# Introducing the measurement function

What is a measurement function? Why is it important? What do we use it for? Music by Bensound

As we saw in the video above, the ultimate aim of uncertainty analysis is to obtain an estimate of the uncertainty associated with the measured value of a measurand. In most cases, the measurand ($Y$) is not measured directly but is instead obtained from a number of input quantities ($X_{i}$) via a mathematical relationship that we will call the ‘measurement function’, as shown below.

Often, we are able to explicitly write the measurement function in terms of an analytic expression of the form:

$$y=f(x_{1}, x_{2}, x_{3} ... x_{n})$$

where y, which is an estimate of the measurand, $Y$, is obtained from estimates, $x_{i}$, of the input quantities, $X_{i}$, via the functional relationship $f$.  There are, however, cases in which it is necessary to define the measurement function in a different way, for example as an iterative solution of a model implemented through code.

Each input quantity may be influenced by one or more error effects (each of which has an associated probability distribution) leading to uncertainty in its estimate, $u(x_{i})$. The aim of uncertainty analysis is to use this information to establish the combined standard uncertainty associated with our estimate of the measurand, $u_{c}(y)$. A summary of the approach towards this aim is shown schematically in the figure below.

In practice, we typically combine individual standard uncertainties mathematically, using the Law of Propagation of Uncertainty, which is given by the ‘Guide to the Expression of Uncertainty in Measurement’ (the GUM), as:

$$u^{2}_{c}(y) = \sum_{i=1}^n c^{2}_{i}u^{2}(x_{i}) + 2 \sum_{i=1}^{n-1} \sum_{j=i+1}^n c_i c_j u(x_i, x_j)$$

where:

• $u_c(y)$ is the combined standard uncertainty associated with our estimate of the measurand, $Y$
• $u(x_i)$ is the standard uncertainty in our estimate of the input quantity $X_i$
• $c_i$ and $c_j$ are sensitivity coefficients
• $u(x_i, x_j)$ is the estimated covariance associated with $x_i$ and $x_j$

In cases in which there is no error correlation between the input quantities, the second term in the Law of Propagation of Uncertainty is not required, and the equation can be simplified to:

$$u^{2}_{c}(y) = \sum_{i=1}^n c^{2}_{i}u^{2}(x_{i})$$

We’ll look at the Law of Propagation of Uncertainty and its constituent terms in more detail as we progress through the following lessons, but first, let’s look at the measurement function in more detail and introduce the concept of a ‘plus zero’ term.