Choosing a Jo
There is a rule, and exceptions to it when deciding the size of a jo (medium length staff used in Aikido), Jodo and other sword/staff practices. The rule (and this comes from ancient times) is that the jo is cylindrical in shape and has specific proportions of 50 1/4" x 15/16". Traditional schools of Jodo and the jo used in the Shindo muso ryu go by this rule. So, old school dojo (koryu), Jodo and Shindo Muso Ryu practitioners should select the 50 1/4" x 15/16" size.
For most aikidoka (Aikido practitioners), a more customized approach is taken which is technically an exception to the rule but since the vast majority of jo usage occurs in Aikido, the customized approach is more largely represented and the jo is sized according to body proportions as shown below.
Wooden Japanese style Jo staff, hanbo, yawara, wooden hiking sticks and canes for martial art and self defense purposes in thicknesses close to one inch offer the best strength and speed but "why so little difference?" The answer is readily apparent when you compare them in your hand - 15/16" feels decidedly thinner than 1". Your hand knows it and the math proves it: while the diameters are only 6% different, the volume of wood differs by 12%. Here's why:
At first glance, you may think 1/16" diameter change is very small. We can calculate [1-.94]/1 x 100 = 6%
The wood volume however is written as V= π(d/2)²h where d is the diameter and h is the length of the staff. Notice the term (d/2)². This means that the volume goes up exponentially as a function of the diameter. If you do the computation, you'll see that a 1" staff has 12% more material than a 15/16" staff. A skilled martial artist doesn't need to crunch numbers to make a determination and as a buyer, it's pretty easy to make a subjective decision without comparing the two physically or getting out a calculator: 1" seems medium thick - a kind of neutral diameter, not to thick, not too thin. 15/16" seems slightly thinner, faster, a little more precise. Indeed, 15/16" is the diameter of the classic Japanese Jo.
While koryu (old school) staffs and the traditional Aikido jo are generally 15/16" and hikers seeking a lighter weight stick also may prefer the thinner option, 1" is better for martial artists seeking slightly more girth and mass without sacrificing too much maneuverability and speed.