Percentage Difference Calculator
Percentage difference, change and error side by side — because they are three different answers.
Updated
For percentage change this is the starting point.
All three answers
40.0%difference
Symmetric: |40 − 60| against their mean of 50.00.
- Percentage change
- +50.0%
- Percentage error
- 50.0%
- Absolute difference
- 20.00
- Mean of the two
- 50.00
- 60 as a percentage of 40
- 150.0%
Difference and change are not the same number. Difference divides by the mean of the two values, so swapping them gives the same answer. Change divides by the first value, so it does not: 40 to 60 and 60 back to 40 give different percentages.
In short
What is the percentage difference between 40 and 60?
Forty percent, using the mean of the two as the base: 20 divided by 50. But the percentage change from 40 to 60 is 50 percent, and from 60 back to 40 it is 33.3 percent. Three answers, three different bases.
Most people searching for percentage difference have a before and an after, which is percentage change.
How to use the percentage difference calculator
One phrase, three calculations, and picking the wrong one is easy because all three produce a plausible percentage. The distinction is entirely about what sits underneath the division — which number is treated as the base — and that follows from what the two values are to each other rather than from anything in the arithmetic.
Percentage difference treats the two values as peers. Neither came first, neither is correct, so the base is their mean and the answer is symmetric: swap the inputs and nothing changes. Two suppliers quoting different prices, two machines producing different outputs, two measurements of the same thing by different instruments — these are difference questions.
40%
Difference between 40 and 60
base is the mean, 50
+50%
Change from 40 to 60
base is the first value
−33.3%
Change from 60 to 40
and it is not the same
Percentage change treats the first value as the starting point. It has a direction and a sign, and it is not symmetric — which is the fact that surprises people. Going from 40 to 60 is a rise of 50 percent, but coming back down from 60 to 40 is a fall of only 33.3 percent, because the base grew on the way up. Nothing is wrong with either figure; they are answering different questions.
Percentage error is the third case and the narrowest. It applies when one value is a known-correct reference — a true measurement, a published figure, an accepted constant — and the other is what you got. It is the absolute version of change, since being wrong by ten percent in either direction is usually equally wrong.
- Two peers, no order
- percentage difference
- A before and an after
- percentage change
- A correct value and a measured one
- percentage error
- Two figures that are themselves percentages
- use percentage points
- Something compared with zero
- none of them work
That last row is not a technicality. Nothing can be expressed as a percentage of zero, so a rise from zero has no percentage at all — not infinity, not one hundred percent, none.
Compare two quantities as a ratio instead
When the question is really about proportion rather than change, a ratio says it more directly and without a base to argue about.
Open the ratio calculator →The same pairs of values through all three calculations, showing how far apart the answers can be for one perfectly ordinary comparison.
| First → second | Difference | Change |
|---|---|---|
| 40 → 60 | 40.0% | +50.0% |
| 60 → 40 | 40.0% | −33.3% |
| 100 → 110 | 9.5% | +10.0% |
| 110 → 100 | 9.5% | −9.1% |
| 50 → 100 | 66.7% | +100.0% |
| 100 → 50 | 66.7% | −50.0% |
| 20 → 25 | 22.2% | +25.0% |
| 1 → 2 | 66.7% | +100.0% |
| 200 → 210 | 4.9% | +5.0% |
| 75 → 90 | 18.2% | +20.0% |
Why the trip back is smaller than the trip out
This is the part that feels wrong and is not. A rise of 50 percent followed by a fall of 50 percent does not return you to where you started: 100 becomes 150, and half of 150 is 75. The base moved. Every percentage change is measured against wherever the value was at the time, which means changes do not cancel by subtraction.
- 40 up to 60
- 20 ÷ 40 = +50%
- 60 back down to 40
- 20 ÷ 60 = −33.3%
- The gap is the same
- 20 either way
- The base is not
- 40 going up, 60 coming down
- Difference, symmetric
- 20 ÷ 50 = 40%
Percentage difference sidesteps the asymmetry by refusing to privilege either value: it divides by the mean, which is the same number whichever way round you enter them. That is its whole reason for existing.
It also means percentage changes multiply rather than add. A 10 percent rise followed by a 10 percent rise is a 21 percent rise overall, not 20, because the second rise applies to a base that already grew. Over several periods the gap widens quickly, which is why compound growth is quoted as a rate rather than a sum.
A convention rather than a law
Percentage difference dividing by the mean is the common convention, and it is what this page uses, but it is worth knowing it is a convention rather than something derived. Some fields divide by one of the two values, or by the larger, and get a different number from the same pair. When it matters, state the base you used rather than assuming the reader shares yours.
Do
- Name the base whenever the audience might assume a different one
- Use percentage points when both figures are percentages
- Use change when one value came before the other
- Treat a rise and the matching fall as different percentages
Don't
- Use percentage difference and percentage change interchangeably
- Average a set of percentage changes by adding them up
- Compute percentage error against an accepted value of zero
- Assume every source divides percentage difference by the mean
Percentage error has the same softness in its sign convention: the common form takes the absolute value, but some references keep the sign so that over and under can be told apart. Neither is wrong. What is wrong is quoting a figure without saying which, then comparing it against somebody else who chose the other.
The formula, worked line by line
Three formulas that differ only in their denominator, which is the entire subject of this page compressed into one sentence.
“The numerator is the same gap every time. The denominator is the decision, and it is the one nobody states.”
Once the base is named the ambiguity disappears completely. "Sixty is fifty percent more than forty" and "forty and sixty differ by forty percent" are both true, both precise, and describe the same two numbers.
percentage difference = |a − b| ÷ ((a + b) ÷ 2) × 100
percentage change = (b − a) ÷ a × 100
percentage error = |b − a| ÷ |a| × 100
b as a percentage of a = b ÷ a × 100
percentage points = b − a, when both are already percentages- Supplier A
- £1,800
- Supplier B
- £2,200
- Mean of the two
- £2,000
- Difference ÷ mean
- 400 ÷ 2,000
- Percentage difference
- 20%
Neither quote is the "original", so there is no starting point to measure change from — this is a difference question. Had you been renegotiating from £1,800 up to £2,200, the same pair would be a 22.2 percent increase instead.
Averaging percentage changes
Adding a set of percentage changes and dividing by the count gives an answer that is almost always wrong, because each change was measured against a different base. Growth of 50 percent then 50 percent is not an average of 50 percent — it is 125 percent overall. The correct average is the geometric mean of the growth factors, not the arithmetic mean of the percentages.
Where each one is normally used
- Finance and business reporting
- change, almost always
- Laboratory and engineering measurement
- error, against an accepted value
- Comparing two independent readings
- difference
- Interest rates, unemployment, market share
- percentage points
- Quality control
- change against a target, error against a specification
The pattern is consistent: if one number has authority over the other, the base is that number. If neither does, the base is the mean.
That single test resolves nearly every case. Ask which of the two values, if either, is the reference — and if the honest answer is neither, you want difference.
Questions people ask
Sources
Where the constants and formulas on this page come from. Each line names the figure it backs.
Percentage error is calculated against an accepted reference value, and sign conventions vary between signed and absolute forms.
NIST Dataplot Reference Manual — percentage error — National Institute of Standards and Technology
A change between two percentages should be described in percentage points, distinct from a relative percentage change.
Style guide — percentages and percentage points — Office for National Statistics
Successive percentage changes compound multiplicatively rather than adding together.
Percentage change and compound growth — Wolfram MathWorld
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