Numerator, denominator and absolutes
What you'll learn
- always name the denominator: 20% is a share OF WHAT
- tell absolute change from relative: 100 → 120 is both "+20" and "+20%"
- never confuse percentage points with percent: 30% → 15% is −15 pp and −50% relative
- understand why a percentage without a denominator is dangerous — and a bare absolute is too
Every percentage is a fraction with two floors
"Percentages are the favourite weapon of people who do not want to be checked. Half of Kotomarket's reports are written in percentages with no denominator." — from L.'s journal
Any share is built the same way:
numerator how many units of the quantity satisfy the condition
share = ──────────── = ───────────────────────────────────────────── × 100%
denominator how many units of the quantity there were IN ALL
Example: 200 out of 1000. Share = 200 / 1000 × 100% = 20%.
The question to ask yourself and others every single time: 20% of what? That "what" has a name, a size and a definition — exactly like the metric from the first lesson.
Here are three different "20%" figures that are easy to conflate in one conversation:
| phrase | numerator | denominator |
|---|---|---|
| "20% of users bought" | 200 buyers | 1,000 users active that day |
| "20% of orders were cancelled" | 200 cancelled orders | 1,000 orders that day |
| "20% of revenue comes from new users" | 200,000 ₽ from new users | 1,000,000 ₽ total |
One number, three meanings, three counting units. On hearing "conversion is 20%", an analyst silently completes it: "20% is this many OUT OF that many, counted this way".
The denominator decides more than the numerator
The same numerator over different denominators tells different stories:
| "720 paying users" — what percentage is that? | denominator | share |
|---|---|---|
| of everyone who opened the app | 12,000 | 6.0% |
| of those who reached the cart | 1,800 | 40.0% |
| of all registered users | 90,000 | 0.8% |
Three honest numbers about one and the same fact. That is why a report says not "conversion is 40%" but "cart-to-payment conversion is 40% (720 out of 1,800)". A parenthesis with two absolutes is the cheapest insurance against an argument.
Absolute and relative change
It was 100, it is now 120. That change can be named by two different numbers, and both are correct:
- absolute change: 120 − 100 = +20 (items, people, roubles — in the metric's own units);
- relative change: (120 − 100) / 100 × 100% = +20% (as a share of the previous value).
The coincidence of "+20" and "+20%" here is accidental — it happened only because the base was exactly one hundred. Check on another example: 12,000 → 14,700. Absolute change +2,700; relative (14,700 − 12,000) / 12,000 × 100% = +22.5%. Nothing alike.
The formula for relative change is always the same:
new − old
relative = ─────────────────── × 100%
old
The denominator is the old value, the comparison base. Divide by the new one and you get a different number — the most frequent arithmetic error in reports.
When a percentage lies, and when an absolute does
A percentage without a denominator is dangerous. "Conversion in this segment grew 50%" sounds like a victory until you learn it went from 2 buyers per 100 to 3 per 100. Three buyers is not a discovery, it is noise.
A bare absolute is dangerous too. "We lost 6,000 people at the first step" sounds catastrophic, but if 12,000 entered that step, it is exactly the same 50% as last month and six months ago — background, not an event.
Hence a rule worth pinning above your desk:
Never conclude from absolutes without a denominator, or from percentages without a volume. Look at both numbers at once.
In practice this means every figure in a conclusion looks like "40% (720 out of 1,800)" or "−1,080 people (−60% of the cart)". Two forms side by side — and it becomes far harder to fool yourself.
Percentage points are not percent
The nastiest trap appears when the metric itself is already measured in percent. Conversion was 30%, now it is 15%. How much did it fall?
Both answers are correct — but they are different answers:
- −15 percentage points (pp) — the plain difference of two percentages: 15 − 30 = −15;
- −50% relative — the relative change: (15 − 30) / 30 × 100% = −50%. Conversion halved.
was: ██████████████████████████████ 30%
now: ███████████████ 15%
difference: −15 pp
ratio: −50% of the previous value
Confusing the two is not a trifle. "Conversion dropped 15%" can mean "from 30% to 15%" (halved, a catastrophe) or "from 30% to 25.5%" (down a tenth, unpleasant but survivable). The two readings differ threefold.
The rule is simple and mechanical:
If a metric is measured in percent, call its difference percentage points and its ratio percent. "Up 5 pp" and "up 5%" are different statements.
A small self-check. The Android share of users was 40%, now it is 44%.
- difference: +4 pp;
- relative change: 4 / 40 × 100% = +10%.
Both numbers belong side by side in the report: "Android share 40% → 44% (+4 pp, +10% relative)". Nobody asks again, and nobody gets it wrong.
was and now and watch the two ways of describing one change diverge.abs_change — how many more people, rel_change — how many percent more relative to week 10.pp_change — the change in percentage points, rel_change_conv — the relative change in percent.Key takeaways
- every percentage is a fraction: a numerator (the volume of the part of interest) and a denominator (the full volume of the quantity); say the denominator out loud
- the same numerator over different denominators tells different stories: 720 paying users is 6% of the entry and 40% of the cart
- absolute change is a difference; relative change is that difference divided by the OLD value
- if the metric is in percent: the difference is percentage points, the ratio is percent (30% → 15% is −15 pp and −50%)
- never conclude from absolutes without a denominator or from percentages without a volume — look at both
Next: the , a construct where the denominator changes at every step, and where ignoring it yields a conversion of 400%.