The evaluation of measurement uncertainty is a mandatory requirement of ISO/IEC 17025. This article provides a practical, step-by-step guide to calculating measurement uncertainty using the bottom-up approach, as described in the Guide to the Expression of Uncertainty in Measurement (GUM).

Step 1: Specify the Measurand

The first step is to clearly define what you are measuring. This is called the measurand. You should write down the equation that you use to calculate the measurand from the input quantities.

Step 2: Identify and Quantify Uncertainty Sources

Next, you need to identify all the potential sources of uncertainty in your measurement process. These can be broadly categorized as Type A or Type B.

  • Type A uncertainties are those that are evaluated by statistical methods, typically from a series of repeated observations. The main example is the repeatability of your measurement.
  • Type B uncertainties are those that are evaluated by other means. This includes information from calibration certificates, manufacturer's specifications, reference materials, and published data.

Step 3: Convert to Standard Uncertainties

All uncertainty components must be expressed as standard deviations. For Type A uncertainties, this is simply the standard deviation of your repeated measurements. For Type B uncertainties, you will need to convert the information into a standard deviation. For example, if a calibration certificate gives a coverage probability of 95%, you will need to divide the expanded uncertainty by a coverage factor, k=2, to get the standard uncertainty.

Step 4: Combine the Standard Uncertainties

The individual standard uncertainties are combined using the law of propagation of uncertainty, often referred to as the root-sum-of-squares (RSS) method. This gives you the combined standard uncertainty, uc.

Step 5: Calculate the Expanded Uncertainty

The final step is to calculate the expanded uncertainty, U, by multiplying the combined standard uncertainty, uc, by a coverage factor, k. The coverage factor is chosen to provide a level of confidence of approximately 95%. For a normal distribution, k=2 corresponds to a confidence level of 95.45%.

Reporting the Result

The final result should be reported as the measured value plus or minus the expanded uncertainty, along with a statement of the coverage probability. For example, 10.05 ± 0.02 g, with a coverage probability of 95%.