Why 85% Is a Cliff, Not a Line

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

foundations

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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.

See the relationship before changing it

The figure reads from left to right. The blue card is task success. The middle card applies the page rule. The green card is two-failure chance. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

Walk the arrows. Back-to-back failure grows on a curve, so a small success drop can nearly double the pain.

Derive the baseline in four named moves

1
Name the input. The chapter baseline is 85 %.
2
Name the relationship. chance = (1 - success fraction)^2 x 100
3
Substitute with units. (1 - 0.85)^2 x 100 = 2.25%
4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change task success

Try Predict the direction of chance = (1 - success fraction)^2 x 100. Test another task success, then compare two-failure chance.

Observe Back-to-back failure grows on a curve, so a small success drop can nearly double the pain. Reset task success to 85 and compare two-failure chance.

Explain Back-to-back failure grows on a curve, so a small success drop can nearly double the pain.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only task success moves here. Field effects named in the technical boundary stay fixed.

TryChoose Check derivation at the 85% task-success threshold, then compare a 6-point shortfall with the squared penalty.

ObserveThe abandonment penalty accelerates from 2.25% to 4.41% as success drops toward 79%, rather than changing linearly.

ExplainSquaring distance below the 85% threshold makes each additional lost point cost more, which creates the modeled cliff.

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.

Technical boundaries
This illustrative curve does not establish causal behavior or model cohorts, accessibility, task difficulty, measurement error, or confidence intervals.

Work the audit first, then check the displayed derivation.

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