What Is Percentage Error?
Percentage error (also called percent error) measures the accuracy of a measurement or estimate by comparing it to the true or accepted value. It is widely used in science, engineering, and statistics to express how far off a measured value is from the expected value. A lower percentage error indicates a more accurate result.
The Percentage Error Formula
The absolute value signs (| |) ensure the result is always positive, indicating the magnitude of the error regardless of direction.
Step-by-Step Example
In a lab experiment, you measured the boiling point of water as 99.1°C. The accepted value is 100.0°C:
- Find the difference: |99.1 - 100.0| = 0.9
- Divide by true value: 0.9 ÷ 100.0 = 0.009
- Convert to percentage: 0.009 × 100 = 0.9% error
Types of Error
Absolute Error
Absolute error tells you the size of the error in the same units as the measurement. For the example above, the absolute error is 0.9°C.
Relative Error
Relative error is a dimensionless ratio. Multiply by 100 to get percentage error. In the example, the relative error is 0.009.
Comparison Table
| Error Type | Formula | Result (Example) | Units |
|---|---|---|---|
| Absolute Error | |Measured - True| | 0.9 | °C |
| Relative Error | Absolute ÷ True | 0.009 | None |
| Percentage Error | Relative × 100 | 0.9% | % |
Examples from Different Fields
Chemistry Lab
You determined the density of aluminum as 2.68 g/cm³. The accepted value is 2.70 g/cm³:
Error = (|2.68 - 2.70| ÷ 2.70) × 100 = (0.02 ÷ 2.70) × 100 = 0.74%
Physics Experiment
Your measurement of gravitational acceleration was 9.72 m/s². The accepted value is 9.81 m/s²:
Error = (|9.72 - 9.81| ÷ 9.81) × 100 = (0.09 ÷ 9.81) × 100 = 0.92%
Estimation in Everyday Life
You estimated a room was 12 feet long but it measured 13.5 feet:
Error = (|12 - 13.5| ÷ 13.5) × 100 = (1.5 ÷ 13.5) × 100 = 11.1%
Acceptable Error Ranges
| Field | Acceptable Error | Notes |
|---|---|---|
| Pharmaceutical manufacturing | < 1% | Strict regulatory standards |
| Chemistry labs (college) | 1-5% | Depends on experiment |
| Physics labs (college) | 1-5% | Equipment limitations |
| Engineering | 1-3% | Safety margins required |
| Biological sciences | 5-10% | More natural variation |
| Social science surveys | 3-5% | Margin of error in polling |
Sources of Error
- Systematic errors: Consistent errors due to faulty equipment or technique. They always shift results in the same direction.
- Random errors: Unpredictable fluctuations in measurements. They can be reduced by taking multiple measurements and averaging.
- Human error: Misreading instruments, incorrect calculations, or inconsistent procedures.
- Environmental factors: Temperature, humidity, and pressure can affect measurements.
Tips for Reducing Percentage Error
- Calibrate instruments before use to minimize systematic error.
- Take multiple measurements and use the average to reduce random error.
- Use precise instruments with smaller graduation marks.
- Follow standard procedures carefully and consistently.
- Control environmental variables that might affect results.
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