The percentage change formula
Percentage change is (new − old) ÷ |old| × 100, where |old| is the starting value without its sign. From 40 to 50 the change is 10, and 10 ÷ 40 = 0.25, so it is +25%. From 50 to 40 the change is −10, and −10 ÷ 50 = −0.2, so it is −20%.
The gap is the same size both ways, but the starting value is different, which is why the percentages differ. The absolute value keeps the sign meaningful when the starting number is negative: going from −50 to −25 moves upward by 25, which is +50% of the size of the starting value.
Percent change, percent difference and percentage points
These three get confused, and they answer different questions.
- Percentage change has a before and an after. This page calculates it.
- Percent difference compares two values with no before and after, such as two cities' populations. It divides the gap by the average of the two, so swapping them changes nothing: 100 and 120 differ by 20 ÷ 110 × 100 = 18.18%.
- Percentage points are the plain difference between two percentages. From 4% to 5% is 1 point, which as a percentage change is +25%.
Choosing the starting value, and when it breaks
Use the earlier value as the start: year on year growth is (this year − last year) ÷ last year. Starting from the wrong end gives a different number, not a wrong one, so say which you used.
A small starting value makes percentages look dramatic: 2 to 20 is +900%. And from zero there is no percentage at all, because the change is infinite compared with nothing. In that case report the plain change, or say the value is new.
For applying a known percentage rather than measuring one, use the percentage increase calculator or the percentage decrease calculator. The percentage calculator has all five percentage questions together.