Why 85% Is a Cliff, Not a Line

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

Why 85% Is a Cliff, Not a Line

The chapter treats 85% task success as a threshold and says every point below “accelerates abandonment” — a claim about curvature. Because users feel two consecutive failures, the trigger is the failure rate squared: 2.25% at 85% but 4.41% at 79%, nearly double for a 6-point drop. This audit checks the numbers behind that word to show why 85% is a cliff, not a line.

Companion to the chapter Interface Design: Worked Examples — every number here comes from that chapter.

Ada: The chapter says every point below 85% “accelerates abandonment” and treats 85% as a threshold. That word accelerates is a claim about curvature, so let me check the numbers that back it, and stick to the ones that are clean.

Start with the success rate itself, from the Temperature-adjust row:

• 742 / 876 = 0.8470 = 84.7%

Now the abandonment trigger, two consecutive failures. Because failures multiply, the probability is the failure rate squared:

• At 85%: (1 - 0.85)^2 = 0.15^2 = 0.0225 = 2.25%
• At 79%: (1 - 0.79)^2 = 0.21^2 = 0.0441 = 4.41%

A 6-point drop in success rate does not raise the two-failure probability by 6% of anything, it nearly doubles it: 4.41 / 2.25 = 1.96. That is the whole reason 85% behaves like a cliff rather than a gentle slope. The quantity users actually feel, back-to-back failures, is the square of the failure rate, so small movements near the threshold have outsized effect: the curve steepens exactly as you approach the edge. The design-meaning is that a raw success rate is the wrong thing to read linearly; the honest metric is (1 - p)^2, and it says the last few points before 85% are worth far more than their arithmetic size suggests.

Every number above is taken from the chapter’s own material and re-derived step by step.