# When linearizing a data point of an inverse relationship, how does its uncertainty change?

I have a question regarding data analysis and uncertainties regarding my physics experiment. Essentially, I conducted an experiment to determine the relationship between salt concentration and specific heat capacity.

I have collected my values and have an inverse relationship. I want to linearise my data points for a better estimate of random and systematic errors since working with a straight line graph is much easier.

I have a specific heat capacity c of 5340 J kg⁻¹ K⁻¹ with uncertainty Δc 320 J kg⁻¹ K⁻¹ so it can be written like this:

5340 ± 320 J kg⁻¹ K⁻¹.

c ± Δc

Since my data seems to have an inverse relationship, in order to linearise, I just have to find the reciprocal of my specific heat capacity which will give me 1/c:

1/(5340) = 1.87 x 10^-4

1/c = 1.87 x 10^-4

How will my uncertainty change? (In other words, what will 1/(Δc) be?

1.87 x 10^-4 ± 1/(Δc)

• Please use the general formula for propagating uncertainties. Jan 8 '21 at 13:58

The general error propagation for a function $$f(x_1, x_2,...,x_n)$$ is $$\Delta f = \sqrt{\sum_{i=1}^n \left(\frac{\partial f}{\partial x_i} \Delta x_i \right)^2}.$$
If $$f(x) = 1/x$$, we can say that $$\frac{\partial f}{\partial x} = -\frac{1}{x^2}$$ and so $$\Delta f = \frac{\Delta x}{x^2}.$$