What is the tolerance of a magnet’s magnetized direction?
We can draw an arrow showing which way the magnetization direction points. How perfect is this arrow? Is the overall direction of magnetization perfectly aligned with the cylinder’s axis? Or can the magnetization direction differ by some small angle?
Yes, this angle has a tolerance, a generally accepted range for the angle between the cylinder’s axis and the magnetization direction. For sintered neodymium magnets, this number is typically within a few degrees. Let’s say less than 5°.
Why is this so? Again, recall that a magnet is a squished together (sintered) collection of tiny magnetic particles (grains). The magnetization direction of all the individual grains is not all perfectly aligned. Each grain is off-axis by some amount. The total magnetization of the magnet is the sum of all those little bits of magnetization. Since some of those tiny arrows are pointing slightly left or right, the off-axis magnetization components tend to cancel each other out. When this cancellation doesn’t quite sum up to zero, you get a combined magnetization that’s slightly off-axis.
Incidentally, this misalignment of the grains is one of several reasons why neodymium magnets can’t be made with a theoretically perfect N64 rating. Some of the strength is lost to grains slightly cancelling each other out.