I learned college statistics but only at the undergraduate level. I feel like it's hard to apply what I learned to real-life situations, unlike self-contained problem sets.
Here's what I'm trying to do:
I want to estimate the population's distribution with some pieces of information I gathered that are not sufficient to estimate the distribution perfectly.
I took a test and I have some information like:
There were 59 students. (n=59)
Mean is 75
Top score was 96
The lowest was 48
There are 13 students within 60 <= x < 75 range
There are 3 students under 60 (x < 60)
If I assume a normal distribution for the population, I already have the mean so now have to know the standard deviation of the population to get the distribution.
However, instead of standard deviation, I got these bunch of partial information that seem useful but don't know how to actually use them to calculate the population distribution.
Is there a standardized way to update my knowledge of the population based on this little partial information? I feel like this should be related to Bayesian update but this seems totally different from what I read in the textbook.