# 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.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Numbers and shares
- Lesson 2/36, ~24 min
- Source: https://productovi.com/courses/math-basics/percent-vs-percentage-points

## A detour to Halves

**Alex Tutor:** Today's main case comes from another company. **Halves** is an app that splits bills between flatmates, couples and travel groups, with a free tier and a paid plan, Halves Plus. Its product team is testing a new paywall, and you're helping report the result in numbers nobody can misread.

## Up 1.31% or up 35%?

**Sofia:** A test summary says: "Paid conversion went from 3.77% to 5.08%, up 1.31%." Someone in the comments says it’s up 35%. Who’s right?

**Alex Tutor:** That's the Halves paywall test. There are two honest ways to describe the change, and each has its own unit.

**Alex Tutor:** The **absolute change** is new − old: 5.08 − 3.77 = 1.31. Both numbers are already percentages, so their difference is measured in **percentage points**: +1.31 pp.

**Alex Tutor:** Before the relative change, count the points for another rate.

> **Exercise:** A subscription app's monthly churn fell from 6% to 4.5%. What is the absolute change?

**Alex Tutor:** The **relative change** compares the change with where you started, the **base**:

![A short and a slightly taller tower of blue blocks; one ruler counts the added blocks, a second compares that part with the original tower.](https://productovi.com/courses/math-basics/inline/percent-vs-percentage-points-illustration-2.webp)

*How much was added, and how much compared with the start.*

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

*a: the new value; b: the old value, the base. Multiply by 100 to get percent.*

**Alex Tutor:** 5.08 ÷ 3.77 − 1 = 0.3475, so **+34.75%**. The commenter rounded it to 35%. Both +1.31 pp and +34.75% are true.

**Sofia:** So the summary took the size of one and the unit of the other. "+1.31%" is the only wrong version.

![Conversion from 3.77% to 5.08% is +1.31 percentage points in absolute terms and +34.75% relative to the 3.77% base; “up 1.31%” is the only wrong version.](https://productovi.com/courses/math-basics/diagrams/m1-l2-pp-vs-relative.png)

> **Exercise:** At CatChow, an online pet-food shop, checkout conversion went from 2.4% to 3.0% after a redesign. What is the relative change, in percent?

## Two numbers, two units

**Alex Tutor:** In an A/B test the difference in pp is the **absolute lift** and the ratio minus one is the **relative lift**. Report the pp first and label the relative one.

**Sofia:** Why pp first? +35% sounds much better.

**Alex Tutor:** That’s exactly why. On a small base, relative changes look huge: from 0.1% to 0.2% is +100% but only +0.1 pp. Points tell you how many more people out of every hundred actually paid.

![Left: one tiny block becomes two and a figure cheers. Right: a huge tower gains a few blocks that barely show.](https://productovi.com/courses/math-basics/inline/percent-vs-percentage-points-illustration-3.webp)

*From almost nothing, any growth looks enormous.*

**Sofia:** And when is the relative one useful?

**Alex Tutor:** When you compare changes that start from different bases, or when the business plans in percent: "revenue +10%". Just never let a reader guess which one you mean.

> **Exercise:** Conversion rose from 4% to 5%. Which statements are correct? Select all that apply.

## A percent of what?

**Alex Tutor:** Every percent has a base, the number you divide by. Change the base and the same gap gives a different percent.

**Sofia:** Like a lesson price going from 620 to 580 UAH and then back?

**Alex Tutor:** Yes. Down: 580 ÷ 620 − 1 = −6.45%. Back up: 620 ÷ 580 − 1 = +6.90%. The same 40 hryvnias, two bases.

**Alex Tutor:** Now compute the way back on your own calculator.

> **Exercise:** A Klasna lesson price was cut from 620 to 580 UAH, a change of −6.45%. Later it goes back up from 580 to 620 UAH. What is the relative change of that rise, in percent, to two decimals?

