How to Calculate Percentage Change, and Why −10 to −5 Is +50%

Percentage change answers one question: how big was the move, compared with where it started? The formula fits on one line, and most mistakes come from using the wrong starting point, the wrong sign or the wrong kind of comparison.

This guide works through the formula, then the cases that catch people out: falls, negative starting values, two changes in a row, and the difference between percent and percentage points.

The formula

$$\text{\% change} = \frac{\text{new} - \text{old}}{|\text{old}|} \times 100$$

In words:

1. Subtract the old value from the new value. That's the change.
2. Divide the change by the old value (its size, ignoring any minus sign).
3. Multiply by 100.

A positive answer is a rise and a negative answer is a fall.

Example: rent. Rent goes from $1,800 to $2,070 a month. The change is 2,070 − 1,800 = 270. Then 270 ÷ 1,800 = 0.15, and 0.15 × 100 = +15%.

Example: a price drop. A jacket goes from $85 to $68. The change is 68 − 85 = −17. Then −17 ÷ 85 = −0.2, so the change is −20%, a 20% decrease.

You can check either one in the percentage change calculator, which shows each line of the working.

How the government does it

The US Bureau of Labor Statistics uses the same three steps for the Consumer Price Index, the main US measure of inflation. Its own worked example takes the index from November to December 2021:

	CPI-U index
November 2021	277.948
December 2021	278.802
Change	0.854 index points
Percent change	0.854 ÷ 277.948 × 100 = 0.3%

BLS makes a point worth keeping: index point changes "are affected by the level of the index", while percent changes are not. A 10-point rise from 180 to 190 is 5.6%, while a 10-point rise from 100 to 110 is 10%. The same is true of any number you track. "Up $50" means something different on a $200 bill than on a $2,000 one, and the percentage is what makes them comparable.

Try the CPI numbers: to three decimal places it's 0.307%.

Increase or decrease: one formula

You don't need separate formulas for going up and going down. The sign tells you which happened. "Percentage increase" and "percentage decrease" are the same calculation with the sign dropped, because the word already says the direction:

• $85 to $68 is a 20% decrease, or a change of −20%.
• $68 to $85 is a 25% increase, or a change of +25%.

Those two aren't the same size, and that's the most common surprise in this whole topic. Both moves are $17, but the first is measured against $85 and the second against $68. So a 20% fall is undone by a 25% rise, not a 20% one.

Negative starting values: why −10 to −5 is +50%

Take a small business that lost $10,000 last year and $5,000 this year. Its result went from −10,000 to −5,000. It improved by $5,000. What's the percentage change?

Plug the numbers into the formula without the absolute value and you get:

$$\frac{-5{,}000 - (-10{,}000)}{-10{,}000} = \frac{5{,}000}{-10{,}000} = -50\%$$

That says the result fell by half, which is backwards: the number went up. The minus sign on the bottom flips the direction.

Dividing by the size of the starting value fixes it:

$$\frac{-5{,}000 - (-10{,}000)}{\lvert -10{,}000 \rvert} = \frac{5{,}000}{10{,}000} = +50\%$$

The loss shrank by half, and the sign now matches the direction the number moved. That's the convention our calculator uses, and it's the usual fix described in references on relative change. Try −10 to −5.

Three cases where you should stop and think instead of reaching for the formula:

• When the sign flips. From −10 to +5 the formula gives +150%. The arithmetic is right, but "up 150%" says nothing useful about going from a loss to a profit. Write "from a $10,000 loss to a $5,000 profit" instead.
• When the starting value is 0. Any change from 0 is an infinite percentage, so there's no answer. Give the change itself: "up 12 units".
• When zero is arbitrary, as with temperature. A drop from 20 °C to 10 °C isn't a "50% fall" in any physical sense, because 0 °C is just the freezing point of water, not "no heat". On the kelvin scale, where 0 is absolute zero, 20 °C and 10 °C are 293.15 K and 283.15 K, a change of about −3.4%. For temperatures, give the change in degrees.

Percentage change vs percentage difference

Percentage change has a direction: there's a before and an after, and you divide by the before. Sometimes there isn't one. Two shops sell the same item for $48 and $52. Neither price is the "starting" one.

For that, use percentage difference: the gap divided by the average of the two values.

	Formula	$48 and $52
% change, 48 → 52	(52 − 48) ÷ 48 × 100	+8.33%
% change, 52 → 48	(48 − 52) ÷ 52 × 100	−7.69%
% difference	|52 − 48| ÷ 50 × 100	8%

Percentage difference gives the same answer whichever way round you put the numbers. Use change for anything over time (prices, salaries, traffic), and difference for side-by-side comparisons. Compare 48 and 52.

Percent vs percentage points

When the thing that changes is already a percentage, there are two honest answers. An interest rate going from 4% to 5% has risen by:

• 1 percentage point (5 − 4), or
• 25 percent (1 ÷ 4 × 100).

Both are true. "Rates rose 1%" is ambiguous, so say which you mean. We have a whole post on percentage points vs percent with more examples.

Four mistakes to avoid

1. Dividing by the new value. The base is always where you started. From $68 to $85, the change is 17 ÷ 68 = +25%. Dividing by 85 gives 20%, which is the answer to a different question (how far it would have to fall to get back).

2. Adding changes that happen one after another. A price that rises 10% and then falls 10% doesn't end where it started. Each change multiplies:

$$1.10 \times 0.90 = 0.99$$

So the overall change is −1%. The fall is 10% of a bigger number than the rise was. The same goes for growth: +20% one year and +30% the next is 1.2 × 1.3 = 1.56, a total of +56%, not +50%.

3. Averaging percentage changes. An investment that gains 50% and then loses 50% has an "average change" of 0%, but 1.5 × 0.5 = 0.75. It's down 25%. To summarise several periods, multiply the factors first, then turn the total back into a percentage.

4. Reversing a percentage the wrong way. A price is $54 after 20% tax was added. The price before tax is not $54 minus 20% ($43.20). It's 54 ÷ 1.2 = $45, because the 20% was 20% of $45, not of $54. The reverse percentage calculator does the division for you.

An aside: why economists sometimes use log changes

Mistakes 1 to 3 all come from the same fact: ordinary percentage changes aren't symmetric. Up 25% and down 20% cancel out, and two changes don't add.

Leo Törnqvist, Pentti Vartia and Yrjö Vartia showed in 1985 that the log change, 100 × ln(new ÷ old), is the only measure of relative change that is symmetric and adds up across periods. With logs, $68 → $85 is +22.3 and $85 → $68 is −22.3, and consecutive changes simply add. You'll meet log changes in economics papers and growth charts. For everyday use, the ordinary percentage change is what people expect, as long as you avoid the mistakes above.

Try it

The percentage change calculator works exactly, with fractions rather than rounded decimals, and shows every line of the working. Its siblings cover the other questions in this post:

• Percentage increase, with a switch to add a percentage
• Percentage decrease, with a switch to take a percentage off
• Percentage difference
• Reverse percentage

Sources

• U.S. Bureau of Labor Statistics. Calculating percent changes (CPI). bls.gov
• U.S. Bureau of Labor Statistics. CPI math calculations (index point vs percent change). bls.gov (PDF)
• Törnqvist, L., Vartia, P., & Vartia, Y. O. (1985). How should relative changes be measured? The American Statistician, 39(1), 43–46. doi:10.1080/00031305.1985.10479385
• Relative change, including the absolute-value convention for negative reference values. Wikipedia
• NIST. SI units: temperature (kelvin and degree Celsius). nist.gov