Percentage Error Calculator

Calculate percentage error, absolute error, and relative error between an observed (experimental) value and the true (theoretical) value.

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Results

\begin{aligned} E_{pct} &= \dfrac{|x_{obs} - x_{true}|}{|x_{true}|} \times 100\% \\ &= \dfrac{|(9.8) - (10)|}{|10|} \times 100\% \\ &= ?\% \end{aligned}
\begin{aligned} E_a &= x_{obs} - x_{true} \\ &= (9.8) - (10) \\ &= ? \end{aligned}
\begin{aligned} E_r &= \dfrac{|x_{obs} - x_{true}|}{|x_{true}|} \\ &= \dfrac{|(9.8) - (10)|}{|10|} \\ &= ? \end{aligned}
Observed (Experimental) Value 9.8
True (Theoretical) Value 10

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What is percentage error?

Percentage error is the absolute difference between a measured value and the accepted true value, expressed as a percentage of the true value. It quantifies measurement accuracy and is widely used in science, engineering, and quality control. Because the difference is taken in absolute value, percentage error is always non-negative; the related absolute error keeps the sign and so records the direction of the deviation.

Formulas

The percentage error is the absolute difference divided by the true value, expressed as a percentage:

E%=∣xobs−xtrue∣∣xtrue∣×100%E_{\%} = \frac{|x_{\text{obs}} - x_{\text{true}}|}{|x_{\text{true}}|} \times 100\%

The absolute error (signed) shows the direction of the deviation:

Ea=xobs−xtrueE_a = x_{\text{obs}} - x_{\text{true}}

The relative error (dimensionless fraction) is:

Er=∣xobs−xtrue∣∣xtrue∣E_r = \frac{|x_{\text{obs}} - x_{\text{true}}|}{|x_{\text{true}}|}

Percentage error equals relative error multiplied by 100.

Worked example

A student measures the acceleration due to gravity as 9.8 m/s². The accepted value is 9.81 m/s².

E%=∣9.8−9.81∣∣9.81∣×100%=0.019.81×100%≈0.10%E_{\%} = \frac{|9.8 - 9.81|}{|9.81|} \times 100\% = \frac{0.01}{9.81} \times 100\% \approx 0.10\%

The absolute error is $9.8 - 9.81 = -0.01$ m/s², which is negative — indicating the measurement was an underestimate.

Sign conventions

Two conventions for percentage error coexist. The absolute-value form used above is always non-negative and reports only the magnitude of the deviation. A signed form omits the absolute value, (xobs−xtrue)/xtrue×100%(x_{\text{obs}} - x_{\text{true}}) / x_{\text{true}} \times 100\%, so a negative result marks an underestimate and a positive result an overestimate. This calculator reports the non-negative magnitude as the percentage error and exposes the direction separately through the signed absolute error.

When each error type applies

  • Percentage error – comparing accuracy across different experiments or scales, such as a chemistry titration against a physics timing experiment.
  • Absolute error – when the unit matters and the signed direction of the deviation is needed.
  • Relative error – the dimensionless form of percentage error, useful in calculations that chain multiple measurements.

Frequently Asked Questions (FAQ)

What is the formula for percentage error?

Percentage error = |observed − true| / |true| × 100%. The vertical bars denote absolute value, so the result is always non-negative. For example, a measured gravitational acceleration of 9.8 m/s² against a true value of 9.81 m/s² gives percentage error = |9.8 − 9.81| / |9.81| × 100% = 0.01/9.81 × 100% ≈ 0.10%. The signed version (without absolute value) is the absolute error and can be negative — indicating an underestimate.

What is the difference between absolute error and relative error?

Absolute error = observed − true value. It has the same units as the measurement and can be positive (overestimate) or negative (underestimate).

Relative error = |absolute error| / |true value|. It is dimensionless and expresses the error as a fraction of the true value. Percentage error is relative error multiplied by 100. Relative error is more useful for comparing measurements on different scales — a 1 mm error is very different when measuring a 10 mm bolt versus a 10 m bridge.

Can percentage error be negative?

Percentage error as defined here is always non-negative because it uses the absolute value of the difference.

However, some textbooks define it without the absolute value: (observed − true) / true × 100%, which can be negative — a negative result means the observed value was less than the true value (an underestimate). This calculator uses the absolute-value convention because the magnitude of the error is usually more informative than its sign. Use the absolute error field to see the direction.

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