Errors during the measurement process

3.6.1 Combined effect of systematic and random errors

If a measurement is affected by both systematic and random errors that are quantified as ±x (systematic errors) and ±y (random errors), some means of expressing the combined effect of both types of error is needed. One way of expressing the combined error would be to sum the two separate components of error, i.e. to say that the total possible error is e = ±(x + y). However, a more usual course of action is to express the likely maximum error as follows:

It can be shown (ANSI/ASME, 1985) that this is the best expression for the error statistically, since it takes account of the reasonable assumption that the systematic and random errors are independent and so are unlikely to both be at their maximum or minimum value at the same time.

3.6.2 Aggregation of errors from separate measurement system component

A measurement system often consists of several separate components, each of which is subject to errors. Therefore, what remains to be investigated is how the errors associated with each measurement system component combine together, so that a total error calculation can be made for the complete measurement system. All four mathematical operations of addition, subtraction, multiplication and division may be performed on measurements derived from different instruments/transducers in a measurement system. Appropriate techniques for the various situations that arise are covered below.

Error in a sum

If the two outputs y and z of separate measurement system components are to be added together, we can write the sum as S D y C z. If the maximum errors in y and z are šay and šbz respectively, we can express the maximum and minimum possible values of S as:

Smax = (y + ay) + (z + bz); Smin = (y – ay) + (z – bz); or S - y + z ± (ay + bz)

This relationship for S is not convenient because in this form the error term cannot be expressed as a fraction or percentage of the calculated value for S. Fortunately, statistical analysis can be applied (see Topping, 1962) that expresses S in an alternative form such that the most probable maximum error in S is represented by a quantity e, where e is calculated in terms of the absolute errors as:

Thus S = (y + z) ± e. This can be expressed in the alternative form:

                                       S = (y + z)(1 ± f) where f = e/(y + z)                           (3.22)

It should be noted that equations (3.21) and (3.22) are only valid provided that the measurements are uncorrelated (i.e. each measurement is entirely independent of the others).

Example 3.6 A circuit requirement for a resistance of 550 is satisfied by connecting together two resistors of nominal values 220 and 330 in series. If each resistor has a tolerance of ±2%, the error in the sum calculated according to equations (3.21) and (3.22) is given by:

Thus the total resistance S can be expressed as

S + 550Ω ± 7.93Ω or S = 550 (1 ± 0.0144)Ω , i.e. S = 550Ω ± 1.4%

Error in a difference

If the two outputs y and z of separate measurement systems are to be subtracted from one another, and the possible errors are ±ay and ±bz, then the difference S can be expressed (using statistical analysis as for calculating the error in a sum and assuming that the measurements are uncorrelated) as:

                  S = (y – z) ± e or S = (y – z)(1 ± f)

where e is calculated as above (equation 3.21), and f = e/9y – z) 

Example 3.7

A fluid flow rate is calculated from the difference in pressure measured on both sides of an orifice plate. If the pressure measurements are 10.0 bar and 9.5 bar and the error in the pressure measuring instruments is specified as ±0.1%, then values for e and f can be calculated as:

This example illustrates very poignantly the relatively large error that can arise when calculations are made based on the difference between two measurements.