Estimates and measurements are rarely perfect. Percent error tells how far off they are compared with the true value. It works like percent change, with the actual value as the whole.
Key idea
$$\text{percent error} = \frac{|\text{estimate} - \text{actual}|}{\text{actual}} \times 100\%$$
Why it works
Being off by 2 cm on a 10 cm pencil is a big miss: 20% of its length. Being off by 2 cm on a 200 cm door is tiny: 1%. Dividing by the actual value measures the error against the size of the real thing. The absolute value bars mean we only care how far off we were, not whether we guessed too high or too low.
Worked examples
Worked example: A guess
Jo guesses there are 72 marbles in a jar. There are 80. What is the percent error?
- Error: $|72 - 80| = 8$.
- Divide by the actual value: $\frac{8}{80} = 0.10$.
- Percent error: $10\%$.
Answer: 10%
Worked example: A measurement
A student measures a board as 12 cm long. It is really 12.5 cm. What is the percent error?
- Error: $|12 - 12.5| = 0.5$.
- $\frac{0.5}{12.5} = 0.04 = 4\%$.
Answer: 4%
Worked example: Rounding
An estimate of 50 people turns out to be 47. Find the percent error to the nearest tenth of a percent.
- Error: $|50 - 47| = 3$.
- $\frac{3}{47} \approx 0.06383$.
- That is about $6.383\%$, which rounds to $6.4\%$.
Answer: About 6.4%
Common mistake
Dividing by the estimate instead of the actual value. In the first example, $\frac{8}{72} \approx 11.1\%$ is wrong. The actual value is the standard we compare against, so it is always the denominator.