Measurement Uncertainty Calculator

Measurement uncertainty is a parameter that characterises the range of values within which the true value of a measurand can be expected to lie. It reflects doubt about the result of any measurement and accounts for all sources of error — both random (Type A) and systematic (Type B). A result reported without uncertainty is incomplete from a metrological standpoint.

Type A uncertainty is evaluated by statistical analysis of a series of repeated measurements — it is derived from the standard deviation of the readings. Type B uncertainty comes from other means such as manufacturer specifications, calibration certificates, reference data, or expert judgment. Both are combined to give the total combined standard uncertainty.

Combined standard uncertainty (uc) is calculated as the square root of the sum of squares of all individual uncertainty components: uc = √(uA² + uB²). This root-sum-of-squares (RSS) method assumes the uncertainty sources are independent and uncorrelated.

The coverage factor k scales the combined standard uncertainty to give the expanded uncertainty at a desired confidence level. k=2 corresponds to approximately 95% confidence (the most common choice in calibration certificates), while k=3 gives approximately 99.7% confidence. The GUM (Guide to the Expression of Uncertainty in Measurement) recommends k=2 for most practical applications.

Expanded uncertainty (U) is obtained by multiplying the combined standard uncertainty by the coverage factor: U = k × uc. It defines an interval around the measurement result that is expected to contain the true value with a specified level of confidence (usually 95% when k=2). This is the value typically reported on calibration certificates.

The Type A (statistical) uncertainty is calculated as the standard deviation of the mean, which equals the standard deviation of the individual readings divided by the square root of the number of readings (√N). As you increase N, the denominator grows and the uncertainty decreases. This is why taking 10 readings produces a lower Type A uncertainty than taking 3 readings.

The divisor depends on the assumed probability distribution of the systematic uncertainty source. A rectangular (uniform) distribution — the most common assumption when only limits are known (e.g. instrument resolution or tolerance band) — uses a divisor of √3 ≈ 1.732. A normal distribution uses 1 (or k if coverage factor is given). A triangular distribution uses √2 ≈ 1.414. Consult your calibration certificate or instrument documentation for guidance.

Relative expanded uncertainty expresses the expanded uncertainty as a percentage of the mean measured value (U / mean × 100%). It allows easy comparison of uncertainty across measurements of different magnitudes and is widely used in quality control and conformity assessment to judge whether a measurement system is fit for purpose.