rise in unemployment in a labor force of 160 million is about 1.6 million people, and that number is more informative than either “one point” or “25 percent”.
  • ►Notice when one side of a debate uses points and the other uses percentages for the same change; that is the tell that both are choosing, and you should hold both.

From the Defense Playbook

Ask for Absolute Numbersseconds

When someone quotes a relative change ("cuts your risk by 50 percent", "doubles the danger"), ask what the risk was before and after in plain counts out of the same number of people, because the same fact sounds enormous as a ratio and modest as a difference.

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

References

  1. Huff, D. (1954). How to Lie with Statistics. W. W. Norton
    Early treatment of percentages of percentages and of choosing the base that makes a change look large or small.
  2. Best, J. (2001). Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists. University of California Press
    How advocates describe the same change in rates in the form that serves their claim.
  3. Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2007). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8(2), 53-96
    The general finding that relative changes are misread as absolute ones and the recommendation to report both.
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