# Deriving Var from mu and two Ps

#### billylyman

##### New Member
If I know the mean of a normal distribution and two probabilities F1=P(Z>z1) and F2=P(Z>z2), I should be able to derive the Var of the distribution, no? Can't figure how. Do I need to manipulate the CDF algebraically, or is there an easier (non-calculus intensive) way?

In real terms, I know that a school's average class size is 33, that 17% of their classes are less than 20, and 5% are more than 50. I want to figure the Var from this, assuming normal distribution (which is not an obviously useful or accurate assumption, I realize, but want to do so anyway).

#### hlsmith

##### Less is more. Stay pure. Stay poor.
What do you mean by var, do you mean variance?

What do we know about the area for +/- 1 standard deviation, now how about 2 standard deviations.

#### BGM

##### TS Contributor
Basically the important fact you need to know is that

$$X \sim \mathcal{N}(\mu, \sigma^2) \iff Z = \frac {X - \mu} {\sigma} \sim \mathcal{N}(0, 1)$$

Therefore if you are given that

$$\Pr\{X \leq x_1\} = p_1 = \Pr\left\{Z \leq \frac {x_1 - \mu} {\sigma} \right\}$$

$$\Pr\{X \leq x_2\} = p_2 = \Pr\left\{Z \leq \frac {x_2 - \mu} {\sigma} \right\}$$

and $$x_1, x_2, p_1, p_2$$ are known,

then you can immediately refer to the quantiles of standard normal (either from table/software) and then set up a simultaneous linear equations for $$\mu, \sigma^2$$ to solve.