Introduction to Uncertainty

Uncertainty is the inevitable error in our measurements. It is the difference between our measurement and the true value.
Uncertainty is inherent in physical measurements.

When calculating with measurements, we need to propagate the uncertainty. That propagated uncertainty tells us how confident to be in our final value.

Let's begin by walking through why uncertainty exists and how to propagate it through calculations. Then we'll follow up with understanding fractional uncertainty, and how to round when using uncertainty.

Measurements

Let's start off with why uncertainty exists in measurements with an example.

Imagine you are laying tiles for a kitchen floor. First, you buy some 10 cm long tiles. Second, you measure your room floor to be 800 cm long. Finally, you lay out 80 tiles. Strangely, it doesn't cover the length of the room. How come?

This is where the uncertainty comes in. Although the tiles were sold as 10 cm long, some are actually longer than 10 cm, and others are shorter. Similarly, the tape measure used to find the length of the room has uncertainty.

So given these uncertainties, how can we know how many tiles we actually need? Unfortunately, we cannot say for sure. We can, however, make an estimate within a degree of probability. For that, we need to propagate the uncertainty and think of our measurements as random variables.

Normal Random Variables

Suppose we measured all of our tiles. We might see some tiles at 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.

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,
TileN(μTile,σTile2)=N(Tilebest,δTile2)=N(10.0,0.22) 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 800 ± 1 cm,
RoomN(μRoom,σRoom2)=N(Roombest,δRoom2)=N(800,12) cm \begin{aligned} Room &\sim N(\mu_{Room}, \sigma_{Room}^2) \\ &= N(Room_{best}, \delta Room^2) \\ &= N(800, 1^2) \text{ cm} \end{aligned}
Note that the probability notation, μX,σX\mu_X, \sigma_X, and the uncertainty notation, Xbest,δXX_{best}, \delta X, are equivalent when modeling your measurements as random normal variables.

Law of Propagation of Uncertainty

Now returning to the question of how many tiles are needed to cross the room. We can calculate this by dividing our random variables for the room's length by the tile's length, Room/TileRoom / Tile. Recall from a statistics course that this will give us a new random variable! With a bit of math, we find that

NumberOfTiles=F(Room,Tile)=Room/TileN(μRoom/μTile,(1μTileσRoom)2+(μRoomμTile2σTile)2) \begin{aligned} NumberOfTiles &= F(Room, Tile) \\ &= Room / Tile \\ &\sim N(\mu_{Room} / \mu_{Tile}, (\frac{1}{\mu_{Tile}} \sigma_{Room})^2 + (\frac{- \mu_{Room}}{\mu_{Tile}^2} \sigma_{Tile})^2) \end{aligned}

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

μNumberOfTiles=Fbest(Room,Tile)=E[Room/Tile]=E[Room]/E[Tile]=μRoom/μTile \begin{aligned} \mu_{NumberOfTiles} &= F_{best}(Room, Tile) \\ &= E[Room/Tile] \\ &= E[Room] / E[Tile] \\ &= \mu_{Room} / \mu_{Tile} \end{aligned}

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

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

δF(X,Y,...)=(FbestXδX)2+(FbestYδY)2+... \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 FbestX\frac{\partial F_{best}}{\partial X} is the partial derivative of FbestF_{best} with respect to XX[1].

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

σNumberOfTiles=δF(Room,Tile)=(FbestRoomσRoom)2+(FbestTileσTile)2=(1μTileσRoom)2+(μRoomμTile2σTile)2 \begin{aligned} \sigma_{NumberOfTiles} &= \delta F(Room,Tile) \\ &= \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}

In case you are curious what 800±110.0±0.2\frac{800 \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,

δXXbest\frac{\delta X}{|X_{best}|}
For example, the fractional uncertainty of 2.00±0.052.00±0.05 is 0.05/2.000.05/2.00 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, δ(XY)(XbestYbest)δXXbest+δYYbest\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, δ(XY)(XbestYbest)=(δXXbest)2+(δYYbest)2\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 introduced earlier. 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±0.151.55±0.21.55 ± 0.\underline{15} \longrightarrow 1.55 ± 0.\underline{2}
because the uncertainty should have one significant figure. Then, we round the best estimate
1.55±0.21.6±0.21.\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.