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

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.

• It is not clear what do you mean... Do you have only the mean? If yes, then the answer is: no.
– Tim
Feb 24, 2017 at 14:52
• As indicated, there is a mean plus three probabilities within the specified ranges from the mean. Feb 24, 2017 at 16:14
• It might be better to link to some of the previous questions to which you refer and say why you think they do not apply to your scientific problem. Feb 24, 2017 at 16:46

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$is known to be. More information can be found in its documentation and any numerical analysis textbook. • Thanks Sycorax. I'm not familiar with root finding. Also, just taking one of the equations, say the first one, would the σ be unique as would there may be an infinity of sigmas (and w1's say) which could provide the same probabilities? Perhaps I mis-understand or miss the point. Feb 24, 2017 at 16:23 • w.l.o.g. consider a standard normal distribution. 68% of the data are within$\mu\pm1$. Your problem is inverted: you know$p$% are within$w$of the mean and want to find$\sigma\$. With the information you have, it's not clear to me why there would be more than 1 answer.
– Sycorax
Feb 24, 2017 at 16:58
• Ok, think that makes sense. In solving expressions with F(.) I need to explicity write out the normal CDF's? Feb 24, 2017 at 18:05
• Ok, will see if I can do it in R - although I'm new to R. Feb 24, 2017 at 18:13
• Many thanks for this. I have been working on the problem looking at R's non-linear solutions. The unitroot I came across and what you have here looks a good solution. I will try this for the three +/- ranges and see if the sigmas do in fact differ. Feb 24, 2017 at 21:12