The better the fit of the data to the line, the lower the uncertainty.
For further information on fitting of lines to data (also known as regression analysis), see: Note that the methods used by isotope geologists (as described by York) are much more complicated than those described by Gonick.
Whether there's a data point on the Y-axis or not, the Y-intercept of the line doesn't change as the slope of the isochron line does (as shown in Figure 5).
A routine statistical operation on the set of data yields both a slope of the best-fit line (an age) and a variance in the slope (an uncertainty in the age).
However, the methods must be used with care -- and one should be cautious about investing much confidence in the resulting age...
especially in absence of cross-checks by different methods, or if presented without sufficient information to judge the context in which it was obtained.
It is not easily explained, in the general case, in any other way.
The data points would be expected to start out on a line if certain initial conditions were met.
Note that the mere existence of these assumptions do not render the simpler dating methods entirely useless.