How percent difference works
A percent change needs a “before” to measure against. Many comparisons have no before: the same jar of coffee in two shops, two readings of one length, two quotes for the same job. Picking either value as the base gives a different answer. With 40 and 50, the gap of 10 is 25% of 40 but only 20% of 50, and neither figure is more correct than the other.
Percent difference avoids the choice by measuring the gap against the midpoint of the two values. The midpoint of 40 and 50 is 45, so the gap is 10 ÷ 45 = 22.22% of it. Swap A and B and the answer stays the same, which is what “symmetric” means here. For two positive values the result always falls between the sizes of the two one-sided percent changes, and the calculator lists those changes, from A to B and from B to A, beside it so you can see the spread.
The formula
The vertical bars take the size of a number without its sign, so the percent difference is never negative and carries no direction. The divisor averages the sizes of A and B, which keeps it positive if a value is negative; for two positive values it is simply their average. When both values are zero there is nothing to compare and the calculator shows a message instead of a result. If only one value is zero, the one-way change from that value is shown as n/a.
Laboratories call the same quantity the relative percent difference, or RPD, and use it to judge whether two duplicate measurements agree. The US EPA’s data dictionary for its Information Collection Rule database defines RPD as the absolute difference between two reported values divided by their average, times 100.
Choosing between difference and change
| Situation | Better measure | Reason |
|---|---|---|
| Price this year against last year | Percent change | There is a clear earlier value |
| A list price against a sale price | Percent change | The list price is the reference |
| The same item in two different shops | Percent difference | Neither shop is the baseline |
| Two people measuring the same object | Percent difference | Both readings have equal standing |
| Your result against an accepted value | Percent error, a percent change measured from the accepted value | The accepted value is the reference |
If you will write “rose”, “fell”, “more than” or “less than”, you need a direction, and percent change is the right tool. Percent difference only says how far apart two numbers are.
Worked examples
- Gap: 3.99 − 3.49 = 0.50.
- Average: (3.49 + 3.99) ÷ 2 = 3.74.
- Divide and multiply by 100: 0.50 ÷ 3.74 × 100 = 13.37%.
- For comparison, the dearer shop charges 14.33% more than the cheaper one, and the cheaper shop charges 12.53% less than the dearer one.
- Gap: 12.9 − 12.4 = 0.5 cm.
- Average: (12.4 + 12.9) ÷ 2 = 12.65 cm.
- 0.5 ÷ 12.65 × 100 = 3.95%. The one-way changes are +4.03% and −3.88%, close together because the readings are close.
- Gap: 200. Average: 200.
- 200 ÷ 200 × 100 = 100%.
- The one-way changes are +200% (100 to 300) and −66.67% (300 to 100), which shows how much the choice of base matters when values differ a lot.
How the three numbers spread apart
When two values are close, all three measures nearly agree. As the gap widens they drift apart quickly.
| A | B | Percent difference | Change A → B | Change B → A |
|---|---|---|---|---|
| 10 | 11 | 9.52% | +10% | −9.09% |
| 10 | 20 | 66.67% | +100% | −50% |
| 10 | 40 | 120% | +300% | −75% |
| 100 | 300 | 100% | +200% | −66.67% |
The result never goes above 200%. It reaches that ceiling when one value is zero, and also whenever the two values have opposite signs.
Mistakes to watch for
- Reporting a direction. A percent difference of 22.22% between 40 and 50 does not mean either value is 22.22% higher. “B is 25% higher than A” is a percent change statement.
- Dividing by one of the values. Using A or B as the divisor turns the calculation back into a percent change and breaks the symmetry.
- Comparing rates with it. For two percentages, such as 12% and 15%, readers usually want the gap in percentage points (3 points), not a percent difference of 22.22%.
- Comparing values of opposite sign, or a value with zero. The result sits at the 200% ceiling whatever the sizes: 5 and −5 give 200%, and so do 1 and −1,000, which says little that is useful.
Some fields define percent difference with a different divisor, such as the larger of the two values. Check which definition a teacher, standard or report expects before quoting a figure from this calculator.
Sources
- Envirofacts ICR data element TOXRPD, Relative Percent Difference (RPD), US Environmental Protection Agency (archive).
- How to Calculate a Percentage Change (Consumer Price Index methodology), Central Statistics Office (Ireland).