relative change

Also called: relative lift, percent change, percentage change

The change as a share of where it started: new ÷ old − 1, the same as (new − old) ÷ old. From 4% to 5% is +1 percentage point and a +25% relative change.

A change can be described in two ways. The absolute change is new − old, in the units of the number itself. The relative change compares that gap with where you started:

$\Delta_{rel} = \frac{a}{b} - 1 = \frac{a - b}{b}$

where $a$ is the new value and $b$ the old one, the base. Multiply by 100 to get percent. A ratio of 2 is +100%, a ratio of 3 is +200%: the original 100% is already inside the ratio.

When both values are themselves percentages (conversion, churn, margins, fees), the absolute change is measured in percentage points, and the relative change in percent. From 4% to 5% is +1 pp and +25% relative. Both are true; "+1%" is neither, because it mixes the size of one with the unit of the other.

The base decides the answer. A price that falls from 620 to 580 drops 6.45%; going back from 580 to 620 is +6.90%. The same 40 units, two different bases. ROI is a relative change too: (return − spend) ÷ spend.

Example

A paywall test moves paid conversion from 3.77% to 5.08%.

Absolute change: 5.08 − 3.77 = +1.31 pp.
Relative change: 5.08 ÷ 3.77 − 1 = 0.3475 = +34.75%.

A clear summary: "Paid conversion rose from 3.77% to 5.08%: +1.31 pp, about +35% relative." The line "up 1.31%" would understate the effect about 26 times.

Common mistakes

Writing pp as %. "+1.31%" for a change from 3.77% to 5.08% reads as a tiny relative change.
Dividing by the new value. The base of a change is the old value: 4% → 5% is +25%, not +20%.
Forgetting the original 100%. Three times as much is +200%, and "up 156%" means 2.56 times.
Big relative changes on a tiny base. 0.1% → 0.2% is +100% but only +0.1 pp; report the points first.
Unlabelled percentages. Always say whether a number is pp or relative %.

Learn it in the course

Percent vs percentage points · Absolute and relative change, new ÷ old − 1, lift in an A/B test, and why "+1%" can mean two different things.
Products of shares: funnels and metric trees · Overall conversion as a product of step rates, working a funnel backwards from a goal, ARPU = paying share × ARPPU, revenue per visitor, stickiness, diluted effects, ownership dilution and RICE.
Notation: indices, bars, hats and Greek letters · How to read x_i, sorted values, x̄ and ŷ, the Greek letters μ, σ, α, β and λ, Δ for a change, and sample vs population.
Contribution margin and break-even · The unit-economics formulas as pure math: CM = P(1 − f) − T − V, CM%, break-even ⌈F ÷ c⌉, margin of safety, step costs, the volume a discount needs, gross-up and ROI.