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.