The percentage change formula
A positive result is a percentage increase and a negative result is a percentage decrease. Divide by the old value, because the change is measured against where you started.
Example: year-over-year change
A shop had sales of 84,000 last year and 96,600 this year. The difference is 96,600 − 84,000 = 12,600. Divide by the old value: 12,600 ÷ 84,000 = 0.15, or a 15% increase. This is how "change over last year" is calculated.
Percentage increase and decrease
- Increase: the new value is larger. From 80 to 100 is (100 − 80) ÷ 80 = 25%.
- Decrease: the new value is smaller. From 100 to 80 is (80 − 100) ÷ 100 = −20%.
The two are not mirror images. A 25% rise is undone by a 20% fall, because each is measured against a different starting value.
Percentage change versus percentage difference
Percentage change has a direction: something went from an old value to a new one. Percentage difference compares two values with no "before" and "after", so it divides by their average: |a − b| ÷ ((a + b) ÷ 2). For 80 and 100 that is 20 ÷ 90 = 22.2%, compared with a 25% change from 80 to 100. Use difference when comparing two things side by side, such as two stores' prices.
Working backwards
Old value = new value ÷ (1 + change ÷ 100)
If sales rose 15% to 96,600, the old value was 96,600 ÷ 1.15 = 84,000. If you want to know what 84,000 becomes after a 15% rise, multiply by 1.15 to get 96,600.
Negative numbers and zero
The formula uses the absolute value of the old number, so a move from −50 to −40 is an increase of 20%. Percentage change from zero cannot be calculated because you would divide by zero. When percentages start from a very small base they can look huge, so it is often worth reporting the raw difference too.
Percentage points are different
If a rate goes from 4% to 5%, it rose by one percentage point, which is a 25% increase in relative terms. Be clear about which one you are reporting.