These assumptions include: 1) the initial amount of the daughter isotope is known, 2) neither parent or daughter product has migrated into, or out of, the closed rock system, and 3) decay has occurred at a constant rate over time.
But what if one or some combination of these assumptions is incorrect?
According to Brent Dalrymple (20-69), "the trick to the isochron diagram is the normalization of both parent and daughter isotope to a third isotope." This third isotope is the non-decay product isotope of the same element as the daughter element.
In the initial state, the graph of daughter isotope to the third isotope versus parent isotope to the third isotope should result in a straight, horizontal line.
The first of these assumptions, that all rocks and minerals that formed from the same homogenous mixture have the same age, is not disputed (Overn 2005).
The second assumption of isochrons is that the initial ratios of the daughter isotope to the non-decay product isotope of the same element are uniform throughout the sample.
The steeper the slope of the line, the more decay has occurred in a sample and the older the sample is (Dalrymple 20).