What is percent error?

Percent error measures how far an experimental value is from the accepted (theoretical) value, expressed as a percentage: |experimental - theoretical| / |theoretical| × 100. Lower percent error = more accurate measurement.

Can percent error be negative?

The standard formula uses absolute values, so percent error is always positive (just the magnitude of deviation). Some fields use signed percent error to indicate direction: positive means measurement higher than expected; negative means lower. Both forms are valid; specify which you use.

What's a good percent error?

Depends on context. Educational labs: under 5-10%. Industrial measurements: under 1-2%. Scientific research: under 0.1-1%. Precision metrology: under 0.001%. Atomic clocks: under 10⁻¹⁵. Always compare to your application's tolerance requirements.

Why divide by theoretical and not experimental?

Theoretical (accepted) is the reference standard. Dividing by it gives error "as a fraction of what it should be". If you divided by experimental, the percentage would slightly differ depending on which direction the error went. Theoretical provides consistent baseline.

What if theoretical value is zero?

Percent error is undefined (division by zero). For values near zero: percent error becomes very large and unstable. In such cases: use absolute error instead, or normalize by something else (range of measurements, RMS, etc.).

How do I reduce percent error?

Multiple strategies: improve measurement technique (calibrate instruments, use better tools), take multiple measurements and average (random error decreases as 1/√n), eliminate systematic errors (consistent biases), control environmental factors, train operators, follow standard protocols.

Mean Absolute Percentage Error = average of |error_i|/|true_i| × 100 over n measurements. Used in forecasting evaluation: time series, ML models. Lower MAPE = better forecast. Common alternatives: RMSE (root mean square error), MAE (mean absolute error). Each highlights different error characteristics.

Percent Error Calculator

Q: What's the difference between percent error and percent change?

Percent error: compares observed to known true value (one is "right"). Percent change: compares two values without designating one as true (often before/after, old/new). Same formula structure but different meanings.

Enter the experimental (measured) value and the theoretical (accepted) value to compute the percent error, absolute error, and relative error.

Last updated: May 30, 2026

About this calculator

Percent error measures how far an experimental or measured value differs from the accepted or theoretical value, expressed as a percentage. The formula: % error = |experimental − theoretical| / |theoretical| × 100. It's the standard way to assess measurement accuracy in science, engineering, and statistics.

For example: if a student measures gravitational acceleration as 9.7 m/s² but the accepted value is 9.81 m/s², the percent error is |9.7 − 9.81|/9.81 × 100 = 1.12%. The smaller the percent error, the more accurate the measurement.

Percent error differs from related concepts: - **Percent change**: includes direction (positive = increase, negative = decrease). - **Percent difference**: compares two values without designating one as "true". - **Absolute error**: just the magnitude of difference (units of original measurement). - **Relative error**: ratio of error to true value (decimal, not percentage).

Why use percent rather than absolute error? Scale matters. A 1 cm error in measuring a finger is huge (~10%); a 1 cm error in measuring a football field is tiny (~0.01%). Percent error normalizes for scale, making errors comparable across very different magnitudes.

Common applications: laboratory science (chemistry, physics experiments), engineering measurements, quality control, statistical estimates, machine learning model evaluation, financial forecasting accuracy, and any context where comparing measured-to-true values matters.

Formula

**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.

How to use this calculator

Enter measured (experimental) value.
Enter theoretical (accepted, true) value.
Calculator returns percent error.
For percentage form: multiply ratio by 100.
For absolute error: just |experimental − theoretical|.
Aim for under 5% for good measurements; under 1% for precision work.

Worked examples

Physics lab measurement

**Scenario:** Measured gravitational acceleration: 9.65 m/s². Accepted value: 9.81 m/s². Percent error? **Calculation:** % Error = |9.65 - 9.81|/9.81 × 100 = 0.16/9.81 × 100 ≈ 1.63%. **Result:** ~1.6% error — typical for a careful student lab. Sources: timing imprecision, friction, air resistance. Below 5% considered good for educational experiments.

Chemical yield

**Scenario:** Reaction should produce 50 g but produces 42 g. Percent error in yield? **Calculation:** % Error = |42 - 50|/50 × 100 = 8/50 × 100 = 16%. **Result:** 16% lower than theoretical. Often called "percent loss" rather than percent error in chemistry. Equivalent percent yield: 42/50 × 100 = 84%. Indicates significant loss — investigate side reactions, incomplete reaction, transfer losses.

Precision manufacturing

**Scenario:** Part specified as 25.00 mm ± 0.05 mm. Measured: 25.03 mm. Within tolerance? **Calculation:** % Error = |25.03 - 25.00|/25.00 × 100 = 0.12%. Tolerance: ±0.05/25 × 100 = ±0.2%. **Result:** 0.12% < 0.2% tolerance — within spec. Pass. Larger errors (over 0.2%) would fail QC. Critical for precision parts (aerospace, optics, watchmaking).

When to use this calculator

**Use percent error for:**

- **Lab reports**: standard error analysis in science. - **Engineering**: tolerance verification. - **Calibration**: instrument accuracy assessment. - **Quality control**: pass/fail criteria. - **Forecasting**: model evaluation. - **Statistical sampling**: poll/survey accuracy. - **Machine learning**: prediction quality. - **Educational science**: experiment evaluation.

**When NOT to use percent error:**

- **Theoretical = 0**: division by zero (undefined). - **Comparing two measurements**: use percent difference instead. - **Direction matters**: use signed percent error or percent change. - **Different units**: must use same units for comparison.

