# Given two insights into the distribution of a normally distribution of data, can I determine the mean?

If I was told, 81% of students have a gpa higher than a 3.0 and 44% of students have a gpa over a 3.5. Assuming the data is normally distributed, how would I go about determining the mean of the distribution.

• Add the self study tag. Apr 5 '19 at 3:48
• @MichaelChernick this isn't a textbook problem. My school doesn't release the mean gpa, and I got some of this data from a printout and want to see if I can figure it out.
– John
Apr 5 '19 at 4:11
• It is a test type question which means that it requires the self study tag, It doesn't have to be taken from a text book or an exam. Apr 5 '19 at 4:31

You have two values to consider, let $$X$$ be the GPA: $$P(X<3)=0.19, P(X<3.5)=0.56$$ Standardizing inner expressions: $$P(Z<(3-\mu)/\sigma)=\phi((3-\mu)/\sigma)=0.19$$ $$P(Z<(3.5-\mu)/\sigma)=\phi((3.5-\mu)/\sigma)=0.56$$

Using z-table, we can find the associated $$z$$ values as $$(3-\mu)/\sigma \approx -0.875$$ and $$(3.5-\mu)/\sigma \approx 0.15$$. When you solve these, you’ll not only obtain the mean but also the standard deviation.