# How do I find the individual relative uncertainties of cubed terms in an equation and use this to find the absolute uncertainty of a value?

I'm currently conducting a physics project for school and I am really stuck on uncertainties. I am trying to calculate the overall uncertainty in Young's Modulus, with the Bending Beam experiment using the equation $$E=(mgl^3)/(4bd^3s).$$ In this equation, $$E$$ represents Young’s Modulus, $$m$$ represents the mass of the load in kg, $$g$$ represents the gravitational constant (which is taken as 9.8ms-2 in all experiments in this project1), $$L$$ represents the length of the beam between supports in m, $$b$$ represents the breadth of the beam in m, $$d$$ represents the depth of the beam in m and $$s$$ represents the distance that the centre of the beam sags by in m. Right now, I have uncertainties in $$b$$, $$d$$ and the gradient of my graph: $$m/s$$. However, I don't understand how to get the uncertainty in $$d^3$$. My teacher told me that I should cube the combined uncertainty in $$d$$ and divide it by my value of $$d^3$$, but this gives me an extremely large value. I have attached pictures of my Excel spreadsheet with my results. I realise this might be a really stupid question but I would appreciate any help.

Here, $$D^3$$ is the mean value of depth cubed.

Here, the combined uncertainty is the square root of $$(scale^2 + calibration^2 + random^2)$$. The uncertainty in $$D^3$$ is calculated the way my teacher told me, by cubing the combined uncertainty. And the relative uncertainty is the uncertainty in $$D^3$$ divided by $$D^3$$. If this is the correct uncertainty can someone please explain why it is so significant and how I should evaluate this.

psychgiraffe's answer above is completely correct, but I get the sense that you're in high school and might not be up on the calculus required to apply it. (If I've misjudged your level, feel free to ignore this answer!)

If $$f$$ is a monomial depending on several variables raised to power (i.e., $$f = x^a y^b z^c \cdots$$) then the general equation given in the other answer is equivalent to$$\left(\frac{\Delta f}{f} \right)^2 = \left( a \frac{\Delta x}{x} \right)^2 + \left( b \frac{\Delta y}{y} \right)^2 + \left( c \frac{\Delta z}{z} \right)^2 + \cdots.$$ Effectively, when you multiply or divide measured quantities, you add together the squares of the relative uncertainties to find the overall relative uncertainty. By comparison, when you add or subtract measured quantities, you add the squares of the absolute uncertainties to find the overall absolute uncertainty: $$f = ax + by + cz + \cdots \quad \Rightarrow \quad (\Delta f)^2 = (a \Delta x)^2 + (b \Delta y)^2 + (c \Delta z)^2 + \cdots$$

See this page for more explanation of how these uncertainty propagation rules shake out in practice.

• this makes so much sense. thank you! yes i am still in high school, i was just struggling to find an answer on the internet that wasn’t over complicated, this is great!
– Rose
Commented Mar 21 at 21:35

Here is how I would do it. There are lots of different rules for calculating uncertainties of powers, multiplications etc but you only need to remember this general formula:

$$\left(\Delta f\right)^2 \approx \left(\frac{\partial f}{\partial x}\right)^2 \left(\Delta x\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2 \left(\Delta y\right)^2 + \left(\frac{\partial f}{\partial z}\right)^2 \left(\Delta z\right)^2 + \ ...$$

where $$f = f(x,y,z,...)$$ is the quantity you are trying to determine the uncertainty of, and $$\Delta x$$ is the absolute uncertainty of quantity $$x$$.

If $$f = f(D) = D^3$$ then you get $$\frac{\Delta (D^3)}{D^3} = 3\frac{\Delta D}{D}$$. So the absolute uncertainty in $$D^3$$ is:

$$\Delta (D^3) = 3D^2 \Delta D$$

As a side note, the best option is to plot a graph whose gradient is E and then carry out some kind of python plotting that gives you the uncertainty in the gradient. This in general leads to a more reliable error for your quantity (and it also avoids you having to do a horrible error propagation yourself). I believe scipy modules exist for this (e.g. see here). If you have some independent variable which you are changing (e.g. the length of the beam), this method is probably best.

In College Physics I experiments I was taught that when quantities are multiplied percentage errors are simply added. If the quantity is squared, the error is doubled. If the quantity is cubed, the error is tripled, if the quantity is square-rooted the error is halved. Error measurements were only taken to 2 significant figures.

Here’s an example, say you are measuring a 1cm cube, where each measurement has a +/-10% error. The largest cube would be $$1.1^3$$ = 1.331 $$cm^3$$, the smallest would be $$0.9^3$$ = .729 $$cm^3$$. Then total error is (.331+1-.729)/2= +/- 0.301 $$cm^3$$ which is three times the error in one measurement (your error measurements are not accurate enough to consider three decimal places.) That is +/- 30% error.
If you take the cube of the error $$0.1^3$$ = $$.001$$ that is obviously false. You can't possibly get a more accurate result when you are compounding errors.
Perhaps your teacher meant take the cube of the (measurement plus the error).