I don't know how I ever lived without my chronograph. I'm wanting an Ohler with a strain gauge.
They are hugely helpful tools. You first of all can verify your velocities from which you can calculate energies. This is a good thing for pistols and revolvers for self defense or whatever. But, with the help of a good ballistics program (available for free on the net, google PCB1.8 or "ballistics program"), you can, with your calculated mean velocity, work out trajectory drop tables, energy at range, wind drift, etc, and calculate point blank range sight in. In fact, you can calculate your 3" point blank range, then if all you have is a 100 yard range, just zero to whatever the drop table says you are at 100 yards with that point blank range sight in. Over the years and from experience, most calibers are about 2-2.5" high at 100 for their 3" point blank range.
Now, if you know nothin' about this stuff, point blank range is the range at which the bullet will neither be above or below 3" (for a 3" point blank range, used for a deer sized kill zone). IOW, if you zero the gun for its optimum point blank range, you can just hold on the deer all the way out to that range and you don't have to worry about drop or hold over, just aim and shoot. If you zero the gun at 100 yards like a lot of hunters do, fine in the woods where you're not likely to get a longer shot, and you DO get a longer shot, you're going to have to worry about hold over. Hope that makes sense.
Standard Deviation, without writing out the formula, is the square of the square root of the average deviation divided by N-1 (degrees of freedom), or something like that. Been a while since I took statistics, like 35 years. IOW, it's a measurement of variance, the accuracy of the load. The smaller the number, the less variance of velocity from the mean (average) and the less elevation error you're going to have. A high standard deviation will mean you're going to string your shots vertically. For rifles or handguns, I've found really GREAT SDs are under 20. Anything over 30 is not good. With, say, an SD of 35, you’re going to start to see some shot stringing in elevation.
Now, then, lets say you are loading a marginal caliber for deer hunting, say a 7.62x39, and want to find out what range it falls below the magic 1000 ft lbs. You can do that if you know the mean velocity at the muzzle (or 15 feet in front of it) and run it through an exterior ballistics program, another good use for such info. Heck, I wrote my own exterior ballistics program 25 years ago before computers were common using a Timex Sinclair computer and the BASIC compiler built in to it. That program is quite accurate and does the statistical analysis that PCB1.8 (the free program) doesn't. If I'm serious about the load, I'll go plug in my old Tandy Color 3 computer and run it through my program. I guess I could break down and buy a better program, but actually, I can calculate standard deviation long hand, the math hasn't left my senile old brain yet.
Before chronographs were common, ammo companies would exaggerate claims of performance and there was on way to verify velocities of the load much less work out a drop table. The dedicated reloader/rifleman would just have to take the velocity claims in the manuals on leap of faith. When sighting in, if you wanted to shoot to a 3" point blank range, you needed a 300 yard range and you just burned a lot of powder experimenting. With chronograph info and a decent exterior ballistics program, press a few buttons and BOOM, there's your answer. Think of it as computer modeling the path of the bullet.