Uncertainty – Estimated Range of Measurement Error – Measurement

Measurement

Measurement is the process of assigning a value to a physical quantity—such as length, mass, temperature, or time—using instruments or sensors. It is the backbone of science, engineering, aviation, and industry, providing the data needed for design, safety, compliance, and decision-making. Every measurement involves comparing the property of interest (the measurand) to a known standard, often using the International System of Units (SI) to ensure consistency.

All measuring instruments, from simple rulers to advanced laser interferometers, have inherent limitations: resolution, sensitivity, calibration, and environmental susceptibility. A measured value thus reflects both the true value and the limitations of the process. For example, in aviation, accurate airspeed and altitude measurements are critical for flight safety, relying on pitot tubes, barometric sensors, and altimeters—all of which introduce their own uncertainties.

Metrology, the science of measurement, emphasizes traceability: every measurement must be linked to national or international standards through a documented chain of calibrations. For example, a micrometer used to check an aircraft part’s thickness must itself be regularly calibrated against certified standards.

Measurement is not simply “reading a number.” It is a controlled process that requires awareness of instrument limitations, environmental effects, and strict procedures. In aviation, compliance with standards such as ICAO Annex 5 and ISO/IEC 17025 ensures that measurements are accurate, repeatable, and internationally comparable. The integrity of measurements is maintained through regular calibration, documentation, and systematic uncertainty analysis.

Error

Error is the difference between a measured value and the true value of the measurand. The true value itself is unknown and, in practice, cannot be obtained with absolute certainty. Thus, error represents an unknowable offset that is always present in any measurement.

Errors are generally classified as:

• Systematic Errors: Consistent, repeatable errors caused by faulty equipment, calibration issues, or environmental factors. For instance, an altimeter with an incorrect reference setting will systematically misreport altitude.
• Random Errors: Unpredictable fluctuations arising from short-term environmental changes, instrument noise, or human factors. For example, electronic noise may cause small, unpredictable variations in sensor readings.
• Gross Errors: Obvious mistakes or blunders, such as misreading an instrument by a factor of ten. These are typically excluded from formal uncertainty analysis.

Error should not be confused with uncertainty. While error is the unknown deviation from the true value, uncertainty is the estimated range within which the true value lies, given all known influences.

Example Table: Types of Measurement Errors and Their Sources

Uncertainty

Uncertainty is the quantified range within which the true value of a measurement is believed to lie, expressed with a specific confidence level (such as 95%). Uncertainty does not indicate poor measurement—it is a sign of good practice, acknowledging and documenting the limitations of the measurement process.

Uncertainty is typically reported as:

Measured Value ± Uncertainty (confidence level)

For example: 1450 ± 15 kg/h (95% confidence)

Uncertainty includes all identifiable sources: instrument limitations, calibration, environmental effects, and operator influences. The Guide to the Expression of Uncertainty in Measurement (GUM), referenced by ICAO and ISO standards, provides the methodology for calculating and reporting uncertainty.

In aviation, uncertainty quantification underpins safety, compliance, and quality. For example, when verifying the thickness of an aircraft skin panel, the uncertainty must be sufficiently small to ensure regulatory safety margins are met even at the lowest possible true value within the uncertainty range.

Measurand

The measurand is the specific physical quantity being measured. Its definition must be precise and unambiguous, including the unit of measurement, reference conditions, and measurement method.

Aviation Example:
“Runway surface friction coefficient under wet conditions at 20°C, measured with a continuous friction measuring device.”

Ambiguity in the measurand’s definition can result in inconsistent or misleading results. For instance, “aircraft weight” could refer to operating empty weight, maximum takeoff weight, or zero fuel weight—each with different implications. Regulatory bodies like ICAO and EASA emphasize clear, unambiguous definitions to ensure safety and consistency.

Standard Deviation (s)

Standard deviation (s) quantifies the spread or dispersion of repeated measurements around their mean. It is a key statistical tool for understanding random variability in measurement.

