Measurement Uncertainty – Estimated Range of Possible Error in Measurement

What Is Measurement Uncertainty?

Measurement uncertainty defines the quantified range within which the true value of a measured parameter is estimated to reside, given all known sources of error and variability. No measurement—regardless of instrument or method—is perfectly exact. The International Vocabulary of Metrology (VIM) describes it as a non-negative parameter characterizing the dispersion of values attributed to a measured quantity, based on the information available. Uncertainty is typically expressed as a “±” value, such as 23.4 ± 0.3°C, often accompanied by a confidence level (e.g., 95%).

Measurement uncertainty reflects the reality that all results are subject to limitations and variability from sources such as instrument precision, environmental conditions, calibration, and even operator technique. In regulated fields like aviation, science, and manufacturing, quantifying uncertainty is essential for safety, compliance, and quality assurance. It allows stakeholders to understand the reliability and comparability of measurements, supporting robust decision-making and risk management. International standards (e.g., ISO/IEC 17025, ICAO Annex 5) mandate the estimation and reporting of measurement uncertainty, underscoring its universal importance.

How Is Measurement Uncertainty Used?

Measurement uncertainty is foundational to the integrity of reported data. By attaching a quantified uncertainty to every measurement—whether airspeed, altitude, or runway length—organizations provide transparency about the reliability of results. For instance, a pitot-static airspeed indicator might display 250 ± 2 knots, where the uncertainty encompasses instrument, environmental, and method-related factors.

Uncertainty is crucial in:

• Regulatory compliance: Proving that measurements meet safety and performance standards.
• Operational safety: Ensuring that error margins are understood and managed.
• Quality assurance: Supporting calibration, maintenance, and certification in aviation and industry.
• International harmonization: Enabling meaningful comparison and pooling of data from different sources.

Without clear uncertainty estimates, measurements cannot be confidently used for safety-critical decisions, certification, or comparative studies. Uncertainty transforms raw data into actionable information by clarifying its limitations and reliability.

Key Concepts Explained

Measurement

A measurement assigns a numerical value and unit to a physical quantity (e.g., length, mass, temperature) using an instrument or method. All measurements are subject to limitations—no reading is perfect. Influences include instrument accuracy and precision, environmental conditions, and operator interpretation. In aviation, for example, measurements might determine altimeter calibration, runway length, or atmospheric pressure—all of which are regulated to ensure safety.

Error vs. Uncertainty

• Error is the unknown deviation from the true value (which is itself unknown).
• Uncertainty is the estimated interval where the true value is expected to lie, given all known influences.

Only uncertainty is reportable and meaningful in scientific, regulatory, or operational contexts.

Accuracy vs. Precision

• Accuracy: How close a measurement is to the true value.
• Precision: How closely repeated measurements agree with each other.

A system can be precise but inaccurate (consistently wrong), or accurate but imprecise (average is correct but readings are scattered). Both high accuracy and precision are required for reliable measurement systems.

Types and Sources of Measurement Uncertainty

Measurement uncertainty arises from two main categories:

Systematic Uncertainty (Systematic Error)

• Caused by consistent, repeatable biases (e.g., miscalibrated instruments, uncorrected temperature effects).
• Systematic errors can be identified and corrected, but any uncorrected bias must be included in uncertainty reporting.
• Example: All runway length measurements are 1 m too long due to a calibration error.

Random Uncertainty (Random Error)

• Caused by unpredictable fluctuations (e.g., electronic noise, environmental variations, operator reading differences).
• Leads to scatter in repeated measurements, characterized by standard deviation.
• Reduced (but not eliminated) by taking multiple measurements and averaging.

Common Sources

• Instrument resolution and drift
• Environmental conditions: temperature, humidity, vibration
• Operator technique and interpretation
• Calibration uncertainty
• Data processing or procedural steps

Estimating and Expressing Measurement Uncertainty

Single Measurement

• Analog devices: Assign ± half the smallest scale division.
• Digital devices: Assign ± the value of the least significant digit.
• Also consider manufacturer’s specification and calibration report.

Example:
A thermometer marked every 0.1°C reads 22.5°C. Uncertainty: ±0.05°C.

