Scope and assumptions
Percentage error compares a measured/estimated value to a known true value, using the true value as the base. Illustrative.
When a measurement misses the mark, the useful question isn't how far off you were in raw units, it's how far off relative to what was actually right. That ratio is percentage error, and the base is always the true value, never your own reading. Swap the two and the number quietly lies.
The magnitude against the standard
Percentage Error = |Experimental − True| ÷ True × 100
Use the absolute value for magnitude; drop it if you want the signed direction (over/under).
| True | Measured | Error | % Error |
|---|---|---|---|
| 100 | 105 | 5 | 5 ÷ 100 × 100 = 5% |
| 50 | 47 | 3 | 3 ÷ 50 × 100 = 6% |
| 200 | 210 | 10 | 5% |
| 80 | 80 | 0 | 0% |
Why the true value sets the scale
Error is "how far off from reality," so reality (the true value) sets the scale. If you divided by the measured value instead, a reading of 105 vs true 100 would read 4.76% — understating the miss against the standard you're judged against.
Signed or absolute?
- Absolute: |105 − 100| ÷ 100 = 5% (magnitude only)
- Signed: (105 − 100) ÷ 100 = +5% (over); (95 − 100) ÷ 100 = −5% (under)
Signed form tells you direction; absolute tells you how big the miss is.
Error, difference, uncertainty — same shape, different base
| Measure | Base | Use |
|---|---|---|
| % error | true value | compare to a known standard |
| % difference | average of two peers | no known truth |
| % uncertainty | measured value | truth unknown, spread of readings |
They share the × 100 shape but answer different questions.
Where the error gets miscalculated
- Dividing by the measured value — base must be the true value.
- Forgetting the absolute value — when you only want magnitude.
- Confusing with percentage difference — error uses true as base, difference uses the average.
- Mixing % error with % uncertainty — error compares to a known true value; uncertainty bounds an unknown one.
- Rounding the % before a pass/fail check — keep precision until the decision.
Related advanced operations
For the rest of the advanced percentage operations, start at the hub: Percentage Difference Formula. Adjacent: Percentage Uncertainty and Reverse Percentage Increase.
FAQ
Should I use signed or absolute error? Use absolute when you only care how big the miss is (pass/fail specs); use signed when direction matters (over vs under, e.g., a calibration bias you want to correct). Pick based on whether the sign carries information.
What if I don't know the true value? Then you can't compute percent error at all — you'd use percent uncertainty or percent difference instead. Error specifically requires a known standard to compare against.
Is a smaller percent error always better? Not necessarily — a 1% error on a tiny measurement and a 1% error on a huge one feel different, and a systematic (signed) error can be worse than a larger random one. Context decides.
How do I combine errors from multiple measurements? Percent errors don't simply add. If measurements are independent, you typically combine their absolute uncertainties in quadrature (root-sum-square); percent error is a reporting figure, not something you aggregate by addition.
Can percent error be over 100%? Yes — if your measured value is more than double (or below zero relative to) the true value, e.g., true 50 measured 110 → 120% error. It isn't capped.
Sources: Khan Academy, Error and percent error; NIST, Measurement uncertainty guidelines.
Disclaimer: Illustrative. In formal metrology, uncertainty has its own propagation rules; consult your field's standard for reporting. Informational only.
Sources and references
- Khan Academy — Error and percent errorhttps://www.khanacademy.org/math
- NIST — Guidelines for evaluating measurement uncertaintyhttps://www.nist.gov/pml/nist-technical-note-1297
Maintenance and corrections
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