Error Percentage Calculator
Error (%)
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How it works
Percentage error measures the discrepancy between a measured or calculated value and a reference (true) value: Error% = |(measured - true) / true| × 100%. This quantifies accuracy in measurements, estimations, and predictions.
**Error vs. uncertainty vs. accuracy vs. precision** Error: deviation of a specific result from the true value. Uncertainty: estimated range within which the true value is expected to lie. Accuracy: closeness to the true value (low error). Precision: repeatability — low scatter in repeated measurements regardless of accuracy. A biased instrument can be precise (consistent) but inaccurate (systematically wrong).
**Propagation of error** When a calculated quantity depends on multiple measured quantities, errors propagate through calculations. For addition/subtraction: absolute uncertainties add. For multiplication/division: relative (percentage) uncertainties add in quadrature. For f(x,y): σ_f = √[(∂f/∂x)²σ_x² + (∂f/∂y)²σ_y²] (error propagation formula).
**Systematic vs. random error** Systematic error: consistent offset in all measurements (calibration error, instrument bias). Cannot be reduced by averaging more measurements — must be corrected. Random error: statistical scatter around the true value. Reduces by averaging: uncertainty ∝ 1/√n for n independent measurements. Standard deviation describes random error distribution.
**Root Mean Square Error (RMSE)** RMSE = √[Σ(predicted - actual)²/n] is the standard metric for regression model accuracy. It's in the same units as the original data. RMSE penalizes large errors more than mean absolute error (MAE) — preferred when large errors are particularly undesirable. Normalized RMSE (divide by mean or range) enables comparison across datasets.
Frequently Asked Questions
- % Error = |measured - true| / |true| × 100%. For a voltmeter reading 4.97V where the true value is 5.00V: error = |4.97 - 5.00| / 5.00 × 100% = 0.6%. Instrument specifications state accuracy as ±(% of reading + % of full scale) or just ±% of reading. A multimeter specified as ±0.5% of reading ±2 digits at 5.000V: uncertainty = 0.005 × 5.000 + 0.002 = 0.027V. Compare this uncertainty to the observed error to determine if the instrument is within spec.
- For multiplication/division: relative errors add in quadrature (for independent errors): δf/f = √((δA/A)² + (δB/B)²). If A has 2% error and B has 3% error, then A×B has √(4+9) = 3.6% error. For addition/subtraction: absolute errors add in quadrature: δ(A+B) = √(δA² + δB²). If both A and B have ±0.1 unit uncertainty, A+B has ±0.14 unit uncertainty. For worst-case analysis (not statistical): just add absolute values — δ(A+B) = δA + δB. Statistical (quadrature) is more realistic for independent random errors; worst-case is required for systematic errors.
- Accuracy: closeness to the true value — low systematic error. Precision: reproducibility — low random error (small spread of repeated measurements). A dartboard analogy: accurate and precise = tight cluster on bullseye. Precise but inaccurate = tight cluster off-center (systematic bias). Accurate but imprecise = spread around bullseye (random error). Inaccurate and imprecise = spread away from bullseye. Accuracy is improved by calibration (removing systematic bias). Precision is improved by better measurement technique, averaging more readings, and reducing noise. High precision does not imply high accuracy — a well-calibrated, noisy instrument may be more accurate than a precise but miscalibrated one.
- Percentage error quantifies whether a manufactured part meets specifications. If a shaft is specified as 25.00 ± 0.05 mm and measured at 25.03 mm: error = 0.03/25.00 = 0.12%. The tolerance is ±0.2% — within specification. For process control (SPC): a process is considered capable if its 3σ spread is within the tolerance band (Cp ≥ 1.0, Cpk ≥ 1.33 for Six Sigma). Percentage error from nominal (mean value) is the offset metric; standard deviation of measurements is the variability metric. Both must be within acceptable limits for a capable, centered process.