Random sampling and proportional distribution for age range data: why do results vary?

Hi everyone

I am looking at age-at-death determination of archaeological populations. Due to the nature of the data, these estimations usually span several years, but not all of them have the same degree of precision.
For example
Individual 1: 5-7 years old
Individual 2: 6-10 years old
Individual 3: 2-15 years old
Individual 4: 8-12 years old.

I am trying to figure out the approximate population structure.

The traditional approach would proportionally distribute each individual to all years included in the age determination, e.g. individual 1: range length = 3 years: 5 years old: 1/3; 6 years old: 1/3; 7 years old: 1/3

My new approach randomly samples the age range of every individual repeatedly, generating (probably wrong) absolut ages at death. By repeaded sampling the entire population and consequtive calculation of means and SDs for each year, I approximate the true population structure.
My main goal was to find a way to express the uncertainty associated with the resulting population structure, which I can do with the SDs.

When I compare the two approaches, the results of proportional distribution and means from random sampling differ quite a lot from each other. I would have assumed them to be very similar. Can someone explain, why the results are so different?

Thank you very much and please, ask if anything is unclear.