# Measurement function

## Examining the uncertainty analysis tree

On the previous page, we introduced the concept of an uncertainty analysis tree, which is focussed on the measurement function. On this page, we’ll examine this diagram, which is reproduced below, in more detail.

As can be seen from the diagram above, this uncertainty analysis tree is centred on the hypothetical measurement function:

y=f(x_1,x_2,x_3) + 0

Notice from the diagram that there are three effects that contribute to the uncertainty in the first term, u(x_1), and that this uncertainty in x_1 is propagated to an uncertainty in the measurand ( y ) using a ‘sensitivity coefficient’,c_1 . We’ll examine sensitivity coefficients in detail in the next recipe of this series.

Moving on, we can see that the second term has its own measurement function:

x_2=x_2(X_a,X_b)

The uncertainty in the terms in this measurement function, u(X_a) and u(X_b) , is again propagated to an uncertainty in the measurand (in this case x_2 ) via a sensitivity coefficient, and the uncertainty in x_2 , u(x_2) , is propagated to an uncertainty in the measurand (y) via the sensitivity coefficient c_2 .

A number of effects contribute to the uncertainty in the third term, u(x_3) , but it is not possible to quantify them separately. Instead, their combined uncertainty has been evaluated and propagated to the measurand using the sensitivity coefficient c_3 .

Finally, the uncertainty associated with the ‘plus zero’ term, u(0) has been evaluated and is propagated to the measurand via the sensitivity coefficient c_0=1 .

It should be noted at this point that the uncertainty analysis tree presented above is a simplified example, designed to introduce the basic concepts involved. In later recipes, we’ll look at a real-life example of a diagram of this type and see how we can expand our approach, but before we do, let’s look at errors, uncertainty and sensitivity coefficients in more detail in the next recipe.