In this article
Percent error tells you how far a measured or estimated value is from the true (accepted) value, as a percentage of that true value. It turns a gap like '0.21 off' into something you can compare across experiments: 2% off is easy to judge, whatever the units.
The formula in three steps
- 1Find the gap9.6 − 9.81 = −0.21. Drop the sign: 0.21.
- 2Divide by the true value0.21 ÷ 9.81 = 0.0214.
- 3Multiply by 1000.0214 × 100 = **2.14%**.
The result says your measurement was about 2.14% below the accepted value. Some teachers keep the sign to show whether you were over or under; the calculator can show both.
Eight examples
| Measured | True value | Gap | Percent error |
|---|---|---|---|
| 9.6 | 9.81 | 0.21 | 2.14% |
| 101.2 | 100 | 1.2 | 1.20% |
| 48 | 50 | 2 | 4.00% |
| 0.52 | 0.5 | 0.02 | 4.00% |
| 298 | 299.792 | 1.792 | 0.60% |
| 3.14 | 3.14159 | 0.00159 | 0.05% |
| 72 | 70 | 2 | 2.86% |
| 1.9 | 2.0 | 0.1 | 5.00% |
Look at rows 3 and 4. A gap of 2 and a gap of 0.02 both give 4%, because each is 4% of its own true value. That is the point of using a percentage.
Even 3.14 for pi, which looks rough, is only 0.05% off the real value.
Percent error, percent difference and percent change
Three formulas look alike and are used for different jobs. The difference is what you divide by.
| Name | When to use it | Divide by | Example |
|---|---|---|---|
| Percent error | Compare a measurement with a known true value | The true value | 9.6 vs 9.81 → 2.14% |
| Percent difference | Compare two measurements, neither is 'right' | Their average | 9.6 vs 9.81 → 2.16% |
| Percent change | How much something grew or shrank over time | The old value | 50 to 60 → +20% |
Percent error and percent difference are close when the numbers are close, but they are not the same thing.
Percent change divides by 50, error by 60 and difference by the average, 55. Always say which one you mean.
Why the direction matters
- Measured 48, true 50
- Gap 2 ÷ 50 = 0.04
- This is the correct percent error
- Same gap 2 ÷ 48 = 0.0417
- Gives a slightly different, wrong answer
- Small here, bigger when the error is large
The true value is the fixed reference, so it always goes at the bottom of the fraction.
How big an error is acceptable?
There is no universal limit. It depends on the instrument and the purpose. A school pendulum experiment with a stopwatch often lands in the 1% to 5% range. A high-precision lab measurement aims for a tiny fraction of 1%.
Bands vary by teacher and by experiment. Treat them as a conversation starter, not a grade.
Accuracy against precision
Taking several readings and averaging them reduces random scatter. It does not remove a built-in bias.
Common sources of error
- Reading the instrument, such as looking at a ruler from an angle.
- Zero error, where a scale or timer does not start at zero.
- Rounding too early, which adds error to every later step.
- Reaction time with hand-held stopwatches.
- Wrong true value, for example using 9.8 for gravity in a place where 9.78 is more accurate.
Questions people ask
What is the percent error formula?
Percent error = |measured value − true value| ÷ |true value| × 100. It expresses the gap as a percentage of the accepted value.
Can percent error be negative?
With the absolute value bars it is always positive. Some courses omit the bars to show direction: negative means the measurement was below the true value.
What is the difference between percent error and percent difference?
Percent error compares a measurement with a known true value and divides by that value. Percent difference compares two measurements and divides by their average.
What is a good percent error?
It depends on the experiment. For many school labs under 5 percent is good and under 2 percent is excellent, while precision science aims for far smaller values.
Why can't I calculate percent error when the true value is zero?
Because the formula divides by the true value. Dividing by zero is undefined, so report the absolute error instead.
Is percent error the same as percent change?
No. Percent change compares an old and a new value over time and divides by the old value. Percent error compares a measurement with a true reference value.