Errors during the measurement process

Error in a product

If the outputs y and z of two measurement system components are multiplied together, the product can be written as P = yz. If the possible error in y is ±ay and in z is ±bz, then the maximum and minimum values possible in P can be written as:

                   Pmax = (y + ay)(z + bz) = yz + ayz + byz + aybz;

                   Pmin = (y – ay)(z – bz) = yz - ayz - byz + aybz

For typical measurement system components with output errors of up to one or two per cent in magnitude, both a and b are very much less than one in magnitude and thus terms in aybz are negligible compared with other terms. Therefore, we have Pmax = yz(1 + a + b); Pmin = yz(1 - a – b). Thus the maximum error in the product P is ±(a + b). Whilst this expresses the maximum possible error in P, it tends to overestimate the likely maximum error since it is very unlikely that the errors in y and z will both be at the maximum or minimum value at the same time. A statistically better estimate of the likely maximum error e in the product P, provided that the measurements are uncorrelated, is given by Topping (1962):

Note that in the case of multiplicative errors, e is calculated in terms of the fractional errors in y and z (as opposed to the absolute error values used in calculating additive errors).

Example 3.8 If the power in a circuit is calculated from measurements of voltage and current in which the calculated maximum errors are respectively ±1% and ±2%, then the maximum likely error in the calculated power value, calculated using (3.23) is

Error in a quotient

If the output measurement y of one system component with possible error šay is divided by the output measurement z of another system component with possible error šbz, then the maximum and minimum possible values for the quotient can be written as:

For a << 1 and b << 1, terms in ab and b2 are negligible compared with the other terms. Hence:

 Thus the maximum error in the quotient is ±(a + b). However, using the same argu[1]ment as made above for the product of measurements, a statistically better estimate (see Topping, 1962) of the likely maximum error in the quotient Q, provided that the measurements are uncorrelated, is that given in (3.23).

Example 3.9

If the density of a substance is calculated from measurements of its mass and volume where the respective errors are ±2% and ±3%, then the maximum likely error in the density value using (3.23) is ± square root 0.022 + 0.0032 = ±0.036 or ±3.6%.