So in a number of places, for example here, I've seen stated without explanation that the standard error of the SMR (at 95% confidence threshold) is simply

$$ SE_{SMR} = 1.96 * \frac{\sqrt O}{E} $$

Where $O$ is the observed count of fatalities and $E$ is the expected count (the origin of this expectation is irrelevant). I haven't been able to find a derivation of this anywhere nor can I see how, for example, it derives from any of the means for calculating a binomial proportion confidence interval. Can someone explain this?

Edit: This question was asked here a couple of years ago, but again states this formula without really explaining why it is correct. The relation to the Poisson distribution helps (h/t @whuber) but doesn't make it completely clear, for me anyway.

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    $\begingroup$ I believe I covered this at stats.stackexchange.com/a/493749/919. $\endgroup$
    – whuber
    Dec 5 '20 at 18:37
  • $\begingroup$ @whuber Thanks. That it comes from assuming a Poisson distribution definitely helps, but for standard error shouldn't we we in addition be dividing by $\sqrt{N}$? And how does the inv. pdf of the normal distribution enter in exactly? $\endgroup$ Dec 7 '20 at 15:15
  • $\begingroup$ This is all standard -- which means you can find many posts about these topics both here, on the Web, and in elementary textbooks. The Normal distribution is an approximation for large counts. $\endgroup$
    – whuber
    Dec 7 '20 at 16:19

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