Finding mean and standard deviation of normal distribution given 2 points.

Question

In general, how do do you calculate the mean and standard deviation of a normal distribution given 2 values on the distribution with their respective probabilities?

For Example:

Suppose that the ages of students in an intro to statistics class are normally distributed. We know that 5% of the students are older than 19.76 years. We also know that 10% of students are younger than 18.3 years.

What are the mean and standard deviation of the ages?

In my attempts to solve a similar problem I can't see how to calculate the mean or standard deviation without first knowing one of the two. I can find the z-score for 95% and 10%, and if I could somehow derive the values for 5% or 90% I could then average the 5% and 95% or 10% and 90% values to then find the mean, but I don't see a way to do so. Is it even possible to solve this problem or is there not enough information?

Answer 1

Let's take the example in question. Assume that the mean is $\mu$ and that the standard deviation is $\sigma$. If we have two z-values $z_1$ and $z_2$ corresponding to our two observations, 19.76 and 18.3 then we can solve the following equations for $\mu \ \text{and} \ \sigma$. $$\frac{19.76 - \mu}{\sigma} = z_1 \\ \frac{18.3 - \mu }{\sigma} = z_2$$ We have two equations in two unknowns, solving which, we can find $\mu$.

Answer 2

From your z-score table the data at $95\%$ is at about mean +$1.65$ standard deviations. Taking $\mu$ as the mean and $\sigma$ as the standard deviation, this tells us that $\mu+1.65\sigma=19.76$ You should be able to write a similar equation from the other piece of data. That gives two equations in two unknowns.