Errors in Measurement | Classification of Errors

To understand errors in measurement, first distinguish a true quantity value, a reference value and a measured value:

True Value

A true quantity value is a value consistent with the definition of the quantity. In the traditional error approach it is treated as unique but unknowable in practice. Averaging many measurements instead estimates an expected value under stated assumptions. Measurement work therefore uses an appropriate reference quantity value and reports its uncertainty.

Measured Value

A measured value is a quantity value that represents a measurement result. It may come from one indication or from calculations, corrections and repeated observations. A complete result generally includes the measured value and its measurement uncertainty.

Measurement error is the measured quantity value minus the reference quantity value. The term static error is sometimes used when the input is constant, but the same sign convention applies.

Using the page’s notation, dA = Am – At, where dA is the error, Am is the measured value and At represents the chosen reference value. A positive result means the measured value is above that reference.

Error is known only when a suitable reference value is known with negligible or stated uncertainty. Measurement uncertainty is different: it characterises the dispersion of values attributed to the measurand from the information available.
The following terms describe error limits and derived results.

Limiting Errors or Guarantee Errors

A limit of error, now commonly called maximum permissible error, is an extreme measurement-error value permitted by a specification or regulation under stated conditions. For example, an ammeter specification may give positive and negative limits as a percentage of reading, range or both. It is a compliance limit, not the actual error of every reading and not a complete uncertainty statement.

Relative Error or Fractional Error

Relative error is error divided by a specified non-zero reference value. Its sign follows the error; its magnitude may be stated as a fraction or percentage. In the notation below:

dA is the error and A is the reference magnitude.
The following first-order rules estimate resultant limiting error. They assume small input errors and use worst-case magnitudes unless a statistical uncertainty method is stated:

(a) Sum of two quantities: let the measured inputs be a₁ and a₂, with A = a₁ + a₂. The first-order relative increment is:

Multiplying and dividing the first contribution by a₁ and the second by a₂ gives:

For worst-case limits, add the magnitudes of the absolute input errors for either a sum or a difference. The displayed signed differential shows how each input affects the result, but cancellation of unknown error signs must not be assumed. For standard uncertainties, combine sensitivity coefficients with variances and include covariance for correlated inputs.
(b) Product of two quantities: for inputs a₁ and a₂, let A = a₁.a₂. Logarithmic differentiation gives the first-order relative relation:

For worst-case small-error limits, add the magnitudes of the relative errors in measurement of the factors. Derived quantities such as power factor require their actual measurement model. For a power A = aⁿ, the first-order relative error magnitude is approximately |n| times the relative error magnitude of a.

Types of Errors

Formal metrology distinguishes systematic and random types of errors in measurement. The practical headings below also separate mistakes, instrument effects, environmental influences and observation problems. A mistake is not itself a measurement error and should be detected rather than included in an uncertainty calculation.

Gross Errors

Gross errors are better described as mistakes or blunders: misreading 21 as 31, using the wrong range, transcribing a value incorrectly or applying the wrong formula. Repetition alone does not make a mistake valid. Prevent and detect these problems through controlled procedures:

1. Record units, range, settings, time and raw indications directly. Use independent calculation checks and preserve an audit trail.
2. Use range and consistency checks, repeat suspect observations and investigate outliers. Independent review can find a transcription or method error, while simple averaging cannot remove it.

Systematic Errors

Among these kinds of errors, systematic measurement error remains constant or changes predictably across replicate measurements. Known systematic effects may be corrected, but the correction and reference values still contribute uncertainty. Common sources include:

Instrumental Errors

Instrumental effects can arise from calibration bias, zero offset, drift, friction, hysteresis, finite input impedance, loading or operation outside rated conditions. Use the measuring instruments within specification, verify zero and range and apply only traceable corrections supported by calibration. Recalibrate, repair or replace an instrument when its performance is outside limits. These are systematic errors in measurement, not gross mistakes.

Environmental Errors

Environmental influences include temperature, pressure, humidity, vibration, supply variation and external magnetic field or electric fields. Their effects may be systematic, random or both. Control or measure relevant conditions and include remaining effects in the uncertainty evaluation:

• Maintain temperature and humidity within the instrument’s rated or reference conditions and allow adequate stabilisation time.
• Use suitable shielding, grounding, lead routing and separation when external magnetic or electric fields can affect the result.

Observational Errors

Observation mistakes include reading the wrong scale or digit. Parallax is an angle-dependent reading shift on an analogue pointer instrument. A mirrored scale helps the observer align the pointer with its reflection; a digital display removes pointer parallax but can still be misread. Depending on viewing practice, observational types of errors may appear systematic or random.

Random Errors

Random measurement error varies unpredictably across replicate measurements. Noise, small environmental fluctuations and reading variability can contribute. Repeated observations can estimate and reduce the random component of the mean, but they do not remove systematic error. Report the resulting precision and combine all relevant uncertainty components under the measurement model; do not label every unexplained discrepancy as one of these kinds of error.