What is percent error?

Percent error quantifies the discrepancy between a measured value and an accepted true value, typically expressed as the absolute difference divided by the magnitude of the accepted value, multiplied by 100%.

What is the difference between absolute and signed percent error?

Absolute percent error is always non‑negative and ignores the direction of the error. Signed percent error retains the sign to indicate whether the measurement is above or below the accepted value.

Can the accepted value be zero?

No. Percent error divides by the accepted value’s magnitude, so an accepted value of zero would make the calculation undefined.

How many decimal places should I use?

Match the significant figures of your measurement process or reporting standards. This tool lets you choose from 0 to 8 decimal places.

Does the sign of the accepted value matter?

This calculator uses the magnitude of the accepted value in the denominator to provide consistent percent error magnitudes even when the accepted value is negative.

Is percent error the same as measurement uncertainty?

No. Percent error compares a result to a reference value, while uncertainty quantifies the range of plausible values due to measurement variability.

Percent Error Calculator

Authoritative Resource and Learning Hub

This calculator implements the standard definition of relative and percent error, suitable for laboratory work, quality assurance, and academic reporting. It emphasizes accuracy, accessibility, and transparency.

Data Source and Methodology

IUPAC Compendium of Chemical Terminology (the “Gold Book”), “Relative error,” 3rd ed. (2014). https://goldbook.iupac.org — Defines relative error as (measured − true) / true, with percent error expressed as a percentage.
NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3: Accuracy and Precision (2012). https://www.itl.nist.gov/div898/handbook/

All calculations are rigorously based on the formulas and definitions provided by these sources.

Implementation notes: For robustness when the accepted value is negative, this tool uses the magnitude |accepted| in the denominator for percent and relative error. This yields consistent magnitudes while preserving the sign in the signed percent error.

The Formula Explained

Let measured value be x_m and accepted (true) value be x_t.

Absolute error: E = x_m - x_t

Relative error: r = E / |x_t|

Absolute percent error: |r| × 100%

Signed percent error: r × 100%

Absolute error:
  E = x_m - x_t

Relative error:
  r = \frac{x_m - x_t}{\lvert x_t \rvert}

Absolute percent error:
  \left\lvert \frac{x_m - x_t}{\lvert x_t \rvert} \right\rvert \times 100\%

Signed percent error:
  \frac{x_m - x_t}{\lvert x_t \rvert} \times 100\%

Glossary of Variables

Symbol / Field	Definition
x_m (Measured value)	The observed or experimental result you obtained.
x_t (Accepted value)	A reference or true value from a reputable source (standard, certificate, literature).
E (Absolute error)	Difference between measured and accepted: E = x_m − x_t.
r (Relative error)	Error as a ratio to the magnitude of the accepted value: r = E / \|x_t\|.
Absolute percent error	\|r\| × 100%. A non‑negative percentage indicating error magnitude.
Signed percent error	r × 100%. Indicates direction: positive if measured > accepted, negative otherwise.
Decimal places	Rounding applied to outputs (0–8) for reporting consistency.

How It Works: A Step‑by‑Step Example

Suppose a caliper measurement gives x_m = 9.85 cm and the accepted value is x_t = 10.00 cm.

Compute absolute error: E = 9.85 − 10.00 = −0.15 cm.
Relative error: r = (−0.15)/|10.00| = −0.015.
Signed percent error: r × 100% = −1.5%.
Absolute percent error: |−0.015| × 100% = 1.5%.

Interpretation: The measurement is 1.5% below the accepted value.

Frequently Asked Questions (FAQ)

Is percent error always positive?

“Percent error” often refers to the absolute percent error (non‑negative). This calculator also provides the signed percent error to indicate direction.

Why use |accepted| in the denominator?

Using the magnitude of the accepted value yields consistent error magnitudes even when the accepted value is negative, while the signed version still conveys direction.

Can I enter scientific notation?

Yes. Most browsers accept values like 3.2e‑5 in number inputs. Ensure the accepted value is not zero.

What if my accepted value is an interval?

Percent error compares to a single reference value. If you have a tolerance interval, you may compute error relative to the nominal value and assess whether the result lies within the specified tolerance.

How does this differ from MAPE?

MAPE (Mean Absolute Percentage Error) averages absolute percentage errors over multiple observations, often used in forecasting. This tool reports the error for one value (or computes per‑row values in bulk mode).

How many decimal places should I report?

Match the precision of your measurement system or reporting standard (e.g., lab SOP). Over‑precision can be misleading.

Does rounding affect correctness?

No. Internally, the calculator computes with the raw numbers; rounding is only applied to the displayed results for readability.

Tool developed by Ugo Candido. Content verified by the CalcDomain Editorial Team.
Last reviewed for accuracy on: September 14, 2025.