MISSION · Number sense
Percent Error and Accuracy
Being 2 cm out matters differently on a pencil than on a road
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LEVEL 2 · LEARN
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The idea in one sentence
An error of 2 means nothing until you know 2 out of what, so percent error divides the mistake by the size of the thing being measured.
The formula
\[\text{percent error} = \frac{|\text{measured} - \text{true}|}{\text{true}} \times 100\%\]Measure 48 when the true value is 50:
\[\frac{|48 - 50|}{50} \times 100 = \frac{2}{50} \times 100 = 4\%\]Three parts to keep straight. The absolute error is \(|48 - 50| = 2\), in whatever units you were using. The relative error is \(\frac{2}{50} = 0.04\), with no units at all. Percent error is just the relative error written as a percentage.
The absolute value bars mean percent error is never negative — it measures size, not direction. If the direction matters, say “2 under” or “reads 4% high” instead.
Why divide at all
Being 2 cm out on a 50 cm pencil is 4% and probably careless. Being 2 cm out on a 50 km road is 0.000004% and is a miracle of surveying. Same absolute error, entirely different verdicts.
Dividing by the true value is what lets you compare errors on things of different sizes, and it is why scientific instruments are rated as “accurate to 0.1%” rather than to a fixed number of units.
The divisor is the true value — sometimes called the accepted or theoretical value — not your measurement. Using your own measurement as the reference makes the answer depend on the very thing you suspect is wrong.
Accuracy is not precision
Two words used interchangeably in ordinary speech, and distinguished carefully in measurement.
Accurate means close to the true value. Precise means repeatable — take the measurement five times and get nearly the same answer.
They come apart in both directions. A scale reading 5.02, 5.01, 5.03 kg for a true 4 kg is precise and badly inaccurate; it has a systematic offset. A scale reading 3.6, 4.4, 3.9 kg averages about right but is not precise at all.
Percent error measures accuracy only. Precision shows up as the spread of repeated readings — which is the standard deviation doing a different job.
A systematic error is the dangerous kind, because repeating the measurement will not reveal it. Only checking against a known standard will.
Where it turns up
- Science practicals, where the accepted value is in the textbook and yours is on the bench.
- Estimation. Rounding 487 to 500 introduces a percent error of \(\frac{13}{487} \approx 2.7\%\), which tells you whether the estimate was good enough.
- Manufacturing tolerance. A part specified as \(50 \pm 0.5\) mm is allowing 1% error.
- Percentage difference, which compares two measurements with no “true” one by dividing by their average instead.
Common mix-ups
- Dividing by the measured value. The true value goes on the bottom.
- Forgetting the \(\times 100\), leaving a relative error labelled as a percent.
- Keeping the sign when the question asks for percent error.
- Confusing it with percent change. The arithmetic is nearly identical, but percent change compares one quantity at two times, while percent error compares a measurement against the truth.
- Reporting more digits than the measurement justifies. A reading of 48 does not support an answer of 4.0000%.
Check yourself
Find the percent error in measuring 48 when the truth is 50, say what the divisor should be, and find the percent error in a reading of 9.8 against a true 10.
\(\frac{2}{50} \times 100 = 4\%\). The divisor is the true value. And \(\frac{0.2}{10} \times 100 = 2\%\).
LEVEL 3 · REFLECT
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