Measurement uncertainty in practice: Determining conductivity statistically

Using electrical conductivity σ as an example, Part 4 of our series on measurement uncertainty examines how to determine measurement uncertainty using statistical methods. Here, too, we determine conductivity by measuring the conductor’s length l, side length a and electrical resistance R, then calculating conductivity σ using the familiar formula.

However, instead of taking an analytical approach based on the mathematical model (see Part 3), we now use statistical considerations alone, incorporating reference material. In the following, we consider only the combined root-sum-of-squares measurement uncertainty.

Reference material as the basis for statistical analysis

This approach requires suitable reference material covering the measurement range of the conductivity measurements. Depending on the situation, a single reference sample may be sufficient for the entire measurement range (single-point method). Alternatively, several reference samples can be used to cover the range (multi-point method). Here, we consider the single-point method.

The “true value” (see Part 1) of the reference sample’s conductivity should be known as accurately as possible. Ideally, it is documented in a certificate. This certificate typically includes:

• the method used to determine conductivity
• information on measurement traceability
• the individual measured values
• the determined measurement uncertainty, including the confidence interval and confidence level

Three contributions to combined measurement uncertainty

In this approach, the combined measurement uncertainty of the conductivity measurement consists of three uncertainty contributions:

1. Scatter of the reference measurements: The accuracy (see Part 2) of the inspection method is determined by taking m measurements on the reference sample. The scatter sRef of the measured values provides the basis for the first uncertainty contribution.
2. Uncertainty of the reference material: In the simplest case, the uncertainty uZert associated with determining the reference sample’s conductivity is taken directly from the reference material certificate.
3. Scatter of the sample measurements: The sample under investigation is also measured several times. The scatter sProb obtained from n measurements provides the third uncertainty contribution and takes the properties of the sample into account.

The combined measurement uncertainty of the conductivity measurement is given by the following formula:

The Student’s t-factors tm and tn account for the fact that the numbers of measurements m and n are often relatively small in practice. The calculation becomes considerably simpler if m and n are identical.

When the certificate uncertainty must be calculated

The uncertainty uZert cannot always be taken directly from the reference material certificate. In this case, the p individual measured values specified in the certificate and their scatter sZert must first be calculated. The formula for combined measurement uncertainty is then extended to account for this additional statistical contribution.

Confidence level and coverage factors

The formulas shown can provide both standard and expanded measurement uncertainties. What matters is the confidence level used for the Student’s t-factors tm, tn and tp. In this approach, the Student’s t-factors can directly fulfil the role of a coverage factor, replacing the commonly used coverage factor k.

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