# Confidence interval for Standardized Mortality Ratio

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.

• I believe I covered this at stats.stackexchange.com/a/493749/919.
– whuber
Dec 5 '20 at 18:37
• @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? Dec 7 '20 at 15:15
• 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.
– whuber
Dec 7 '20 at 16:19