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A type A uncertainty estimate is derived from repeated measurements. For example, I may estimate the uncertainty on a measurement by repeating the measurement $N$ times and then calculating some measure of its dispersion, such as the standard deviation or, as below, the Allan deviation (suitable when $y$ might vary slowly over time):

$$u_y = \sqrt{\frac{1}{2(N-1)} \sum_{i=1}^{N-1} (y_{i+1} - y_i)^2}$$

If circumstances prevent me from taking any more measurements, how can I use the distribution of $y$ to estimate the meta-uncertainty $u_u$, (i.e. the uncertainty of the uncertainty)?

u-meta
Source: XKCD

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  • $\begingroup$ I saw now this old question of yours. If you're still interested in the answer, I can provide it, with some references, also for the Allan deviation. It might take some time, but let me know. $\endgroup$
    – Massimo Ortolano
    Jun 15, 2022 at 14:37
  • $\begingroup$ @MassimoOrtolano Thanks for the offer. In reality, I have no practical need for an answer any more, as I am meanwhile working on a different topic. $\endgroup$
    – gerrit
    Jun 15, 2022 at 15:55
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    $\begingroup$ Anyway I've bookmarked this and if I find the time I'll write an answer anyway since it's a very specialized topic, something I'm very fond of, and it might be useful to contribute to the spread of certain topics (I've already written a bit about the Allan variance in this answer). $\endgroup$
    – Massimo Ortolano
    Jun 15, 2022 at 16:52

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