When a calibration certificate states that a pressure gauge reads 4.005 bar when it was set to 4 bar, that single number is incomplete without one more piece of information: how uncertain is that 4.005 bar reading? Could the true value be 4.002 bar? Could it be 4.010 bar? Without knowing the uncertainty, you cannot know whether the deviation is real or just noise in the measurement system.

Measurement uncertainty is the quantification of the doubt that surrounds every measurement. It does not mean the measurement is wrong. It means we are honest about how precisely we know the true value. Every measurement has uncertainty. A calibration certificate without an uncertainty statement is incomplete, and in ISO/IEC 17025 accredited laboratories, reporting measurement uncertainty is a mandatory requirement.

This guide explains measurement uncertainty in practical instrumentation calibration terms from first principles, without unnecessary mathematical complexity. We use a full pressure gauge calibration example to show every calculation step, from reading the raw calibration data all the way to reporting the expanded uncertainty on a calibration certificate. For background on pressure calibration procedure and the as-found/as-left data format, see our article on how to calibrate a temperature transmitter step by step.

What Is Measurement Uncertainty?

Measurement uncertainty is a parameter that characterises the dispersion of values that could reasonably be attributed to the measurand (the quantity being measured). In simpler terms: it is a range around a measurement result that tells you how confident you are about the true value.

Consider weighing yourself on a bathroom scale and reading 72 kg. If you step off and back on three more times and get 71.8, 72.2 and 72.0 kg, you know the scale does not always give the same answer. The uncertainty of that scale is something like ±0.2 kg. You are confident the true weight is between 71.8 and 72.2 kg, but you cannot say exactly where in that range it falls.

The same principle applies to every pressure gauge, temperature transmitter and flow meter you calibrate. The reference standard you use is not perfect. The readings you take have some scatter. The resolution of the instruments limits how finely you can read them. Measurement uncertainty accounts for all of these contributions simultaneously.

Type A and Type B Uncertainty: The Fundamental Division

The GUM (Guide to the Expression of Uncertainty in Measurement) published by BIPM divides uncertainty into two types based on how they are evaluated.

The Worked Example: 0-10 Bar Pressure Gauge Calibration

We will work through a complete measurement uncertainty calculation using a 5-point pressure gauge calibration with one up cycle and one down cycle. This is the standard format for most process plant pressure gauge calibrations.

Instruments used

In this calibration, the UUC is set to the calibration point (the technician adjusts pressure to match the gauge reading), and the reference standard reading is observed and recorded. This means we read the changing standard and must consider the resolution of the standard in the uncertainty budget.

Raw calibration data

Calculating Type A Uncertainty (Ua): Repeatability

With only one up reading and one down reading at each calibration point, we have n = 2 observations per point. The Type A standard uncertainty is the standard deviation of the readings divided by the square root of the number of readings.

For the 2-bar calibration point: readings are 2.003 (up) and 2.005 (down).

Calculating Type B Uncertainty Components

Type B uncertainty components come from external information rather than the calibration readings themselves. For pressure gauge calibration, we consider four Type B components: Ub1 (standard certificate), Ub2 (accuracy of standard), Ub3 (resolution), and Ub4 (hysteresis). Ub5 (zero error) is also included for completeness.

Ub1: Uncertainty from the Standard's Calibration Certificate

The reference standard's calibration certificate states an uncertainty of 0.01 bar at k=2. The standard uncertainty (at k=1) is:

Ub2: Uncertainty from the Accuracy of the Standard

The standard's accuracy is stated as 0.1% of reading. Since this is a percentage of reading, the absolute value changes at each calibration point. A rectangular distribution is assumed, so we divide by sqrt(3).

Ub3: Uncertainty from Resolution

Because the UUC is set to the calibration point and the standard reading is the observed value, we consider the resolution of the reference standard (0.001 bar). The uncertainty due to resolution is half the resolution divided by sqrt(3), using a rectangular distribution.

Ub4: Uncertainty from Hysteresis

Hysteresis is the absolute difference between the down cycle reading and the up cycle reading at each calibration point. A rectangular distribution is assumed.

Ub5: Uncertainty from Zero Error

Zero error is measured after releasing all pressure and comparing the instrument reading with zero. In our example, both the up and down cycles return to exactly 0.000 bar at zero, so:

Combined Standard Uncertainty (uc)

Combined standard uncertainty is not a simple addition of all the individual uncertainty components. Adding them directly would overestimate the total uncertainty because the individual errors are independent and do not all point in the same direction at the same time. Instead, we use the root sum of squares (RSS) method, also called quadrature addition.

Expanded Uncertainty and the Coverage Factor k

The combined standard uncertainty uc represents a confidence level of approximately 68% (one standard deviation). For a calibration certificate to be meaningful and useful, we need to report at a higher confidence level, typically 95%.. To achieve this, we multiply the combined standard uncertainty by the coverage factor k.

The expanded uncertainty is what appears on the calibration certificate. For example, at the 10 bar calibration point, the certificate would state: "The measured value is 10.008 bar. The expanded measurement uncertainty is U = 0.015 bar at k = 2 (approximately 95% confidence level)."

The Uncertainty Budget: Identifying the Dominant Contributors

A useful way to check your uncertainty calculation is to look at what is driving it. Square each individual uncertainty component and express it as a percentage of the total combined variance. This is called the uncertainty budget.

The budget reveals that over 57% of the total uncertainty at the 10 bar point comes from the accuracy of the standard (Ub2). The standard certificate uncertainty (Ub1) contributes another 43%. Repeatability, resolution and hysteresis are all negligible contributors at less than 1% each. This tells us that to significantly reduce the measurement uncertainty, we would need a more accurate reference standard. Improving the calibration procedure itself (more repeat readings, better technique) would have almost no effect.

Summary: The Eight Steps to Calculate Measurement Uncertainty

1. Perform the calibration and record all data Complete the upscale and downscale readings at all calibration points. Record the UUC readings, the standard readings (up and down) and any zero readings.
2. Calculate Type A uncertainty (Ua) For each calibration point: calculate the standard deviation of the repeated readings, then divide by the square root of the number of readings. Ua = S / sqrt(n).
3. Calculate Ub1: Standard certificate uncertainty Take the uncertainty stated on the reference standard's calibration certificate and divide by the stated k factor. Ub1 = Certificate uncertainty / k.
4. Calculate Ub2: Accuracy of the standard Multiply the accuracy percentage by the reading at each calibration point, then divide by sqrt(3). Ub2 = (Accuracy% × Reading) / sqrt(3).
5. Calculate Ub3: Resolution uncertainty Identify which instrument is being read (the changing one). Take half its resolution and divide by sqrt(3). Ub3 = (Resolution / 2) / sqrt(3).
6. Calculate Ub4: Hysteresis uncertainty For each calibration point: take the absolute difference between the down reading and the up reading, divide by sqrt(3). Ub4 = |Down - Up| / sqrt(3).
7. Calculate combined standard uncertainty (uc) Square all individual uncertainties, add them together, and take the square root. uc = sqrt(Ua² + Ub1² + Ub2² + Ub3² + Ub4² + Ub5²).
8. Calculate expanded uncertainty (U) and report Multiply the combined standard uncertainty by the coverage factor k (typically k = 2 for 95% confidence). U = k × uc. Report U alongside the calibration results on the certificate.

Further Reading and External Resources

Frequently Asked Questions: Measurement Uncertainty in Calibration