How to Calculate Percentage Change

Subtract the old value from the new one, divide by the old value, multiply by a hundred. That is the whole formula, and almost every mistake made with it comes from the division rather than the arithmetic. The denominator is always where you started, never where you ended, and never the average of the two.

change % = (new - old) / old x 100

200 to 250   (250 - 200) / 200 = 0.25   ->  +25%
250 to 200   (200 - 250) / 250 = -0.20  ->  -20%

Increases and decreases are not symmetric

Those two lines describe the same pair of numbers and give different answers, and both are correct. Going up by twenty-five percent and coming back down by twenty percent returns you to the start. The asymmetry is not a quirk to be corrected — it falls directly out of the denominator changing — and it is why a stock that falls fifty percent needs to gain a hundred to recover.

Percent and percentage points are different units

If a rate moves from four percent to five percent, that is a rise of one percentage point and of twenty-five percent. Both statements are true and they differ by a factor of twenty-five. Headlines routinely pick whichever sounds larger, which is worth remembering whenever a figure seems surprising: interest rates, tax bands and unemployment are all quoted in percent already, so any change in them has two legitimate descriptions.

Percentage points measure the gap. Percent measures the change relative to where you were.

Changes do not add up

A ten percent rise followed by a ten percent rise is not twenty percent, it is twenty-one, because the second applies to the already-increased figure. Successive percentages multiply rather than add. This is the same mechanism as compound interest, and it is why a discount stacked on a discount never removes as much as the two numbers suggest.

The reversal that catches everyone

To find a price before a twenty percent tax was added, do not subtract twenty percent. Divide by 1.2. Subtracting takes twenty percent of the larger figure and lands you short every time. The same applies to reversing a discount, a markup or any other proportional change: you undo a multiplication by dividing, not by subtracting the same percentage.

• Denominator is the starting value, always.
• A rise and the fall that undoes it have different percentages.
• Percentage points and percent are different units; say which one you mean.
• Reverse a percentage by dividing, not by subtracting.

When percentage change is the wrong tool

It breaks down entirely when the starting value is zero, because any increase from nothing is infinite, and it becomes misleading when the base is very small. A rise from two cases to three is a fifty percent increase, which is technically correct and tells the reader almost nothing. When the base is small, report the raw numbers as well.