How to represent measurement with uncertainty by a normal distribution.

Question

I want to represent a measurement with uncertainty, for example $$b=39\pm 3$$ by a normal distribution:$$f(x)=\dfrac {1}{\sqrt {2\pi \sigma ^{2}}}e^{-\dfrac {\left( x-\mu \right) ^{2}}{2\sigma ^{2}}}$$ I think the $\mu$ should be 39, which is obvious, but I am just confused about what should be $\sigma$? How can we generate the $\sigma$?

Answer 1

If you assume the normal distribution, then the so called 68–95–99.7 rule (see https://en.wikipedia.org/wiki/68–95–99.7_rule). That is, about $68\%$ of the data lie within $1 \sigma$ of the mean; about $95\%$ lie within $2\sigma$; and about $99.7\%$ lie within $3\sigma$.

Thus, pick $\sigma=3$ if $\pm 3$ implies that about $68\%$ of the data are within $\pm 3$; pick $\sigma = 1.5$ if $\pm 3$ implies that about $95\%$ of the data are within $\pm 3$; and so on.

You can tailor the choice of $\sigma$ to fit any percentage of data lying within $\pm 3$ of the mean.