Multiple Measurements

• Calculate mean and standard deviation from repeated measurements.
• Standard error of the mean = standard deviation / √(number of measurements).
• Expanded uncertainty (for ~95% confidence) is typically 2 × standard deviation.

Example:
Readings: 10.2, 10.4, 10.3, 10.1, 10.3
Mean = 10.26; Standard deviation ≈ 0.11
Report: 10.26 ± 0.22 (at 95% confidence)

Confidence Levels

• ±1 standard deviation ≈ 68% confidence
• ±2 standard deviations ≈ 95% confidence
• Always specify the confidence level in reports

Reporting Format

Standard format:
Measured Value ± Uncertainty [Unit] (Confidence Level)

Example:
Runway Length = 2,000 ± 3 m (95% confidence)

This format is required by ISO/IEC 17025, ICAO Annex 5, and other international standards.

Propagation of Uncertainty in Calculations

When results are calculated from multiple measurements, uncertainties must be combined:

Example:

• Altitude 1 = 1,000 ± 2 ft; Altitude 2 = 500 ± 1 ft
• Total = 1,500 ± 3 ft

For multiplication:

• 20.0 ± 0.2 (1%) × 1.00 ± 0.01 (1%) = 20.0 ± 0.4 (2%)

Best Practices and Practical Tips

• Repeat measurements if possible; use statistical analysis for the mean and standard deviation.
• Include calibration uncertainty and instrument limitations.
• Document all sources of uncertainty, including environment and operator effects.
• Report results with significant digits matched to uncertainty.
• Always specify the confidence interval (typically 95%).
• Overestimate rather than underestimate if unsure.

Examples and Use Cases

Example 1: Aircraft Tire Pressure

A calibrated gauge reads 210 psi. Manufacturer’s stated accuracy: ±2 psi. Repeated readings: 209, 211, 210, 212, 209 psi.
Mean = 210.2 psi; Standard deviation = 1.3 psi.
Combined uncertainty (root-sum-square): ≈ ±2.4 psi.
Reported as: 210.2 ± 2.4 psi (95% confidence)

Example 2: Altimeter Calibration

Reference pressure standard: ±0.3 hPa; Altimeter readings’ standard deviation: ±0.2 hPa.
Combined uncertainty: ±0.4 hPa.
Reported as: Altitude = 2,500 ± 0.4 hPa (95% confidence)

Use Case: Runway Length Certification

Laser distance meter (resolution ±0.01 m, calibration ±0.05 m); five readings:
Mean = 2,999.94 m; Standard deviation = ±0.02 m; Total uncertainty = ±0.06 m.
Reported as: Runway Length = 2,999.94 ± 0.06 m (95% confidence)

Glossary of Related Terms

Frequently Asked Questions

What is the difference between error and uncertainty?

A: Error is the unknown deviation from the true value; uncertainty is the estimated range where the true value likely resides, based on all known influences.

Why is measurement uncertainty important?

A: It ensures transparency, supports regulatory compliance, enables meaningful comparison, and underpins safety-critical decisions.

How is measurement uncertainty estimated?

A: By identifying and quantifying all significant sources of error—using statistical analysis for repeated measurements, manufacturer specs for single readings, and combining them via propagation rules.

How should measurement results be reported?

A: Value ± uncertainty, with units and confidence level. Example: 2000 ± 3 m (95% confidence).

What are common sources of uncertainty?

A: Instrument limitations, calibration drift, environmental conditions, operator interpretation, and procedural factors.

Actionable Rules of Thumb

• Always report measured values with uncertainty and confidence level.
• For analog devices, use ± half the smallest division.
• For digital devices, use ± the last digit displayed.
• Overestimate uncertainty if unsure of all sources.
• Combine uncertainties appropriately: add absolute for addition/subtraction, add relative for multiplication/division, multiply relative by exponent for powers/roots.

Quick Reference: Propagation Rules

Summary Table: How to Estimate and Express Measurement Uncertainty

Measurement uncertainty is at the heart of reliable, safe, and transparent measurement practices. Whether calibrating an altimeter, certifying a runway, or conducting laboratory tests, understanding and properly reporting uncertainty ensures confidence and comparability across all technical fields.