**Choosing metric:**

- **Percent error**: when there's a "true" reference value. - **Percent difference**: when comparing two measurements neither is "true". - **Percent change**: when measuring change over time. - **Absolute error**: when scale-free comparison not needed.

**Acceptable error ranges:**

| Application | Typical acceptable | |---|---| | Educational labs | < 5-10% | | Industrial measurement | < 1-2% | | Scientific research | < 0.1-1% | | Metrology standards | < 0.001-0.01% | | Atomic clocks | < 10⁻¹⁵ |

**Common applications:**

- **Experimental physics**: measuring constants, validating theory. - **Analytical chemistry**: quantitative analysis accuracy. - **Manufacturing**: parts within spec. - **Calibration labs**: instrument verification. - **Weather forecasting**: prediction skill. - **Stock market predictions**: model evaluation. - **Medical tests**: diagnostic accuracy. - **Election polls**: pre-election vs actual.

**Real-world error sources:**

- **Random**: scatter due to noise, vibration. - **Systematic**: bias from miscalibration. - **Method**: limitations of measurement technique. - **Human**: parallax, reading mistakes. - **Environmental**: temperature, humidity effects.

Minimize each through: - Careful calibration. - Multiple measurements (averaging). - Better instruments. - Standardized procedures. - Trained operators.

**Software:**

- **Excel**: =ABS(measured-true)/ABS(true)*100. - **Python**: simple formula or numpy. - **R**: built into many stats packages. - **MATLAB**: simple expression. - **Specialized lab software**: built into many systems.

**Pitfalls:**

- **Dividing by zero**: theoretical can't be zero. - **Negative theoretical**: take absolute value. - **Comparing percent errors at different scales**: same percent error can mean different things. - **Confusing percent error with percent change**: different concepts. - **Over-precision**: percent error precision limited by input precision. - **Mismatched units**: ensure both values in same units.

**Statistical reasoning:**

For random samples: as sample size n increases, percent error decreases as 1/√n.

Quadruple sample → halve standard error.

This is why large samples give more accurate estimates.

**Common applications:**

- **High school chemistry**: lab report calculations. - **Engineering**: tolerance management. - **Scientific publications**: error reporting. - **Calibration certifications**: traceability documentation. - **Quality control**: SPC (Statistical Process Control). - **Machine learning model selection**: MAPE, RMSE. - **Weather/climate models**: skill scores. - **Financial forecasting**: revenue/cost prediction accuracy.

**Pitfalls:**

- **Confusing absolute and relative error**. - **Theoretical = 0**: undefined. - **Theoretical not really "true"**: standard value might also have uncertainty. - **Mixing units**: same units for both values. - **Rounding too aggressively**: precision lost. - **Using percent error inappropriately**: when no true value exists, use percent difference.

Common mistakes to avoid

Dividing by experimental value instead of theoretical.
Forgetting absolute value (giving negative percent errors when positive expected).
Confusing percent error with percent change.
Comparing percent errors of vastly different scales without context.
Reporting more significant figures than input precision allows.
Using when theoretical value is zero (division by zero).
Using when "true" value is unknown (use percent difference).
Mixing units between experimental and theoretical.

Frequently Asked Questions

Sources & further reading

Formula

Worked examples

Physics lab measurement

Chemical yield

Precision manufacturing

When to use this calculator

**Use percent error for:**

**When NOT to use percent error:**

**Choosing metric:**

**Acceptable error ranges:**

**Common applications:**

**Real-world error sources:**

Minimize each through: - Careful calibration. - Multiple measurements (averaging). - Better instruments. - Standardized procedures. - Trained operators.

**Software:**

**Pitfalls:**

**Statistical reasoning:**

For random samples: as sample size n increases, percent error decreases as 1/√n.

Quadruple sample → halve standard error.

This is why large samples give more accurate estimates.

**Common applications:**

**Pitfalls:**

Common mistakes to avoid

Dividing by experimental value instead of theoretical.

Forgetting absolute value (giving negative percent errors when positive expected).

Confusing percent error with percent change.

Comparing percent errors of vastly different scales without context.

Reporting more significant figures than input precision allows.

Using when theoretical value is zero (division by zero).

Using when "true" value is unknown (use percent difference).

Mixing units between experimental and theoretical.

Frequently Asked Questions

About this calculator

Formula

How to use this calculator

Worked examples

Physics lab measurement

Chemical yield

Precision manufacturing

When to use this calculator

Common mistakes to avoid

Frequently Asked Questions

What is percent error?

Can percent error be negative?

What's a good percent error?

What's the difference between percent error and percent change?

Why divide by theoretical and not experimental?

What if theoretical value is zero?

How do I reduce percent error?

What is MAPE?

Sources & further reading

Related Calculators

Percentage Calculator

Percentage Change Calculator

Absolute Value Calculator

Long Division Calculator

Rounding Calculator

Fraction Calculator

Inputs

Results

About this calculator

Formula

How to use this calculator

Worked examples

Physics lab measurement

Chemical yield

Precision manufacturing

When to use this calculator

Common mistakes to avoid

Frequently Asked Questions

What is percent error?

Can percent error be negative?

What's a good percent error?

What's the difference between percent error and percent change?

Why divide by theoretical and not experimental?

What if theoretical value is zero?

How do I reduce percent error?

What is MAPE?

Sources & further reading

Related Calculators

Percentage Calculator

Percentage Change Calculator

Absolute Value Calculator

Long Division Calculator

Rounding Calculator

Fraction Calculator

Inputs

Results