Normal distribution around N

Question

I have not done stats for several years and seem to have forgotten the basics.

I am trying to find the standard deviation of a normal distribution given a desired mean and cumulative probability between $a \in \mathbb R$ and $b \in \mathbb R$, with $a < b$.

For example, if I want $\mu = 5, a = 0, b = 10, \text{cumulative(a,b) = 98.76%}$, I would get $\sigma = 2$ ($\text{cumulative(a,b)}$ is the cumulative probabilty from $a$ to $b$). The distribution looks like this:

Right now, I am using http://onlinestatbook.com/2/calculators/normal_dist.html and by trial and error (changing standard deviation and looking at resulting probability) to get desired standard deviation.

What is a formula I can use to get $\sigma$ of a normal distribution such that the probability between $a \in \mathbb R$ and $b \in \mathbb R$, with $a < b$ is $x \in [0,1]$?

Answer 1

If you know what $\operatorname{cumulative}(-\infty,a)$ or $\operatorname{cumulative}(-\infty,b)$ is, you can find $\sigma$ through standardizing your distribution. Consider the standard normal distribution, $Z\sim N(0,1)$ (i.e. a normal distribution with $\mu=0$,$\sigma=1$).

Now, $z=\frac{x-\mu}{\sigma}$. In other words, $\operatorname{invornm}(\operatorname{cumulative}(-\infty,a))=\frac{a-\mu}{\sigma}$. Repeating this with $b$, we get a pair of equations which we can use to solve for $\sigma$. This technique is known as standardizing the normal distribution.

Answer 2

In the more general situation: You need to know two (separate) probabilities $P(X \le a)$ and $P(X \le b)$ for two given numerical values $a$ and $b.$ Then standardizing you can get numerical values for $(a - \mu)/\sigma$ and for $(b - \mu)/\sigma$ from printed normal tables and solve the two equations for $\mu$ and $\sigma,$ in terms of $a,b,$ and those two numerical values.

In your specific problem you could use $\mu$ as the median, giving one of the two probabilities as $0.5,$ then get the probability corresponding to either $a$ or $b,$ using symmetry.