28 Water / Wastewater CLARIFYING MEASUREMENT UNCERTAINTY


While fl ow meters are calibrated under ideal laboratory conditions, the environments into which they are installed vary greatly. Uncertainty analyses are therefore essential to determine whether measurement systems, once installed, are capable of meeting required performance targets. In fi nancial terms the expression of uncertainty allows us to estimate the degree of exposure caused by a measurement.


I


t is a popular misconception that measurement is an exact science. In fact, all measurements are merely estimates of the true value being measured and the true value can never be known. An estimate implies that there is some degree of doubt about the accuracy of that measurement. For example, the repeated measurement of a fi xed quantity will never yield the same result every time.


The degree of doubt about the measurement becomes increasingly important with the requirement for increased accuracy. For example, when considering the relative cost of fl uids, measurement of high-value petroleum fl ow must be much more accurate than water fl ow for either industrial or domestic supply.


Uncertainty of measurement gives an indication of the quality or reliability of a measurement result. The uncertainty of a measurement is the size of this margin of doubt. In effect, it is an evaluation of the quality of the measurement result produced. To fully express the result of a measurement to refl ect its true value, three numbers are required:


1. The measured value - the fi gure indicated on the measuring instrument.


2. The uncertainty of the measurement - the margin or interval around the indicated value inside which you would expect the true value to lie within with a given confi dence level.


3. The level of confi dence attached to the uncertainty - a measure of the likelihood that the true value of a measurement lies in the defi ned uncertainty interval.


For example: 3.4 l/s ± 0.5 l/s at a 95 % Confi dence


The Standard Uncertainty (u) is the basic building block of uncertainty used for general uncertainty calculations. It defi nes a narrow band either side of the mean value (or, if appropriate, single value) within which the true value might be expected to lie.


Unfortunately, the confi dence level attached to this band is low. Assuming a normal distribution, we are only 68 % confi dent that the true value will lie within this interval. Of course, the use of an uncertainty band with a confi dence of only 68 % is unacceptably low for the majority of practical measurement situations. A higher confi dence level, and therefore a larger uncertainty interval, is required. This larger interval is called the Expanded Uncertainty (U), and normally industry requires it to correspond to 95 % Confi dence.


IET JANUARY / FEBRUARY 2023 Figure 3 – Normal Distribution showing Confi dence levels and Coverage Factor


The multiplier between the Standard Uncertainty and the Expanded Uncertainty at a given confi dence level is called the Coverage Factor (k). The value of k depends on the confi dence level you require (Figure 1)


U=u*k


For a normal distribution with a suffi ciently large number of measurements the k values used are shown in Figure 2 with a schematic representation of a Normal Distribution shown in Figure 3.


What is not uncertainty?


Very often people confuse error and uncertainty by using the terms interchangeably. However, uncertainty is the margin of doubt associated with a measurement, whereas error is the difference between the measured value and the true value.


Other situations which are not uncertainty include:


• Mistakes made by operators, as these can be avoided by working carefully through a procedure and checking work.


• Tolerances, as these are acceptance limits chosen for a process or a product.


Figure 1 – Confi dence Levels


Figure 2 - Coverage factor (k) for different normal distribution confi dence levels


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