squiggle/packages/website/docs/Internal/Processing-Confidence-Intervals.md

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---
title: Processing Confidence Intervals
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author: Nuño Sempere
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---
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This page explains what we are doing when we take a 90% confidence interval, and we get a mean and a standard deviation from it.
## For normals
```js
module Normal = {
//...
let from90PercentCI = (low, high) => {
let mean = E.A.Floats.mean([low, high])
let stdev = (high -. low) /. (2. *. 1.6448536269514722)
#Normal({mean: mean, stdev: stdev})
}
//...
}
```
We know that for a normal with mean $\mu$ and standard deviation $\sigma$,
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$$
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a \cdot Normal(\mu, \sigma) = Normal(a \cdot \mu, |a| \cdot \sigma)
$$
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We can now look at the quantile of a $Normal(0,1)$. We find that the 95% point is reached at $1.6448536269514722$. ([source](https://stackoverflow.com/questions/20626994/how-to-calculate-the-inverse-of-the-normal-cumulative-distribution-function-in-p)) This means that the 90% confidence interval is $[-1.6448536269514722, 1.6448536269514722]$, which has a width of $2 \cdot 1.6448536269514722$.
So then, if we take a $Normal(0,1)$ and we multiply it by $\frac{(high -. low)}{(2. *. 1.6448536269514722)}$, it's 90% confidence interval will be multiplied by the same amount. Then we just have to shift it by the mean to get our target normal.
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## For lognormals
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TODO