**Alex Tutor:** Ratios work the same way: twice as much is +100%, three times +200%. When a dashboard at Halves showed sign-ups "up 156%" over a long weekend, that meant 2.56 times the usual number.

> **Exercise:** A usual long weekend brings Halves about 787 sign-ups. This weekend the dashboard says "up 156%". About how many sign-ups is that?

## A percent of a percent

**Alex Tutor:** First, a quick call on a tiny base: which headline goes first?

> **Exercise:** A referral email moved sign-up-to-paid conversion from 0.2% to 0.3%. Which line should open the test summary?

**Alex Tutor:** The classic trap is when both numbers are rates: fees, conversion, churn, margins. Then "+1%" can mean two different things.

**Sofia:** Klasna’s payment fee is 2.5%. If the provider cuts it to 2%, did the fee fall by 0.5%?

**Alex Tutor:** It falls by 0.5 pp. Relative to the old fee, 2 ÷ 2.5 − 1 = −20%: the fee bill shrinks by a fifth. Both are true; "0.5%" is neither.

> **Exercise:** At Klasna, an English lesson leaves 29.76% of its price as contribution margin in the 16-lesson package and 25.09% in the 32-lesson package. So the contribution margin ratio is 4.67% lower in the bigger package.

## ROI is a relative change too

**Alex Tutor:** One more place where a/b − 1 hides. In the product-analytics course, the return on an ad cohort is ROI = (gross profit − spend) ÷ spend.

**Sofia:** That’s (a − b) ÷ b, with gross profit as a and spend as the base. So an ROI of 0% means the cohort has just paid back.

**Alex Tutor:** Exactly, and −100% means nothing came back at all. Because the cohort keeps earning, ROI is always stated at an age: "ROI at month 6".

> **Exercise:** An ad cohort at Halves cost $12,000. By month 6 it has brought $15,000 of cumulative gross profit. What is its ROI at month 6, in percent?

## Your turn: fix the headline

**Alex Tutor:** Back to the paywall test. Rewrite the summary line so that nobody can misread it.

**Sofia:** Both numbers, both units, and the pp first?

**Alex Tutor:** And the starting point, so the reader can check it: from 3.77% to 5.08%.

> **Exercise:** The paywall test summary says: "Paid conversion went from 3.77% to 5.08%, up 1.31%." Rewrite it in one or two sentences so that nobody can misread it, and say in one more sentence what was wrong with the original.

## Where you will meet this

> **In short**
>
> - **Absolute change** = new − old. Between two percentages it is in **percentage points (pp)**.
> - **Relative change** = new ÷ old − 1 = (new − old) ÷ old, in percent of the old value, the **base**.
> - 4% → 5% is +1 pp and +25%. Twice as much is +100%, three times +200%.
> - In a test, report the absolute lift in pp first and label the relative lift.

> **These lessons use it**
>
> - [Case: reading a finished test](/courses/product-analytics/reading-a-test): the effect in pp with its interval, and the relative lift labelled as one.
> - [Capstone: the unexplained spike](/courses/product-analytics/spike-investigation): "up 156%" against a fair baseline.
> - [CAC, payback and ROI](/courses/product-analytics/cac-payback-roi): ROI at a stated month.
> - [Conversion rates and proportion tests](/courses/statistics-for-pm-and-marketers/conversion-rate-tests): a big relative lift on a small sample.
> - [Spread: range, SD and quartiles](/courses/statistics-for-pm-and-marketers/spread): the coefficient of variation, spread as a percent of the mean.
> - [Contribution margin: what one lesson leaves](/courses/finance-for-product/contribution-margin): CM% of the price.

**Alex Tutor:** Next time: rounding. When to do it, and why break-even always rounds up.

- Previous lesson: [Order of operations, ratios and multiples](https://productovi.com/courses/math-basics/order-of-operations-and-ratios.md)
- Next lesson: [Rounding: at the end, and sometimes up](https://productovi.com/courses/math-basics/rounding-and-rounding-up.md)

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