Percentage change calculator

Percentage change, the figure a percent change calculator gives, is (new - old) / |old| x 100. The result carries a sign: positive for a rise, negative for a fall. From 1,600 to 1,880 the change is 280, and 280 / 1,600 = 0.175, so +17.5%. The same two numbers in the other order, 1,880 to 1,600, give -14.89%, because the change of 280 is now measured against 1,880. Which number came first is the one fact the formula cannot do without.

Percentage change is also called relative change. As a ratio, 1,880 is 1.175 times 1,600. The ratio minus 1 is always the change as a decimal, which is a useful check on any figure you are handed.

The bars around the old value are there for negative starting points. A business that lost 400 one year and made 100 the next has improved by 500. Divided by -400 that is -125%, a fall, which is plainly the wrong word. Dividing by the size of the starting value gives +125%. Even so, a percentage change that starts below zero or crosses zero is a weak measure: going from -400 to -100 is +75% by the same rule, and few readers will guess what either figure means. For profit, temperature in Celsius or any other quantity that can be negative, report the plain change of 500 or 300 alongside, or instead.

Percentage changes in sequence multiply. A rise of 17.5% followed by a fall of 17.5% takes 1,600 to 1,880 and then to 1,880 x 0.825 = 1,551, a net change of -3.06%, not zero. To combine a run of changes, convert each to a multiplier, multiply them together and subtract 1: here 1.175 x 0.825 = 0.969375, so -3.0625%.

If the two values are percentages already, there are two changes and both are correct. A mortgage rate going from 3.2% to 4% has risen 0.8 percentage points and 25%. Choose points when comparing rates, and say which one you are using.

Percentage change is the right tool when one number is a before and the other an after, or one is a reference and the other a measurement. When two numbers are peers with no natural starting point (two shops' prices, two people's measurements of the same thing), percentage difference is the measure meant, and it divides by the average instead. For 1,600 and 1,880 it gives 16.09%, between the +17.5% and -14.89% that the two directions of change produce.

A small or zero starting value makes the figure explode or disappear. A change from 0 has no percentage at all, and from 2 to 5 it is +150%, which is accurate and usually less useful than the plain change of 3.