Math
Percentage change vs. percentage points: the mistake in most headlines
“Interest rates rose 2%” and “interest rates rose 40%” can describe the same event. The first means two percentage points (5% → 7%); the second is the relative change of the rate itself. Both are correct; using one when you mean the other is how statistics get misread — and sometimes deliberately spun.
Definitions
Percentage change = (new − old) ÷ old × 100. It is relative to the starting value.
Percentage points = new% − old%. It is the simple difference between two percentages, with no division. Use it whenever both numbers are already percentages.
| Metric | Before | After | Percentage points | Percentage change |
|---|---|---|---|---|
| Interest rate | 5% | 7% | +2 points | +40% |
| Unemployment | 4% | 3% | −1 point | −25% |
| Approval rating | 40% | 50% | +10 points | +25% |
| Tax rate | 20% | 22% | +2 points | +10% |
| Conversion rate | 2% | 3% | +1 point | +50% |
Why percentages do not reverse
A 50% increase followed by a 50% decrease does not return you to the start: 100 → 150 → 75. The base changed. Going from 80 to 100 is a 25% increase, but going from 100 back to 80 is only a 20% decrease. This asymmetry is why a stock that falls 50% needs a 100% gain to recover, and why “average” percentage returns overstate what an investor actually earned.
Three more traps
- Averaging percentages. Two shops with 10% and 30% growth do not average to 20% unless they started the same size; weight by the base.
- Compounding vs adding. Three years of 10% growth is 33.1%, not 30% (1.1³ = 1.331).
- Small bases. A rise from 1 case to 3 cases is “+200%” — true, and nearly meaningless. Always ask for the absolute numbers alongside the percentage.
Sequential discounts do not add up
“20% off, then an extra 10% at checkout” is not 30% off. The second discount applies to the already-reduced price: 100 → 80 → 72, a total saving of 28%. Multiply the remaining fractions (0.8 × 0.9 = 0.72) rather than adding the percentages. The same logic applies to a price rise followed by a tax: a 10% increase and 17% sales tax compound to 28.7%, not 27%.
Markup vs. margin
Retailers trip over this constantly. Markup is the percentage added to cost; margin is the percentage of the selling price that is profit. Buy at 80, sell at 100: the markup is 20 ÷ 80 = 25%, but the margin is 20 ÷ 100 = 20%. To convert, margin = markup ÷ (1 + markup) and markup = margin ÷ (1 − margin). A shop that wants a 50% margin needs a 100% markup — it must double its cost price.
Negative and zero bases
Percentage change is undefined when the starting value is zero (you cannot divide by it) and misleading when it is negative: a loss shrinking from −100 to −50 is a “−50% change” by the formula even though the situation improved. In those cases report the absolute change, or the change relative to a meaningful positive base such as revenue. CalculatorCamp’s percentage change calculator refuses a zero starting value for exactly this reason.
A quick check for any headline
If the thing being measured is itself a percentage (a rate, share, probability), ask whether the change is quoted in points or in percent. If a large relative change is quoted, ask what the base was. CalculatorCamp’s percentage change calculator shows both the relative change and the absolute difference so you can report the one that is honest for your context.
Sources
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