1
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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 $\mu$.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.