Propagation of Uncertainties

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…from CheLabWiki, an online resource for chemical-engineering laboratories located at www.chelabwiki.org; Site Revision #366; 6 January 2009.

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In analyzing experimental data we usually combine measured values to compute values for other quantities. For example, from values measured for the temperature $T$ , pressure $P$ , volume $V$ , and number of moles $n$ of a gas sample we might compute a value for the compressibility factor $Z$ ,

(1)

$Z \ = \ \frac{PV}{nRT}$

The measured quantities each have some uncertainty, and now the question is this: How do those uncertainties combine to yield an uncertainty in the value computed for $Z$ ?

General Expression

Consider a quantity $y$ whose value is to be obtained from the values of measured quantities $x 1$ , $x 2$ , … .

(2)

$y \ = \ f(x_1, x_2, ...)$

This equation is commonly called a measurement equation; it is usually derived from a model of the experimental situation. Each measured $x i$ has some uncertainty $u x i$ > 0. We expand the (unknown) exact value of $y$ in a Taylor series about the computed value, keeping only the first-order terms,^[1]

(3)

$y \ = \ y_{cal} \ + \ \sum_i \left ( \frac{\partial f}{\partial x_i} \right ) u_{xi}$

The partial derivatives measure the sensitivity of $y$ to changes in the $x$ s. We now estimate the uncertainty in $y$ as (Recall that uncertainties are always positive.)

(4)

$u_y \ = \ |y - y_{cal}| \ \approx \ \sum_i \left |\left ( \frac{\partial f}{\partial x_i} \right ) \right | u_{xi}$

Special Cases

When $y$ is obtained by simple sums or differences of the measured $x$ s, then (4) gives the uncertainty in $y$ as the algebraic sum of the individual uncertainties. For example, say we have measured the length $l$ and width $w$ of a rectangle; the measured values have uncertainties $u l$ and $u w$ . Then the uncertainty in the computed perimeter $p$ , from (4), is given by

(5)

$u_p \ = \ 2u_l + 2u_w$

But when $y$ is obtained by a product or quotient of the measured $x$ s, then (4) gives the fractional uncertainty in $y$ as the algebraic sum of the fractional uncertainties in the $x$ s. For example, the uncertainty in the rectangular area $A$ computed from our measured length and width would be

(6)

$\frac{u_A}{A} \ = \ \frac{u_l}{l} + \frac{u_w}{w}$

Likewise, for our example of the compressibility factor (1), the uncertainty in $Z$ would be obtained by

(7)

$\frac{u_Z}{Z} \ = \ \frac{u_T}{T} + \frac{u_P}{P} + \frac{u_V}{V} + \frac{u_n}{n}$

When $y$ involves other functions of the $x$ s (such as powers, exponentials, logs, etc.), then merely apply (4) to obtain the required expression for the uncertainty.^[1] For example, the volume of a right cylinder is

(8)