How to calculate the probability of a normal distribution with unknown mean and unknown variance?

How do you calculate the probability of a normal distribution with unknown mean and unknown variance? If a problem stated, for example, that 15% of the time sales are more than 15,000 and 20% of the time more than 10,000.... What is the probability that the sales fall between __ and _ copies? What is the probability that the sales are greater than _ copies? What is the probability that the sales are less than __ copies?

What method do you use to solve statistical problems like these?

• The information given is sufficient to compute mean and variance. – André Nicolas Nov 3 '13 at 0:18
• A better title would be something like "How to find the mean and standard deviation of a normal distribution, given two quantiles?" – Silverfish Nov 3 '13 at 0:27

You have two unknowns: the parameters for mean ($\mu$) and standard deviation ($\sigma$). If only you knew those, you could solve the "find the probability" questions easily.

You also have two facts: you've been given two quantiles from the distribution, in other words two values of $x$. You know that $z=\frac{x-\mu}{\sigma}$ (more nicely for you, $x=\mu+z\sigma$) and you know how to use a statistical table to switch between probabilities (or percentages) and z-values. That means you can write two equations in which you know $z$ and $x$, so the only two unknowns are the parameters.

Any problem with two unknowns, but where you get given two facts, suggests you should be forming two equations involving the two unknowns and solving them simultaneously. That's exactly what happens here. Equations of the form $x=\mu+z\sigma$ are linear and since the coefficient of $\mu$ is the same (one) in both equations, it's nice and easy to solve - just subtract your two equations from each other to eliminate the $\mu$ and you can now find the $\sigma$. Then substitute $\sigma$ back into one of your equations to find $\mu$. It's probably a good idea to check your arithmetic, by seeing if your $\mu$ and $\sigma$ values work in the second equation too.

A good general rule for all basic normal probability questions is to know how to interrelate facts and unknowns. If you know the relationships, then you should be able to find a route from the facts you have been given, to the thing you have been told to calculate.

Use statistical tables to interrelate probabilities and z-values. If you have been given a probability, work out the z-score. If you have been told to work out the probability, then use the z-score. (If you don't yet know the z-score, you will need to work it out from facts you have been given.)

Use the equation $z=\frac{x-\mu}{\sigma}$ (or its rearrangements $x=\mu+z\sigma$, $\sigma=\frac{x-\mu}{z}$ or $\mu=x-z\sigma$) to interrelate the values of $z$, $x$, $\mu$ and $\sigma$. If you know three of the four, you can always find the fourth. Other times, like in this question, you can be expected to form simultaneous equations.

• @Hagen I find students often don't realize you can translate a statistical problem into an algebraic one: the math and stats can get seen as separate. I don't like teaching "recipes" for "given this, find that" as there are so many combinations of facts you can be presented with! So now I give students a flowchart showing the two interrelationships ($x \mu, z, \sigma$ via the formula and $z\leftrightarrows P$ via tables), tell them to think which ones they know, and work out the route to the thing they were told to find... – Silverfish Nov 3 '13 at 0:56