Measurement Uncertainty

Troy Williams

2021-02-15

Measurement Uncertainty

Typically when you measure things, there is a certain amount of error. In everyday life, this is ignored for the most part. I was thinking about uncertainty in measuring areas. Like everyone else, I learned about uncertainty in measurement in high school during science class. It was further reinforced in university in every lab I took. The problem was, it was addressed as a set of rules to memorize and apply that covered the use of significant figures. Most people do not fully explore what this means exactly and consequently have trouble with the concept outside the typical canned responses. I was having trouble understanding why calculating an area with uncertainty was expressed the way it was.

This page provides many examples and canned formulae for calculating various types of quantities accounting for the uncertainty in measurement. It is a good resource with some good examples. I will focus on the pool volume calculation as it is closest to the area calculation that I want to understand and is typical of the other references I have read.

To solve the volume/area problem, you need to determine the fractional uncertainty of the measurement. To find the area, add the fractional uncertainties together. Multiply this with the nominal value to determine the potential error in the volume.

Here is an example using area. if we have:

\[ l = 4.2 m \pm 0.25 m \rightarrow \Delta l\]

\[ w = 3.45 m \pm 0.33 m \rightarrow \Delta w\]

\[ p_1 = \frac{\Delta l}{l} \approx 0.05952 \]

\[ p_2 = \frac{\Delta w}{w} \approx 0.09565 \]

The nominal area, \(A_n\), is:

\[ A_n = l \cdot w = 4.2 \cdot 3.45 = 14.49 m^2 \]

The error, \(\epsilon\), is:

\[ \epsilon = 14.49 \cdot (0.05952 + 0.09565) = 14.49 m^2 \cdot 0.15517 = 2.2484 m^2 \]

The full uncertainty is:

\[ A = A_n \pm \epsilon \approx 14.49 m^2 \pm 2.25 m^2 \]

\[ A = A_n \pm \epsilon \approx 14.49 m^2 \pm 15.5 \% \]

As you can see, it is a relatively straight forward process. It just takes a few steps. I wanted to understand where this process came from.

Derivation

Why can we sum the uncertainty if we are adding linear measurements? Why can we not merely add/or multiple the uncertainties for an area calculation? We take a look at the question in more detail in this section. We’ll derive the equations from first principles so we can understand why the methods are presented the way they are as described in the first section and the References. Take a look at fig. 1.

Figure 1: The red rectangle represents the nominal area, that is x \cdot y. The blue rectangle represents the maximum area and the red rectangle represents the minimum area.

From fig. 1, we have 3 rectangles defined:

\[ A_n = x \cdot y\](1)

\[ A_{\text{min}} = \left (x - \Delta x \right) \left ( y - \Delta y \right)\](2)

\[ A_{\text{max}} = \left (x + \Delta x \right) \left ( y + \Delta y \right) \](3)

From the diagram and equations above, it can be observed that:

\[ A = A_n \pm \left( \frac{A_{\text{max}} - A_{\text{min}}}{2} \right)\](4)

How does eq. 4 above relate to the process described in the previous section? Let’s expand the equation algebraically and see what happens.

\[ A_{\text{max}} = y x + y \Delta x + x \Delta y + \Delta y \Delta x \](5)

\[ A_{\text{min}} = y x - y \Delta x - x \Delta y + \Delta y \Delta x \](6)

Subtract eq. 6 from eq. 5:

\[A_{\text{max}} - A_{\text{min}} = \left ( y x + y \Delta x + x \Delta y + \Delta y \Delta x \right ) - \left ( y x - y \Delta x - x \Delta y + \Delta y \Delta x \right ) \]

\[ A_{\text{max}} - A_{\text{min}} = y x + y \Delta x + x \Delta y + \Delta y \Delta x - y x + y \Delta x + x \Delta y - \Delta y \Delta x \]

Collect like terms and simplify:

\[ A_{\text{max}} - A_{\text{min}} = 2 y \Delta x + 2 x \Delta y \](7)

The area is:

\[ A = A_n \pm \epsilon = A_n \pm \left ( \frac{2 y \Delta x + 2 x \Delta y}{2} \right ) \]

\[ A = A_n \pm \epsilon = A_n \pm \left ( y \Delta x + x \Delta y \right )\](8)

Fractional Uncertainty

To see if the way fractional uncertainty is given matches what we derived in eq. 8, let’s combine the steps of fractional uncertainty into one equation:

\[ f_x = \frac{\Delta x}{x} \]

\[ f_y = \frac{\Delta y}{y} \]

\[ A = A_n \pm A_n \cdot \left ( f_x + f_y \right ) \]

\[ A = A_n \pm A_n \cdot \left ( \frac{\Delta x}{x} + \frac{\Delta y}{y} \right ) \]

Fractional uncertainty expressed as a single equation:

\[ A = x y \pm x y \cdot \left ( \frac{\Delta x}{x} + \frac{\Delta y}{y} \right ) \](9)

Simplifying eq. 9:

\[ A = x y \pm \left ( y \Delta x + x \Delta y \right ) \](10)

We can see that eq. 8 and eq. 10 are equal.

Summary

The derivation shows that the methods presented in the References are correct but simplified. They force one to assume that the presentation is correct and that they shouldn’t question it. I try to question everything. I try to understand things from a basic level. Primarily to practice my skills but to gain a deeper understanding of basic assumptions.

We could perform the same analysis for ratios of uncertain measurements. I don’t need to understand them at the moment. In the future, when I do, I’ll post another article.

References

Here is a list of documents I used to understand better what is happening and to determine how to derive the equations: