### Introduction to Uncertainty

Uncertainty is a critical component of any physical measurement. Let's walk through what uncertainty is and why it exists. Then, we can learn how to propagate uncertainty, use fractional uncertainty, and round uncertainty.

### Understanding Uncertainty

Let's start off with what uncertainty is. Uncertainty is the inevitable error between our measurement and the true value.

What does that actually mean? Consider this example:

Imagine you are laying tiles for a ballroom. You measure the floor's length to be 2000 cm and your tiles to be 10 cm. You lay 200 tiles, but it doesn't cover the length. How come?

The likely reason is because your tape measure for the room is off by 1 or 2 cm. On top of that, your tiles are not precisely 10 cm long, some are shorter, some longer. This is what we mean by uncertainty - the measured value is different from the true value.

Uncertainty is inherent in physical measurements.

At first, it seems that we should simply use a better tape measure and ensure our tiles are the same length. In reality, this can can only reduce the uncertainty. It is not possible to eliminate uncertainty.

Given these uncertainties, how can we know how many tiles we actually need? Unfortunately, we cannot - we can only estimate. We estimate by thinking of our measurements as random variables.

### Normal Random Variables

What are normal random variables? Suppose we measured all of our tiles. We might see lengths of 10, 10.05, 9.8, or even 10.27 cm. After aggregating all this data, we might find that ~68% of the tiles fall within 0.2 cm of 10 cm. If we were to graph this data, it'd look like a normal distribution.

In this case, we say that the mean of the tile's length is 10.0 cm, and the standard deviation of the tile's length is 0.2 cm. We can write our tile's length as 10.0 ± 0.2 cm,

\begin{aligned} Tile &\sim N(\mu_{Tile}, \sigma_{Tile}^2) \\ &= N(Tile_{best}, \delta Tile^2) \\ &= N(10.0, 0.2^2) \text{ cm} \end{aligned}
Similarly, we can write our room's length as 2000 ± 1 cm,
\begin{aligned} Room &\sim N(\mu_{Room}, \sigma_{Room}^2) \\ &= N(Room_{best}, \delta Room^2) \\ &= N(2000, 1^2) \text{ cm} \end{aligned}
Note that the probability notation, $\mu_X, \sigma_X$, is equivalent to the uncertainty notation, $X_{best}, \delta X$.

### Law of Propagation of Uncertainty

Let's return to the question of how many tiles are needed to cross the room. We can calculate this by dividing our room's length, $Room$, by the tile's length, $Tile$. This will give us a new random variable:

\begin{aligned} NumberOfTiles &= Room / Tile \\ &= N(\mu_{NumberOfTiles}, \sigma_{NumberOfTiles}) \end{aligned}

The new mean, $\mu_{NumberOfTiles}$, is derived as

\begin{aligned} \mu_{NumberOfTiles} &= E[Room/Tile] \\ &= E[Room] / E[Tile] \\ &= \mu_{Room} / \mu_{Tile} \end{aligned}

The new standard deviation, $\sigma_{NumberOfTiles}$, is derived by applying the Law of Propagation of Uncertainty, also known as the generalized form of Adding in Quadrature:

Law of Propagation of Uncertainty:

For any differentiable function $F(X,Y,...)$ where $X,Y,...$ are independent and random, the uncertainty is

$\delta F(X,Y,...)=\sqrt{\left(\frac{\partial F_{best}}{\partial X} \delta X \right)^2 + \left(\frac{\partial F_{best}}{\partial Y} \delta Y \right)^2 + ...}$
where $\frac{\partial F_{best}}{\partial X}$ is the partial derivative of $F_{best}$ with respect to $X$[1].

In our case, $F_{best}(X,Y,...)=F_{best}(Room,Tile)=\mu_{Room} / \mu_{Tile}$. The derivation is as follows:

\begin{aligned} \sigma_{NumberOfTiles} &= \sqrt{\left(\frac{\partial F_{best}}{\partial Room} \sigma_{Room}\right)^2 + \left(\frac{\partial F_{best}}{\partial Tile} \sigma_{Tile}\right)^2} \\ &= \sqrt{\left(\frac{1}{\mu_{Tile}} \sigma_{Room}\right)^2 + \left(\frac{-\mu_{Room}}{\mu_{Tile}^2} \sigma_{Tile}\right)^2} \end{aligned}

Thus, we find

\begin{aligned} NumberOfTiles &\sim N(\mu_{Room} / \mu_{Tile}, (\frac{1}{\mu_{Tile}} \sigma_{Room})^2 + (\frac{- \mu_{Room}}{\mu_{Tile}^2} \sigma_{Tile})^2) \end{aligned}

In case you are curious what $\frac{2000 \pm 1}{10.0 \pm 0.2}$ is, try plugging it into the Uncertainty Calculator :) Want to derive the uncertainty equation for other operators and functions? Check out Proofs.

### Fractional Uncertainty

We use fractional uncertainties because they represent the percent of uncertainty in our estimate. Fractional uncertainty is defined as the uncertainty of an estimate divided by the absolute value of the best estimate,

$\frac{\delta X}{|X_{best}|}$
For example, the fractional uncertainty of $2.00±0.05$ is $0.05/2.00$ and thus the percentage uncertainty is 2.5%.

When first learning to propagate uncertainty for multiplication, many students are taught to simply add fractional uncertainties, $\frac{\delta (X*Y)}{(|X_{best}*Y_{best}|)} \approx \frac{\delta X}{|X_{best}|} + \frac{\delta Y}{|Y_{best}|}$. This is an approximation that is quick to compute, but is actually an upper bound and is less mathematically rigorous than the true formula, $\frac{\delta (X*Y)}{(|X_{best}*Y_{best}|)}=\sqrt{\left(\frac{\delta X}{|X_{best}|}\right)^2 + \left(\frac{\delta Y}{|Y_{best}|}\right)^2}$[1].

The true formula can be derived from the Law of Propagation of Uncertainty. Check out the multiplication section of Proofs to see that.

### Rounding

Now we can propagate uncertainty through any differentiable function, and we understand fractional uncertainty. One last question remains: how do we round our measurements?

The convention is to round the uncertainty to one significant figure. Then, we round the estimated value to the same place value as the uncertainty[2].

For example, we round the uncertainty

$1.55 ± 0.\underline{15} \longrightarrow 1.55 ± 0.\underline{2}$
because the uncertainty should have one significant figure. Then, we round the best estimate
$1.\underline{55} ± 0.2 \longrightarrow 1.\underline{6} ± 0.2$
because the estimate should be written to the same place value as the uncertainty.

### References

[1] Taylor, John. An Introduction to Error Analysis - The Study of Uncertainties in Physical Measurements, Second Edition. University Science Books.

[2] Evaluation of Measurement Data — Guide to the Expression of Uncertainty In Measurement (GUM). September, 2008.