Measurement Uncertainties Basics


In measurement, uncertainty coming from measuring and surroundings will make evaluators need to decide whether to accept or reject the measurement result. This chapter will be talking about the introduction of measurement uncertainty and measurement decision rule, please refer to the following definitions:

  •  Decision Rule

    • Decision rule shall be applied for specification or standard declaration for laboratory

    • Shall consider the respective risk level (e.g. error acceptance, error reject and hypothesis setting) before finalize decision rule.

    • Used for balancing risks for accepting non conforming items (customer risk) or delay conforming items (production risk)

  • Measurement Uncertainty

    • An estimated for measurand’s true value range and potential distribution.

    • If measurement uncertainty is smaller, that means the true value has smaller distribution and the error between measurand & true value is smaller.

    • Better quality for the measurement.

Measurement is all about obtaining measurand value, but measurand’s true value cannot be measured due to the following factors (but not limited to):

For measurement uncertainties, ISO guidance principle recommends the measurement uncertainty by:

    • List all potential factor.
    • Establish measurement mode.
    • Calculate each factor’s standard uncertainty.
    • Calculate combined standard uncertainty (Uc)

If expanded uncertainty is required, then:

    • Calculate valid degrees of freedom (currently only applicable for calibration)
    • Select confidence interval (default is 95%)
    • Calculate coverage factors (k = 2.0 for testing results, calibration pending)
    • Calculate expanded uncertainty U = k*Uc

Therefore, the measurement uncertainty can cover measurand’s real value (within 95% of confidence interval), and can be presented as Y = Measurement value ± uncertainty (Y = y ± U)

Decision Rule and Example

The following graph indicates all scenarios for decision rules between the measurand and true value.

Decision Rule Scenarios

For quick decision rule practices, please refer to the following descriptions to evaluate:

One regulation requires that commercial toys’ plasticizer content cannot exceed, but one company’s inspection result has plasticizer content of 0.07% with extended uncertainty of 0.04%. If we based on the following decision rules, which will pass or fail?

    • Inspection result (ignoring uncertainty) cannot exceed 0.1%? (Pass)
    • Inspection result cannot exceed 0.1% considered measurement uncertainty? (Fail, due to possibility where measurement uncertainty exceeds 0.1)
    • Inspection result cannot exceed 0.1%, but extended uncertainty is moderately low (cannot exceed 1/3 of the upper tolerance). (Fail, since measurement uncertainty is rougly 0.0133333%, the uncertainty included measurement still exceeds 0.1)

Statistical Distribution for Measurement Uncertainties

Statistical distribution is considered as one of the major contributions towards measurement errors, this will strongly impact the accuracy of respective measurement. 

But before going through all the distributions, there are few terms which need to be explained before (see below):

Distribution Terminology

The following are the distributions which are commonly seen in calibration’s measurement uncertaintity.

Standardized Normal Distribution
Rectangular Distribution
Triangular Distribution
U-Shaped Distribution

In conclusion, the statistical distribution depends on where the uncertainty behaves for respective measurement before deciding what is the appropriate model to imply for the uncertainty calculation.

The following table is a summarized reference for calculate standard deviation, uncertainty and degrees of freedom.

Uncertainty Distribution Calculation Setup
Degrees of Freedom (DoF) Explanation

Measurement Uncertainty Calculation Setup

For measurement uncertainty setup, the following process flow will be the general guideline to establish the model for respective calibration’s measurement uncertainty based on environment, master gauges and other variation factors.

Measurement Uncertainty Setup Flow
Measurement Uncertainty Calculation Factor

Measurement Uncertainty Examples

The following examples from TAF’s measurement uncertainty training course are displayed to demonstrate how measurement uncertainty is formulated:

The first one is the resistor calibration when perform cross matching voltage:

Resistor Calibration Example & Condition

After consolidating the uncertainty resources for the resistors, the following uncertainty summary and final measurement value is given below:

Resistor Uncertainty Summary & Final Result

The second one is power sensor comparison with standardized direct current source:

Power Sensor Calibration Example & Condition

After consolidating the uncertainty resources for the power sensor and sources, the following uncertainty summary and final measurement value is given below:

Resistor Uncertainty Summary & Final Result

In conclusion, measurement uncertainty shall be the evaluation method for all calibration laboratories. There is no finite measurement from laboratories, uncertainties from gauges, environment, appraisers and other factors shall be considered before presenting calibration results.

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