How strong is the magnetic field from a neodymium magnet? I was shopping for neodymium magnets online and noticed that the magnets are usually rated by either magnetic field strength or the force. For example, one magnet I saw is rated at 1 Tesla and another was rated at 98 lbs. In either case I am confused on how to interpret this rating because the magnetic field and force vary from the distance to the magnet. It is unclear to me whether this means that it is 1T inside the magnet or just outside the magnet or even a mile away from the magnet.
What is the exact definition of these ratings? How would I, for example, calculate the magnetic field strength 3 centimeters away from a 1T bar magnet?
 A: The ratings you are seeing are probably a standardized pull force and the remanence.
The standard pull force or strength is the force needed to pull the magnet perpendicularly away from the surface of a thick, ground, flat steel plate, when the magnet and steel are initially in full, direct, surface-to-surface contact. Vendors may used different plates, but the pull force values for identical magnets from good vendors should not differ much.  (There are strong commercial incentives to maximize the reported strength, so everyone is likely to choose similarly optimized plates.) Note that the actual pull force in a real application can be significantly less, since it depends on the geometry of the situation, the properties of whatever is being pulled on,  the presence of other forces such as gravity, ….
The remanence (or closed circuit flux density) is the magnetic induction in a closed magnetic circuit where the North and South poles of the magnet are connected by ferrous material so (ideally) no magnetic field leaks out. It is also essentially the magnet field that would be measured in an infinitesimal gap between the North and South poles of two (identical) magnets brought together.
If you really want to know what you are buying, the best vendors provide both remanence and surface field, pull force measured in multiple figurations, the maximum energy product, standard grade, plots of pull force vs distance from magnet surface, and more.
The magnetic field $3$ cm away from a $1$ T bar magnet depends on the shape of the magnet and how "away" is defined.  For a cylindrical magnet, the magnetic field a distance $z$ from a pole face along the symmetry axis is
$$B=\frac{B_r}{2}\left (\frac{D+z}{\sqrt{{R}^2+{(D+z)}^2}} -\frac{z}{\sqrt{R^2+z^2}}\right )$$
where $B_r$ is the remanence, and $D$ & $R$ are the height and radius of the magnet.
If you know the remanence and diameter of a cylindrical magnet, you can estimate the pull force.
The contact pull force for an ideal magnet with flat pole area $A$ is
$$F_{pull}=\frac{B_{r}^2 A}{2 \mu_0}$$
For example, a 1 inch diameter cylindrical neodymium magnet with $B_r=13200$ Gauss would be expected to have $F_{pull}=351\,\textrm{N}=79\,\textrm{pounds}$, and the measured value of $61$ pounds for a real (non-ideal) version of this magnet seems reasonable.
If you can measure the surface magnetic field of a cylindrical magnet, you can check if the magnet's rated remanence seems roughly accurate.  The surface field (or open circuit flux density) is the magnetic field strength measured at the centre of surface of one of the magnet poles when there are no other magnetic or ferromagnetic material nearby. From the above formula, we can see that the surface field and remanence for an ideal cylindrical magnet are related by
$$B_{surf}=\frac{B_r}{2}\left (\frac{D}{\sqrt{R^2+D^2}} \right )$$
As the magnet gets longer, $B_{surf}$ approaches $B_r/2$.
