# Getting standard deviation given probability

#### Ahmed Arnous

##### New Member
Hi all
If you have a mean of 2 and you want to get standard deviation with probability of 90% that the new mean will lie between 1.5 and 2.5. I tried to solve out this question by considering the inverse of Z score
Probability of 90% correspond to z =1.281 then I don't know how to solve out the following equation to get t0 (standard deviation)

Could you help me to solve this equation

#### hlsmith

##### Omega Contributor
I don't follow your question or what you are explicitly asking. If this is a problem for a book or course, can you post the original question.

Thanks

#### Ahmed Arnous

##### New Member
This is a problem in a course and it is about Byesian analysis. The problem is asking about estimating a priori estimate in the population. The mean measurement in the population is around 2, and we will assume there is a 90% probability of it being between 1.5 and 2.5. So the priori estimate follow normal distribution with a mean of 2 and we need to estimate standard deviation

#### hlsmith

##### Omega Contributor
Well if you didn't say "Bayesian", I might have had a guess.

#### JesperHP

##### TS Contributor
Ok so what you are saying - just to make sure we understand you - is that you have a population mean $$\mu$$ which i assume to be normally distributed $$\mu \sim \mathcal N(\mu_0,\tau^2_0)$$ where $$\mu_0=2$$ and $$\tau_0^2$$ is undecided but youre beliefs are that $$Pr(1.5\leq\mu \leq 2.5)=0.9$$ - not 1.2815 as in youre picture because how can the probability be more than 1? - anyway you want to use this information in order to decide which prior to use and therefore need to find the $$\tau_0^2$$ such that $$Pr\left(\frac{1.5-\mu_0}{\tau_0}\leq \frac{\mu-\mu_0}{\tau_0}\leq \frac{2.5-\mu_0}{\tau_0}\right)=0.9$$?