Measuring change in percent
Prices, salaries, populations and stock values are all compared by how much they changed in percent. The calculation is simple, but it hides several traps: a 20 percent drop does not undo a 20 percent rise, and five years of 25 percent total growth is not 5 percent a year. This calculator gives the percent change, the multiplier, the change needed to return to the start, the average change per period and the combined effect of successive changes.
The formulas
- Percent change = (final − start) ÷ start × 100.
- Multiplier = final ÷ start = 1 + percent change ÷ 100.
- Change needed to go back = (1 ÷ multiplier − 1) × 100.
- Average change per period = multiplier^(1 ÷ periods) − 1.
- Successive changes combine by multiplying their multipliers: (1 + t₁) × (1 + t₂) × (1 + t₃) − 1.
Worked example
A value rises from 80 to 100 over 5 years.
- Percent change: (100 − 80) ÷ 80 = +25 percent.
- Multiplier: 100 ÷ 80 = 1.25.
- To fall back from 100 to 80 takes a −20 percent change, since 1 ÷ 1.25 = 0.80.
- Average change: 1.25^(1/5) − 1 = 4.564 percent a year, not 5 percent.
- Separately, a rise of 10 percent followed by a fall of 10 percent gives 1.10 × 0.90 = 0.99: a −1 percent combined change.
Changes from 80 to other values
| Final value | Percent change | Multiplier | Change needed to go back |
|---|---|---|---|
| 40 | −50.00% | 0.50 | +100.00% |
| 60 | −25.00% | 0.75 | +33.33% |
| 80 | 0.00% | 1.00 | 0.00% |
| 100 | +25.00% | 1.25 | −20.00% |
| 120 | +50.00% | 1.50 | −33.33% |
| 160 | +100.00% | 2.00 | −50.00% |
The last column explains why losses hurt investors more than gains help them: after a 50 percent loss, you need a 100 percent gain just to break even.
Percent change versus percentage points
When a rate itself changes, say a mortgage rate going from 6 percent to 7 percent, the change is 1 percentage point, but a 16.7 percent increase in the rate. News reports mix the two, so check which one is meant. Use percentage points for differences between percentages and percent change for the relative movement.
Common uses
- Inflation: if a basket of goods rose from $250 to $265, prices went up 6 percent.
- Salary raises: a raise from $52,000 to $55,000 is 5.77 percent.
- Discounts: a price cut from $80 to $60 is −25 percent, and the original price is 33.33 percent higher than the sale price.
- Growth rates: a company’s sales growing from $2 million to $3 million in 4 years have grown 10.67 percent a year on average.
When the start is zero or negative
Percent change is undefined when the starting value is zero, since you cannot divide by zero. With a negative start, such as a loss turning into a profit, percent change is technically computable but rarely meaningful. In those cases, report the absolute change instead.