Across the break, equal distances no longer mean equal values

What it is

The uniform-scale contract says "equal distance means equal value": a centimetre of axis means the same number of units wherever it sits. A broken axis defaults on it — below the gap a centimetre is ten units, above it perhaps a hundred, while the axis looks unchanged. Three basic readings fail at once: the distance between two points straddling the break has no definition (which side's conversion?); the slope across the break has none (a line with different slopes on either side snaps at the seam); "how much higher" across the break requires the reader to splice two conversions, which readers do not do.

Why it happens

When comparing distances and angles, the visual system does not consult the scale: which of two lines is steeper or which point higher is a direct perceptual verdict, correct only by coincidence when the scale is uniform. The break turns "coincidentally correct" into "necessarily wrong," and unpredictably so — depending on which side of the break the data falls and whether it straddles. The most insidious case is trend reversal: with different break placements the same data can read as accelerating or decelerating, while each chart looks self-consistent "within its side." Readers have no universal correction formula (unlike end truncation's amplification factor); the only sound policy is refusing cross-break comparisons.

Where it stops holding

The failure is precisely inference across the break: comparisons lying wholly on one side remain valid (each side is internally uniform) — which is why broken axes occasionally "seem to work." But that stakes reliability on "readers happen not to need cross-break judgements," while in outlier scenarios the most-wanted comparison is precisely outlier versus majority — the crossing one. In interactive exploration, a broken axis should at least offer hover readouts of true values, bypassing the distance corruption.

Applying it

• Review rule: any broken axis disables by default every cross-break conclusion sentence (steeper / higher / doubled) unless the conclusion rests on labelled values.
• Where a break must stay, give each side its own local grid and ticks, signalling "read segment by segment."
• Verification: have readers compare one cross-break pair and one same-side pair; if cross-break error rates run markedly higher (and shift direction with break placement), the distortion is demonstrated.

Related

• Same group: U3.02.1 A broken axis removes a value interval from the middle of the scale · U3.02.3 Break markers are often too small to notice, equal to unmarked · U3.02.4 Length encodings crossing the break lose all comparability · U3.02.5 Facets or log scales usually replace the broken axis
• Nearby: U3.07.2 Trend judgement depends on line-segment angle · U3.03.1 Log scales suit data spanning orders of magnitude
• Search terms: broken axis · distance distortion · slope judgement · nonuniform scale