I have three probabilities of a value falling within the following ranges +/- 5%, +/- 10% and +/- 20% of the mean value

Question

I have the mean value and my question is: is it possible to calculate the standard deviation assuming a normal distribution? Looking at previous questions which have been asked, this question is different from other the variants about how to calculate a standard deviation. Thanks.

Answer 1

My understanding of the question is that you have the following pieces of information. Notation $\mu, w_1, w_2, w_3$ give the mean and some distance from the mean, respectively. You also have $$\mathbb{P}(x\in [\mu\pm w_1]) = p_1$$ $$\mathbb{P}(x\in [\mu\pm w_2]) = p_3$$ $$\mathbb{P}(x\in [\mu\pm w_3]) = p_3$$

Any one of those equations is sufficient: we're interested in finding the zero of $g(\sigma)=\mathbb{P}(x\in [\mu\pm w_1]) - p_1$ for the standard deviation $\sigma$.

If we assume the normal distribution, as you do in your question, then we just need to know the probability $\mathbb{P}(x\in [\mu\pm w_1])$ which is obviously given by the CDF $F$:

$$g(\sigma) = F(\sigma;\mu+w_1)-F(\sigma;\mu-w_1) - p_1 = 0 $$

where the parameter under inference is $\sigma$. Root finding might work. It might be easier to work on the log scale to avoid $\sigma \le 0$.

The only real challenge with this approach is what to do when each of the three equations gives you a different estimate of $\sigma$, or whether the normal model is reasonable at all.

set.seed(13)
x <- rnorm(10000, 5, 2)
x_mean <- mean(x)

p_est <- length(x[x > -1 & x < 1])/length(x)

G <- function(sigma, mu=x_mean, p=p_est){
    pnorm(1, mean=mu, sd=sigma)-pnorm(-1, mean=mu, sd=sigma)-p_est
}


sigma <- seq(0.5, 3.5, 0.1)

plot(sigma, G(sigma), ylab=expression(G(sigma)), xlab=expression(sigma), type="l")
abline(h=0, col="grey")

uniroot(G, interval = c(0.5, 3.5))
$root
[1] 1.988266

$f.root
[1] -1.201178e-07

$iter
[1] 7

$init.it
[1] NA

$estim.prec
[1] 6.103516e-05

This plot gives a notional idea of what's going on: uniroot is searching for where the function crosses $g(\sigma)=0$. Clearly, the function crosses very close to where the true value of $\sigma$ is known to be. More information can be found in its documentation and any numerical analysis textbook.