Measurement uncertainty basics

Question

$$x = (\overline{x}-K)\pm \Delta x\tag{1}$$ $$\Delta x = s_{\overline{x}} = \sqrt{\frac t {\sqrt n} s_x} \tag{2}$$ $$s_x = \sqrt{\frac 1 {n-1} \sum^{n}_{i=1}(x_i-\overline{x})} \tag{3}$$ $$\frac {\Delta x}{\overline{x}}\times 100\% \tag{4}$$ $$\text{Ablolute: }U = 8.11V\pm0.02 V\tag{5}$$ $$\text{Relative: }U = 8.11(1\pm0.0025) V\tag{6}$$

1. How does omitting the $t$ effect the $\Delta x$ (especially for small $n$)

In the *physical formulae* for my Physics lecture it omits the $t$ in the equation. Since in one of my problems the professor didn't use it either, I assume this is okay at least for this basic level course. I somewhat understand it's meaning.

2. If the relative measurement uncertainty is defined as in $(4)$, how do you derive at $(6)$ from $(5)$?

These values where given in the solution to a problem without further explanation.

(LaTex question: Is there a way to make the overline for the x appear closer to the x?)

Answer 1

Your physics lecturer is either being a bit slack or, more likely, is simply seeking to get an idea of the order of magnitude for the uncertainty. In statistical inference, many tests look at how far a result is from its mean and the meaningful way to look at this difference is almost always as a certain number of sigmas (population standard deviations, if they are known) or data standard deviations (an estimate of sigma derived from the actual measured data if sigma is not known). The notation $t$ suggests the Gosset (Student) $t$-test. The reason is that the statistic expressed in this way can be related to the probability of observing your dataset this far from the mean given the null hypothesis of a certain mean. That is, if you get a low probability from this assumption, this casts doubt on the soundness of the assumption. Deviations expressed as the appropriate number of standard deviations are thus a highly fundamental idea in the doctrine of falsifiability (see my answer here).

So there is a great deal of theory in deciding the appropriate scale factor (in this case a critical $t$ for the $t$-test). But, having said that, these scale factors don't vary too much and you can use the $\Delta x$ to get a feel for whether a deviation is physcially meaningful: to find something one standard deviation away from the mean is not very uncommon, and wouldn't raise eyebrows, two SDs is a bit more uncommon, three SDs mean you really need to look at your null hypothesis carefully whilst when we get to somewhere between 4.5 and 6 sigma, as in the data suggesting the Higgs Boson you're on to something enough to award a Nobel Prize. In this last case, the null hypothesis is that the background experimental variability in the collider experiment could account for the little "bump" that bespeaks the Higgs Boson. It's possible, and it still IS possible. It's just that after so many reproduction of the result it's highly unlikely.

Lastly (6) is gotten from (5) by dividing the "absolute" uncertainty by the nominal mean ($0.0025 \approx 0.02 / 8.11$). "Proportional" or "Fractional" uncertainty would be a better word, but unfortunately "Relative" is also current usage: you can express it either as a percentage, or as an unadorned ratio, or as a "parts per million" or "billion".