Three questions, three denominators
| Measure | Question | Formula | Reference value |
|---|---|---|---|
| Percent error | How far is my result from the correct value? | |measured − accepted| ÷ accepted × 100 | The accepted value |
| Percent change | How much did it grow or shrink? | (new − old) ÷ old × 100 | The earlier value |
| Percent difference | How far apart are two values of equal standing? | |a − b| ÷ ((a + b) ÷ 2) × 100 | The average of the two |
The denominator is the whole story. Percent error has a value that is right by definition, so everything is measured against it. Percent change has a time order, so the earlier value is the base. Percent difference has neither, so the fairest base is the midpoint between the two.
Percent error: a result against the truth
A lab measures the acceleration due to gravity as 9.6 m/s² where the accepted value, NIST's standard gravity, is 9.80665 m/s². The error is |9.6 − 9.80665| ÷ 9.80665 × 100 = 2.11 percent. The sign is normally dropped because the question is how far off, not which way; if a report keeps it, a negative result means the measurement ran low. The percent error calculator does this with either convention. When the accepted value is zero the formula breaks, which is a sign that percent error is the wrong tool and an absolute error in the original units is the right one.
Percent change: later against earlier
A price goes from $80 to $100: (100 − 80) ÷ 80 × 100 = +25 percent. Run it backwards, from $100 to $80, and the change is (80 − 100) ÷ 100 × 100 = -20 percent. The same twenty dollars is a quarter of the smaller number and a fifth of the larger one, which is why a stock that falls 50 percent needs to rise 100 percent to recover. The percentage change calculator keeps the sign and the direction; the percentages guide covers the reverse problem of finding the original value from a change.
Percent difference: two values, no reference
Two thermometers in the same room read 80 and 100. Neither is the standard, so the difference is measured against their average: |80 − 100| ÷ 90 × 100 = 22.22 percent. The answer is the same whichever instrument is named first, which is the point of using the midpoint. It is the right measure for comparing two quotes, two estimates or two repeated measurements; the ratio calculator gives the same relationship as a ratio, 80:100, when a ratio reads better than a percentage.
The same numbers through all three
| Measure | With 80 and 100 | Result |
|---|---|---|
| Percent error, if 100 is the accepted value | |80 − 100| ÷ 100 × 100 | 20% |
| Percent change, from 80 to 100 | (100 − 80) ÷ 80 × 100 | +25% |
| Percent change, from 100 to 80 | (80 − 100) ÷ 100 × 100 | -20% |
| Percent difference | |80 − 100| ÷ 90 × 100 | 22.22% |
Four honest answers between 20 and 25 percent for one pair of numbers. None is wrong; each answers a different question, and a report should say which it used.
Percent versus percentage points
When the values are themselves percentages, a second ambiguity appears. A savings rate that moves from 4 percent to 5 percent has risen by 1 percentage point and by 25 percent; both are true, and the second sounds four times as dramatic. Rates, shares and poll results are clearest in points; the relative change belongs in a second sentence if it is needed at all. The percentage calculator handles the plain "what is x percent of y" questions that sit underneath all of this, and the rounding calculator keeps the result to a sensible number of decimals; a percent error quoted to six places from a two-digit measurement claims a precision it does not have.
Which one to use
- One value is known to be correct: percent error.
- One value comes before the other in time: percent change, with its sign.
- Two values of equal standing: percent difference.
- Two percentages: say percentage points unless the relative change is the point.
Frequently asked questions
What is the difference between percent error and percent change?
Percent error compares a measured or estimated value with a known correct value, and divides the gap by the correct value. Percent change compares a later value with an earlier one, and divides by the earlier value. The formulas look alike, but percent error has a reference that is right by definition and is usually reported without a sign; percent change has a direction and keeps its sign.
When do I use percent difference?
When neither value is the reference: two measurements of the same thing, two competing estimates, two prices from two shops. Percent difference divides the gap by the average of the two values, so it gives the same answer whichever one you start from. Percent change does not have that symmetry.
Why does going up 25 percent and down 20 percent land in the same place?
Because percent change is measured against the starting value, and the starting value differs each way. From 80 to 100 is up 25 percent of 80; from 100 back to 80 is down 20 percent of 100. The same 20 units are a bigger share of the smaller number.
What is the difference between percent and percentage points?
A percentage point is the plain difference between two percentages; a percent change is that difference relative to the starting percentage. If a rate rises from 4 percent to 5 percent, it is up 1 percentage point and up 25 percent. Interest rates, poll results and tax rates are usually compared in points; growth is reported in percent.
Can percent error be negative?
It is usually reported as an absolute value, so no. Some fields keep the sign to show whether a measurement runs high or low; if you do, say so, and use the same convention throughout a set of results.