# Measurement function

## Introducing the ‘plus zero’ term

On the previous page, we introduced the measurement function, and saw that input quantities are sensitive to errors from one, or more, error effects.

However, when conducting an uncertainty analysis, there is an additional factor that we must also consider: the extent to which the measurement function describes the true physical state of the measurement process. We can account for this factor by including ‘plus zero’ term in the measurement function:

y=f(x_1,x_2,x_3…xn)+0

Here, this ‘plus zero’ term does not alter the value of the measurand, but will have an associated uncertainty in recognition of the fact that all measurement functions are approximations to the physical process they describe. In other words, this term considers the extent to which the equality of the measurement function may not hold. For example:

• If the measurement function is a linear equation, the ‘plus zero’ term considers the extent to which the instrument may be non-linear
• If the measurement function is a spectral integral determined numerically using a trapezium or rectangular rule, the ‘plus zero’ term considers the extent to which this rule acts as an approximation of the integrated quantity
• If there is an assumption that quantities or effects cancel each other out, the ‘plus zero’ term considers the uncertainty in the extent to which they cancel
• If there is an assumption that the effects of stray light are negligible, the ‘plus’ zero’ term considers the extent to which the assumption is valid

Once we have a clear picture of the extent to which the measurement function describes the true physical state of the measurement process and the effects that influence each input quantity, we can determine the uncertainty in the measurand through the process of uncertainty analysis. We’ll examine uncertainty analysis in detail later in a separate lesson, but before we do, it will be useful to examine the benefits of representing the measurement function graphically in the form of an uncertainty analysis tree, which we’ll look at on the next page.