Percentage change, percentage difference and percentage points
Three ways to compare numbers with percentages — and the common mistakes that make statistics misleading.
· Awaiting human editorial review
Percentage change has a direction
Percentage change compares a new value with an original one: (new − original) ÷ original. Going from 80 to 100 is a 25 % increase. Going back from 100 to 80 is a 20 % decrease, not 25 %, because the starting point is different. This asymmetry is why a 50 % loss needs a 100 % gain to recover.
Percentage difference is symmetric
When neither value is the starting point — comparing two prices, two measurements or two products — percentage difference divides the gap by the average of the two values. Comparing 40 and 60 gives 20 ÷ 50 = 40 % whichever way round you put them.
Percentage points are for comparing percentages
If an interest rate rises from 4 % to 5 %, it has risen by one percentage point. In relative terms it has risen by 25 %. Both statements are true, but they answer different questions, and confusing them is one of the most common ways statistics are misreported.
Finance often uses basis points (hundredths of a percentage point) to avoid the ambiguity: a rise from 4 % to 5 % is 100 basis points.
Undoing a percentage
To find a price before a 20 % discount, divide by 0.8 rather than adding 20 % back. Adding 20 % to 80 gives 96, not the original 100. The reverse percentage calculator handles discounts, tax and markups this way.