For a set of n measurements ( x_1, x_2, …, x_n ):

[ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2} ]

Example: Five thickness measurements (mm):

Sum of squared deviations = 0.001
Standard deviation ( s = \sqrt{0.001/4} = 0.016 ) mm

In uncertainty analysis, standard deviation of repeated measurements forms the Type A standard uncertainty.

Standard Uncertainty (u)

Standard uncertainty (u) is the uncertainty of a measurement, expressed as a standard deviation. It is the universal unit for combining different sources of uncertainty.

• Type A evaluation: Standard uncertainty is the standard deviation of the mean, ( s/\sqrt{n} ), from repeated measurements.
• Type B evaluation: Standard uncertainty is estimated from calibration certificates, manufacturer’s specs, or expert judgment, converted to a standard deviation using probability distributions.

All uncertainty components must be converted to standard uncertainty before they can be combined.

Combined Standard Uncertainty (uc)

Combined standard uncertainty (uc) is the total standard uncertainty arising from all significant sources, calculated using the root-sum-square (RSS) method:

[ u_c = \sqrt{u_1^2 + u_2^2 + … + u_n^2} ]

This assumes the sources are independent. If any are correlated, covariance terms are added. Each uncertainty component—whether from instrument calibration, environmental variation, or operator technique—must be identified and included.

Aviation Example:
When calibrating a precision altimeter, the combined uncertainty may include reference standard uncertainty, temperature variation, instrument resolution, and human reading error.

Expanded Uncertainty (U)

Expanded uncertainty (U) is the value obtained by multiplying the combined standard uncertainty by a coverage factor (k), typically ( k = 2 ) for 95% confidence:

[ U = k \cdot u_c ]

Expanded uncertainty is what appears on calibration certificates and test reports, communicating the range within which the true value is expected to lie with the chosen level of confidence.

Example:
Measurement = 120.0 V, combined standard uncertainty = 0.5 V, ( k = 2 )
Reported as: 120.0 ± 1.0 V (95% confidence)

The coverage factor may be adjusted for non-normal distributions or limited degrees of freedom.

Type A and Type B Uncertainty Evaluations

• Type A: Based on statistical analysis of repeated measurements. Quantifies the random variability using standard deviation and standard error.
• Type B: Based on information other than direct repetition (manufacturer’s specs, calibration data, literature, expert judgment). Standard uncertainty is calculated using probability distributions.

All uncertainties, whether Type A or B, must be expressed as standard uncertainties before they are combined.

Probability Distributions in Uncertainty Evaluation

Probability distributions describe how likely different values of an uncertainty component are. Choice of distribution directly affects standard uncertainty calculation.

• Normal (Gaussian): Used for random, statistically evaluated uncertainties.
• Rectangular (Uniform): Used for uncertainties where all values in a range are equally likely (e.g., instrument resolution). Standard uncertainty = maximum range / √3.
• Triangular: Used when central values are more likely but extremes are possible. Standard uncertainty = max range / √6.
• U-shaped: Rare, used when extremes are most likely.

Proper selection of distribution is vital for accurate uncertainty analysis.

Uncertainty Budget

An uncertainty budget is a structured table listing all significant sources of uncertainty, their types, estimated values, probability distributions, and standard uncertainties. It provides transparency, traceability, and justification for the reported uncertainty.

Components typically include:

• Instrument calibration and drift
• Repeatability (Type A)
• Environmental variability
• Operator influence
• Manufacturer’s specifications (Type B)

Example: Uncertainty Budget for Fuel Flow Calibration

An uncertainty budget is required for all accredited calibration and testing activities under ISO/IEC 17025 and ICAO standards.

Summary

Uncertainty is an inescapable aspect of measurement. Rather than a weakness, it is a hallmark of rigorous, reliable, and transparent science and engineering. In aviation and high-stakes industries, comprehensive uncertainty analysis ensures compliance, safety, and quality—underpinning every decision from maintenance to navigation and certification.

By identifying, quantifying, and documenting all sources of uncertainty, organizations can ensure that their measurements are trustworthy, their compliance is defensible, and their operations remain safe and efficient.