I have census block data that I have clipped using my 1/4 mile buffer. For each of my census polygons it has kept the original population, is there a way to get the population recalculated just within the buffer?
Long story short--no matter what you do you'll be introducing new errors into your results. It's unavoidable, and inevitably involves some sort of interpolation.
One of the most simple is binary dasymetric (this is better, in my opinion, than say, areal weighting) areal interpolation.
Here's a couple of examples: http://www.colorado.edu/geography/leyk/data/Hamid/dasy%20lit/eicherBrewerDasy.pdf https://anitagraser.com/2012/11/18/improving-population-density-maps-using-dasymetric-mapping/
It was also recommended to me that I take the total population of the block vs the percentage of the block within the buffer and apply the math to the total.