Percent Error Calculator

**Percent Error:** % Error = |experimental − theoretical| / |theoretical| × 100 Where: - experimental = measured or observed value - theoretical = accepted or expected value - | | = absolute value (always positive) **Result is always positive (when using absolute value).** **Worked example:** Theoretical value: 9.81 m/s² (g on Earth). Experimental: 9.70 m/s². % Error = |9.70 − 9.81| / |9.81| × 100 = 0.11 / 9.81 × 100 = 0.01121 × 100 = 1.12% **Signed percent error (sometimes used):** % Error (signed) = (experimental − theoretical) / |theoretical| × 100 Positive: measurement higher than expected. Negative: measurement lower than expected. For above: (9.70 − 9.81)/9.81 × 100 = -1.12%. **Related error metrics:** | Metric | Formula | |---|---| | Absolute error | |experimental − theoretical| | | Relative error | |error| / |theoretical| | | Percent error | relative error × 100% | | Mean absolute error | average of |errors| | | RMS error | √(average of squared errors) | **Worked example: temperature measurement** Theoretical (true): 100°C (boiling water at sea level). Measured: 98.5°C. Absolute error: 1.5°C. Relative error: 1.5/100 = 0.015. Percent error: 1.5%. **Significance of percent error:** | Percent Error | Quality | |---|---| | < 0.1% | Excellent | | 0.1-1% | Very good | | 1-5% | Good | | 5-10% | Acceptable for many purposes | | 10-30% | Poor — investigate | | > 30% | Likely error or wrong technique | Tolerances vary by field. Chemistry typically demands ~1-5%. Field measurements may accept 10-20%. **Comparison to percent change:** Percent change is for measuring change over time or between groups. % Change = (new − old)/old × 100. Sign indicates direction. Percent error specifically compares observed to "true" value. **Why divide by theoretical (not experimental)?** Theoretical is the reference. Comparing error relative to "what should be" gives consistent meaning. If you divided by experimental instead, you'd get a slightly different number depending on which way the error went. **Sources of measurement error:** - **Systematic**: consistent bias (wrong calibration, faulty equipment). - **Random**: scatter around mean (electronic noise, parallax). - **Personal**: observer error. - **Method**: limitation of measurement technique. - **Environment**: temperature, pressure, vibration. Reducing percent error involves identifying and minimizing each source. **Common scientific applications:** - **Chemistry lab**: percent yield calculations. - **Physics experiments**: measuring constants. - **Engineering**: tolerance specifications. - **Calibration**: instrument accuracy. - **Quality control**: manufacturing precision. - **Survey research**: poll accuracy. - **Weather forecasting**: prediction error. - **Machine learning**: model evaluation metrics. **Worked examples by field:** **Chemistry (percent yield):** Theoretical yield: 10.0 g. Actual yield: 8.5 g. % Error = |8.5 - 10|/10 × 100 = 15% (actually called "percent loss" in this context). Percent yield = actual/theoretical × 100 = 85%. **Physics (constant determination):** Theoretical: c = 299,792,458 m/s. Experimental: 299,000,000 m/s. % Error = |299,000,000 - 299,792,458| / 299,792,458 × 100 = 0.26%. **Manufacturing tolerance:** Specification: 50.0 mm ± 0.1 mm. Part measured: 50.07 mm. % Error = |50.07 - 50.00|/50.00 × 100 = 0.14%. If tolerance is 0.1 mm = 0.2%, then 0.14% is within tolerance. Acceptable. **Polling:** True value (population proportion): 52% support a candidate. Poll measurement: 49%. % Error = |49 - 52|/52 × 100 ≈ 5.8%. For polls with ±3% margin of error, this is just outside expected range. **Statistical sampling:** Random samples have inherent variability. Percent error decreases as sample size increases: - Standard error decreases as 1/√n. - Quadruple sample → halve error. For 1000-person poll: ~3% margin. For 10,000: ~1%. For 100,000: ~0.3%. **Engineering tolerances:** Common precision tolerances: - **±5%**: rough machining, sand casting. - **±1%**: standard machining. - **±0.1%**: precision components. - **±0.01%**: optical instruments. - **±0.001%**: scientific standards. Each step up costs roughly 10× more. **Programming:** def percent_error(experimental, theoretical): return abs(experimental - theoretical) / abs(theoretical) * 100 Trivial in any language. **Pitfalls:** - **Dividing by zero**: theoretical can't be 0. - **Sign confusion**: standard uses absolute value (always positive). - **Confusing with percent change**: different concepts. - **Wrong reference**: divide by theoretical, not experimental. - **Reporting too many digits**: precision of percent error limited by input precision. - **Wrong formula**: some use percent difference (without "true" value). **Common applications:** - **Lab reports**: standard error analysis. - **Quality control**: pass/fail criteria. - **Engineering specs**: tolerance verification. - **Calibration**: instrument check. - **Forecasting**: model accuracy. - **Sports analytics**: prediction quality. - **Financial estimates**: revenue/profit projections. - **Machine learning**: regression model evaluation. **Mean Absolute Percentage Error (MAPE):** For multiple measurements: MAPE = average of percent errors = (1/n) × Σ |error_i|/|true_i| × 100 Common machine learning metric for time series forecasting. **Software:** - **Excel**: =ABS(B1-A1)/ABS(A1)*100 - **Python**: from numpy: np.abs(exp - theo) / np.abs(theo) * 100. - **R**: similar. - **MATLAB**: abs(exp - theo) / abs(theo) * 100. **Comparison of error metrics:** | Metric | What it measures | |---|---| | Absolute error | Magnitude (units of original) | | Relative error | Fraction (dimensionless) | | Percent error | Fraction × 100 | | Squared error | Magnitude² (for variance) | | Standard deviation | Spread in original units | | Coefficient of variation | Standard deviation / mean (%) | Choose based on what's appropriate for the context. **Common pitfalls:** - **Negative theoretical**: take absolute value in denominator. - **Both zero**: undefined; both small: huge sensitivity to rounding. - **Inappropriate comparison**: comparing percent errors of different quantities can mislead. - **Treating percent error as the only quality metric**: also need accuracy, precision, traceability.