Percentage Error

Define the reference and the sign convention

Write $x$ for the measured or estimated value and $r$ for the reference value. The signed difference is

A negative value means the result is below the reference. The absolute error magnitude is $|x-r|$. This corresponds to the metrology definition of measurement error, which allows a reference quantity value rather than requiring an exactly known “true” value.

In this lesson, percentage error means the non-negative magnitude

The quotient must compare values expressed in the same unit. A signed normalised error can instead be reported as $(x-r)/|r|$ times 100%, but label it explicitly because sign conventions vary.

Calculate the difference before dividing

A length is reported as 19.6 cm against a 20.0 cm reference. The signed error is $-0.4$ cm; its magnitude is 0.4 cm. Thus

An alternative reading of 20.4 cm has the same percentage error but the opposite signed error. The percentage magnitude alone does not identify the direction.

Converting both readings to millimetres leaves $4/200=0.02$, so the relative error is unchanged. Multiplicative changes of unit cancel between numerator and denominator.

Recover permitted values from a relative limit

Suppose a mathematical specification allows an error magnitude of at most 3% relative to a positive reference of 80 mm. Write $x$ for the numerical reading in millimetres. Then

Solving the absolute-value inequality gives

Thus the permitted measurements run from 77.6 mm to 82.4 mm.

The endpoints are included because the phrase was “at most”. If the requirement were “less than 3%”, both endpoints would be excluded. This is a stipulated tolerance calculation, not a general rule for deciding whether a physical product is acceptable.

Conversely, a bound of 0.2 mm relative to 50 mm represents $0.2/50\times100\%=0.4\%$. A bound is a permitted or stated range; it is not the same thing as an observed error.

Distinguish error from percentage difference

Sometimes two results are compared without choosing one as the reference. For positive values, a commonly used symmetric percentage difference divides the absolute difference by their arithmetic mean:

For 48 and 52, this gives $4/50\times100\%=8\%$. Treating 52 as the reference instead gives $4/52\times100\%\approx7.69\%$. These are not conflicting calculations. They use different denominators to answer different questions.

State the formula because “percentage difference” is not used consistently in every field. Do not apply this positive-value formula unchanged to arbitrary signed quantities or to a pair whose mean is zero.

Recognise when a percentage is a poor description

At a zero reference, percentage error is undefined. Near a very small reference, a tiny absolute difference can produce a huge percentage. Report the absolute difference and its units when that is more meaningful.

Relative comparisons can also depend on the origin of a scale. Readings of 21 °C and 20 °C differ by 1 °C, producing a numerical ratio of $1/20=5\%$. The corresponding Kelvin readings differ by 1 K but give $1/293.15\approx0.003411$, or about 0.341%.

The physical difference did not change. The denominator’s origin did. For thermodynamic temperature ratios, use an absolute scale; for other applications, explain the chosen reference convention. Percentage arithmetic does not automatically make every scale suitable for ratio interpretation.

Propagate stated bounds through the actual calculation

Suppose positive lengths are bounded by $9.9\le L\le10.1$ cm and $4.9\le W\le5.1$ cm. For $A=LW$, multiplication is increasing in both positive inputs, so

Relative to the central product $10\times5=50$ cm², these limits correspond to a decrease of 2.98% and an increase of 3.02%. Simply adding the input percentages, 1% and 2%, gives a useful small-error approximation of 3%, but not both exact endpoints.

The calculation assumes every combination of the stated bounds is permissible. If the quantities are constrained together, the attainable range may be narrower. Statistical uncertainty propagation is another method with further assumptions; it should not be silently substituted for deterministic bounds.

Keep error separate from uncertainty and bias

An observed difference from a reference is one comparison. Measurement uncertainty describes the dispersion of values attributed to the measurand under the available information. JCGM’s uncertainty definition makes that distinction explicit.

A small percentage error in one trial does not demonstrate that an instrument will remain accurate, and opposite errors can cancel in a sample mean. Repeated observations and a suitable reference are needed to investigate systematic and random effects. Keep the interpretation proportional to the evidence.

Related topics

Use Estimation and Approximation for numerical bounds, then Mathematical Modelling to distinguish calculation error from an unsuitable model.