For simple measuring tools, calculating the uncertainty of a measurement is easy. After finding the resolution of the measuring instrument, which is the smallest possible measurement that can be definitively measured, the zeroorder uncertainty (ZOU) of that tool, and therefore the uncertainty of any measurement it makes, is simply one half of the resolution.
1
𝑍𝑂𝑈 =
2 𝑟𝑒𝑠
EXAMPLE:
A metric ruler has normal measurement incrementation, and so it measures up to 30 cm total, and each centimeter marking has 10 mm of gradiation between it and the next full centimeter. Therefore, the smallest possible measurement that can be made with this ruler is 1 mm, because that is the smallest measurement that is explicitly defined. Therefore, the resolution of the ruler is 1 mm. (res = 1 mm). Using the above equation, the zero-order uncertainty of this ruler is 0.5 mm. Therefore, any measurement taken by this ruler must be amended by “± 0.5 mm”, because in truth any measurement taken by it could actually be within 0.5 mm of the measurement on either side, since the measurement is not explicitly defined in this range.
The zero-order uncertainty is very easy to calculate, but it is very limited, because it only defines primary measurements taken directly by a measuring instrument. Any calculations that involve that measurement are using uncertain values, so the calculated value is obviously uncertain as well. But, how uncertain is it? With the simple measurement, you can give the zero-order uncertainty as the range that the true value must lie between. With calculated values using measurements, though, it is not so simple. You can still define a range that the true value must exist between, but it is not as simple as just halving the resolution of an instrument.
Let us assume that there is some arbitrary measure y that is defined as a function of three other measurements:
x, t, and g. 𝑦 = 𝑓(𝑥, 𝑡, 𝑔)
This is analogous to a volume measurement of a rectangular prism, which is easily recognizable: V=lwh. The volume measurement (V) depends on three other measurements (l, w, and h). Returning to our arbitrary measurements, let us assume that y depends on x, t, and g according to the following function:

𝑥 𝑦 = √ 𝑔𝑡
2
3
+ (𝑥 + 𝑡)
3
2 − sin⁡(2𝑔)
The values of x, t, and g are measured directly with measuring instruments of known zero-order uncertainties. To make this easier, the zero-order uncertainties of each measurement are given below: 𝑥 = ±0.05⁡ 𝑡 = ±0.1⁡ 𝑔 = ±0.005
The units of each measurement will not be bothered with.
What can be done with all of this information? To calculate the uncertainty of a dependent value based on its independent contributors, the following formula is used:
𝑈 𝑦 𝑛
𝜕𝑦
= √∑(
𝜕𝑥 𝑖=1 𝑖
𝑈 𝑥 𝑖
) 2
In this formula, U y
is the uncertainty of the dependent variable, while 𝑈 𝑥 𝑖
is the uncertainty of the i th independent variable, and
𝜕𝑦
𝜕𝑥 𝑖
is the partial derivative of the dependent variable with respect to the i th independent variable. In words, the above formula is the square root of the sum of the squares of the partial derivative of the dependent variable with respect to each independent variable multiplied by the individual uncertainty of that independent variable.
Returning to our arbitrary y from above, we can apply the formula we have just introduced. It will be easier to first find the partial derivatives of the dependent variable y with respect to each independent variable:
𝜕𝑦
𝜕𝑥
=
1
2
( 𝑥 𝑔𝑡
2
3
)
−
1
2
(
2𝑥 𝑔𝑡 3
) +
3
2
(𝑥 + 𝑡)
1
2

𝜕𝑦
=
1 𝑥
2
( 𝑔𝑡 3
)
−
1
2
(
−3𝑥 𝑔𝑡 4
2
) +
3
(𝑥 + 𝑡)
1
2
𝜕𝑡 2 2
𝜕𝑦
𝜕𝑔
=
1
2
( 𝑥
2 𝑔𝑡 3
)
−
1
2
( 𝑡
−𝑥
3 𝑔
2
2
) + 2cos⁡(2𝑔)
Plugging these into our formula, we can find the uncertainty of y for any values of x, t, and g:
𝑈 𝑦
= √((
1
2 𝑥
( 𝑔𝑡
2
3
)
−
1
2 2𝑥
( 𝑔𝑡 3
) +
3
2
(𝑥 + 𝑡)
1
2 ) (0.05))
2
+ ((
1
2 𝑥
( 𝑔𝑡
2
3
)
−
1
2
(
−3𝑥 𝑔𝑡 4
2
) +
3
2
(𝑥 + 𝑡)
1
2 ) (0.1))
2
+ ((
1
2 𝑥
( 𝑔𝑡
2
3
)
−
1
2 −𝑥
( 𝑡 3 𝑔
2
2
) + 2 cos(2𝑔)) (0.005))
2
Although very arduous looking, this is what is necessary to be able to calculate the uncertainty of y when it is defined by the function that was written earlier. A simpler example will use the volume formula for a right cylinder:
𝑉 = 𝜋𝑟
2 ℎ where r is the radius of the cylinder, and h is the height of the cylinder. The approach is the same as it was for finding the uncertainty of the arbitrary y above. First, let’s define some zero-order uncertainties for the tools used to measure r and h. Both measurements were taken with a digital caliper of resolution 0.001 mm, so the zero-order uncertainties for these two measurements are both ±0.0005 mm. Now let us find the partial derivatives necessary to use the uncertainty formula:
𝜕𝑉
𝜕𝑟
= 2𝜋ℎ𝑟
𝜕𝑉
= 𝜋𝑟 2
𝜕ℎ
Using these partial derivatives with the zero-order uncertainties already defined, the uncertainty of any calculated volume for this cylinder can be found using the following function of r and h:

𝑈
𝑉
= √((2𝜋ℎ𝑟)(0.0005)) 2 + ((𝜋𝑟 2 )(0.0005)) 2
So, we can now know how uncertain any volume measurement we give of this cylinder is when it is calculated from measured values of its radius and height. For a full example, let’s say some sample measurements are taken of the cylinder’s radius and height:
Radius (mm) Height (mm)
2.004
2.002
1.998
1.998
5.003
5.010
5.005
5.007
Taking the averages of these measurements, the average radius is 2.001 mm ± 0.0005 mm and the average height is
5.006 mm ± 0.0005 mm. We will use the average values as representations of the whole cylinder. Using the volume formula, the volume is calculated to be:
𝑉 = 𝜋(2.001⁡𝑚𝑚) 2 (5.006⁡𝑚𝑚) = 62.97⁡𝑚𝑚 3
The uncertainty of this volume measurement can also be calculated from the formula we just developed:
𝑈 𝑣
= √((2𝜋(5.006⁡𝑚𝑚)(2.001⁡𝑚𝑚)(0.0005⁡𝑚𝑚)) 2 + ((𝜋(2.001⁡𝑚𝑚) 2
So, the final calculated volume of the cylinder is written most correctly as:
)(0.0005⁡𝑚𝑚)) 2 = 0.0321⁡𝑚𝑚
3
62.97⁡𝑚𝑚 3 ± 0.0321𝑚𝑚 3
You cannot precisely say that the volume of the cylinder is exactly 62.97 cubic millimeters, but you can definitively and accurately say that the volume of the cylinder is between 62.94 and 63.00 cubic millimeters, which are the two extremes of the uncertainty range. Try experimenting with using more precise measuring instruments. That is, try making the zero-order uncertainties of the radius and height measurements smaller. How does this affect the final uncertainty of the volume? How does having larger zero-order uncertainties affect the final volume uncertainty? What does this say about the benefits or detriments of using more precise measurement tools?