Functional Relationshipو The input quantities X1, X2, X3, …, Xn are themselves measured quantities which may further depend upon some other quantities, including corrections and correction factors for systematic effects, thereby leading to a complicated functional relationship, which may rather be difficult if not impossible to write down explicitly. Further, it may not be an algebraically defined function and may be a portion of it is determined experimentally or it exists only as an algorithm that is calculated numerically. Sometimes •f =•xp is determined experimentally by measuring the change in Y by incorporating a change in Xp. In this case, knowledge of f is or a portion of it is correspondingly reduced to an empirical first order Taylor’s expansion. So the function f may be taken in a broader sense.
If the mathematical model does not satisfy the degree of accuracy desired, then additional input quantities may be included in the function f to eliminate the inadequacy.
For example for ordinary day-to-day weighing in a market place, mass of the commodity is taken as the nominal mass of the weight. For a better degree of accuracy, we take into account the actual mass of the weight. For still better accuracy we apply air buoyancy correction for which we may take density of airas 1:2 kg=m3. For still better accuracy, we calculate values of the density of air, and those of weight and commodity by measurements. To improve the accuracy further, we may like to know the actual composition of air or measure the density of air inside the balance at the time of weighing only. In the first case it is a simple relation, a correction due to mass of weight is applied in the second case, air buoyancy is added in the third case, a relationship of air density with environmental conditions is added further and a few measurements of density of weight and commodity are to be taken. To improve further experiment for determination of air density in situ is carried out and added in the relationship.
In another example, power dissipated across a given resistor is given by
To improve accuracy variation of resistance of R with temperature is to be considered giving
Relationship becomes more and more complex, if dependence of ˛ with temperature and measurement of temperature are taken into account.
Source: measurement uncertainty
