Percentage Error
Percentage error is a measure of how inaccurate a measurement or calculation is compared to the true or accepted value. It expresses the difference between an observed (measured) value and a true (known) value as a percentage of the true value.
Purpose: Percentage error is used to:
- Compare measured values against known or accepted values
- Assess the validity and accuracy of experimental results
- Evaluate the precision of measuring instruments
- Determine the reliability of calculations and predictions
Reasons for Error: Errors can occur due to:
- Human error: Mistakes in reading instruments, recording data, or performing calculations
- Device limitations: Inherent inaccuracies in measuring equipment
- Environmental factors: Temperature, humidity, or other conditions affecting measurements
- Systematic errors: Consistent biases in measurement methods
- Random errors: Unpredictable variations in measurements
Significance: A small percentage error (typically <5%) indicates high accuracy, while a large percentage error suggests significant deviation from the true value. The acceptable error threshold depends on the specific application and field of study.
Computing Percentage Error
The calculation of percentage error involves several steps, each building upon the previous one:
1. Absolute Error: The absolute difference between the observed and true values:
Absolute error = |Vobserved - Vtrue|
This gives the magnitude of the error without considering direction (positive or negative).
2. Relative Error: The absolute error divided by the true value:
Relative error = |Vobserved - Vtrue| / Vtrue
This normalizes the error relative to the true value, making it easier to compare errors across different scales.
3. Percentage Error: The relative error multiplied by 100%:
Percentage error = (|Vobserved - Vtrue| / Vtrue) × 100%
This expresses the error as a percentage, which is more intuitive and easier to interpret.
Example Calculation:
Observed value = 56.891
True value = 62.327
Absolute error = |56.891 - 62.327| = 5.436
Relative error = 5.436 / 62.327 = 0.08722
Percentage error = 0.08722 × 100% = 8.722%
Note: These equations assume that the true value is known. When the true value is unknown, statistical methods such as standard deviation may be used to estimate uncertainty. For such cases, consider using our Standard Deviation Calculator.
Negative Percentage Error
While percentage error is typically expressed as a positive value (due to the absolute value in the formula), it's important to understand signed percentage error for a complete analysis:
Signed Percentage Error: When the absolute value is not taken, the formula becomes:
Signed percent error = ((Vobserved - Vtrue) / Vtrue) × 100%
Example:
Observed value = 7
True value = 9
Signed percent error = ((7 - 9) / 9) × 100%
= (-2 / 9) × 100%
= -22.222%
Interpretation:
- Negative percentage error: Indicates the observed value is smaller than the true value. The measurement or calculation underestimates the actual value.
- Positive percentage error: Indicates the observed value is larger than the true value. The measurement or calculation overestimates the actual value.
- Zero percent error (0%): The ideal outcome, indicating perfect accuracy with no difference between observed and true values.
Understanding the sign of the error is crucial in many scientific and engineering applications, as it helps identify systematic biases and provides direction for corrections.