Let's say we have a central range $(24.5, 27.2)$ and there is a 90% probability value x is within this range, as in $P(24.5 < x < 27.2 = 0.9.$ The range is normally distributed.
The mean value is = $$\frac{(24.5 + 27.2)}{2} = 25.85$$
How do I calculate the standard deviation?
Best Answer
Suppose $\sigma$ is the standard deviation. Then we know that the part of the distribution we are integrating to get 90%${}=0.90$ is from the $Z$-score $\frac{24.5 - 25.85}{\sigma} = \frac{-1.35}{\sigma}$ to $Z$-score $\frac{27.2 - 25.85}{\sigma} = \frac{1.35}{\sigma}$. A way to proceed is from a table of cumulative standardized normal values -- find the pair of symmetric $Z$-scores for 5% and 95%, between which we find 90% of the probability mass. Call these $Z_{0.05}$ and $Z_{0.95}$. Then either solve $\frac{-1.35}{\sigma} = Z_{0.05}$ or solve $\frac{1.35}{\sigma} = Z_{0.95}$ for $\sigma$ (since both will yield the same $\sigma$).