Measurement Uncertainty – A Necessity for Every Laboratory!

How to Calculate the Uncertainty of a Measurement? [PSU example]

First, you must identify all sources of uncertainty in a measurement. Say you want to measure a PSU’s efficiency; you have to take the following into account besides the environmental conditions:

The equipment that you will use:

• AC Source
• DC Loads
• Power Analyzer
• Test Fixture
• Cabling

Then, you have to calculate the uncertainty of each of the above, which will actually affect the efficiency measurement. The individual uncertainty levels combined will provide the total uncertainty.

Uncertainty Evaluation Procedure

The main steps to evaluating the overall uncertainty of measurement are:

1. Identify all sources of uncertainty: Determine the factors contributing to uncertainty, like instrument precision, environmental conditions, or measurement methods.
2. Estimate individual uncertainties: Assign uncertainty value to each source, often based on instrument specifications or calibration.
3. Determine the type of uncertainty: Classify each uncertainty as systematic (consistent) or random (variable).
4. Calculate the combined uncertainty:
   • For addition or subtraction, Add the absolute uncertainties.
   • For multiplication or division, Add the relative (percentage) uncertainties.
5. Use error propagation formulas: Apply the correct formulas when combining uncertainties for complex calculations, especially for functions of multiple measurements.
6. Account for correlated uncertainties: If any measurements are related, adjust for the correlation when combining uncertainties.
7. Express the overall uncertainty: Determine the total uncertainty, often as a range (e.g., 10.0 ± 0.2).
8. Report the result: Present the measurement with its associated uncertainty, indicating the precision of the result.

Standard Uncertainty Definition

Standard uncertainty is the estimated uncertainty of a measurement expressed as a standard deviation. It represents the range within which the true value of a measurement is likely to lie, with a certain level of confidence (typically 68% for a normal distribution). The symbol for standard uncertainty is usually represented by u(x), where x is the measured quantity. It indicates the uncertainty associated with that particular measurement.

Standard Uncertainty Calculation Example – Type A

Say we have a number of repeated readings from a sample (e.g., a PSU’s efficiency at 100% load, conducted on different days by different operators). We calculate the mean value () and the standard deviation (s). From these, we can get the estimated standard uncertainty u of the mean (x) with the following formula:

Eq1: Where N is the number of measurements

Standard Uncertainty Calculation Example – Type B

If we don’t have enough information, we can still get an estimate of the upper and lower limits of the uncertainty. We will have to assume the distribution of the values. For example, the standard uncertainty for a uniform distribution is:

Eq2: Where a and b are the limits of the rectangular distribution (the minimum and maximum values).

We can also use the following type:

Eq3: Where a is the semi-range (or half-width) between the upper and lower limits.

Uniform distribution patterns are the most common, but you should keep in mind that uncertainties imported from calibration certificates are typically normally distributed (spread factor k=2).

Can Type A and Type B Uncertainty Levels be Combined?

Yes, they can, and in some cases, you actually have to combine them to get the combined standard uncertainty.

Combined standard uncertainty refers to the overall uncertainty in a measurement that arises from multiple independent sources. It is the result of combining individual uncertainties from different sources, such as instrumental errors, calibration errors, or measurement techniques.

Eq4: Where u₁, u₂,…,u_n are the standard uncertainties from each source.

This is the simplest scenario and applies when the uncertainty levels are independent of each other. If the uncertainty levels are correlated, a more complex approach is needed, which is beyond the scope of this article.

Coverage factor (k)

The coverage factor (k) is a multiplier used to calculate an interval around a measurement result within which the true value is expected to lie with a certain confidence level. It is typically used in conjunction with the standard uncertainty to define a confidence interval.

For example:

• k = 1 corresponds to a 68% confidence level of a normal distribution.
• k = 2 corresponds to a 95% confidence level (most commonly used).
• k = 3 corresponds to a 99% confidence level.

The result is often expressed as:

• Measurement ± k × u_c

Where u_c is the standard uncertainty.