# Lab Manual

## Propagation of Uncertainties
***

### The Sophisticated Approach
We can use calculus to tell us how much a function varies if we change one of its parameters. For the following, we’ll assume that we are trying to compute some value, Y , using the inputs A, B, and C. This could be the perimeter of a rectangle, in which case the perimeter, Y , is the sum of four variables:

$$Y(A, B, C, D) = A + B + C + D$$  

In the case of a volume measurement, the volume, Y , would be the product of length, width, and height:

$$Y(A, B, C) = ABC$$

The uncertainties in our measured values will be given by δA, δB, and δC. The uncertainty in the
computed value is, then, δY .

If we have a function which is the sum or difference of a number of variables,
$$Y = A \pm B \pm C$$

Then we can compute the uncertainty in Y with the following formula:  

\begin{equation}
(\delta Y)^2 = (\delta A)^2 + (\delta B)^2 + (\delta C)^2
\tag{R1}
\end{equation}

If the function is some kind of product of our three variables,  
$$ Y = ABC \quad \textrm{or} \quad Y = \frac{AB}{C} \quad \textrm{or} \quad Y = \frac{A}{BC} \quad \textrm{or} \quad Y = \frac{1}{ABC}$$  

Then we can compute the uncertainty in Y via:

\begin{equation}
(\frac{\delta Y}{Y})^2 = (\frac{\delta A}{A})^2 + (\frac{\delta B}{B})^2 + (\frac{\delta C}{C})^2
\tag{R2}
\end{equation}

Finally, if our function is a power of one of our variables (for example, the area of a circle),

$$Y = A^\alpha$$  

Then we can compute the uncertainty by  
\begin{equation}
\frac{\delta Y}{Y} = |\alpha| \frac{\delta A}{A}
\tag{R3}
\end{equation}

Let’s try our volume example. Because V = xyz, we use ([R2](#mjx-eqn)),  

\begin{align*}
\frac{\delta V}{V} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left(\frac{\delta y}{y}\right)^2 + \left(\frac{\delta z}{z}\right)^2} \\
= \sqrt{\left(\frac{0.5}{100}\right)^2 + \left(\frac{0.5}{50}\right)^2 + \left(\frac{0.5}{20}\right)^2} \\
\frac{\delta V}{1 * 10^5} = 0.027386 \\
\delta V = 2738.6
\nonumber
\end{align*}

Therefore we may write  
$$V = (1.000 \pm 0.027) * 10^5 cm^3 \quad \textrm{or} \quad V = (0.1000 \pm 0.0027) m^3$$

Or, in more modern notation,  
$$V = 1.000(27) * 10^5 cm^3 \quad \textrm{or} \quad V = 0.1000(27) m^3$$

Although this method seems complicated, it gives the best estimate of our uncertainties. After you’ve used it a few times, it will become second nature.

### Combining Formulae

What happens if we have something like the volume of a cylinder:  
$$ V = \pi r^2 h$$  

Here we must work with both ([R3](#mjx-eqn)) and ([R2](#mjx-eqn)). The key is to look at the formula as the product of $r^2$ and *h* ($\pi$ is exact and has no uncertainty). This requires us to use ([R2](#mjx-eqn)). But to get $\delta r^2$, we’ll need ([R3](#mjx-eqn)).

Let’s start with ([R2](#mjx-eqn)):  

\begin{equation}
(\frac{\delta V}{V})^2 = (\frac{\delta r^2}{r^2})^2 + (\frac{\delta h}{h})^2
\nonumber
\end{equation}  

Now, ([R3](#mjx-eqn)) tells us that  

\begin{equation}
\frac{\delta \left(r^2 \right)}{r^2} = 2 \frac{\delta r}{r}
\nonumber
\end{equation}

So we can make the substitution:

\begin{equation}
\left(\frac{\delta V}{V}\right)^2 = \left(2 \frac{\delta r}{r} \right)^2 + \left(\frac{\delta h}{h}\right)^2
\nonumber
\end{equation}

Although it is tempting to multiply through by $V^2 = (\pi r^2 h)^2$ on both sides before we compute, it’s usually easier if we don’t. The fractional errors ($ \delta x/x$) are used in so many places that it makes sense to use those more often than not.

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