Analysis (analytical geometry, calculus and differential equations) describes changing quantities, like fluid flow, dynamics, and kinematics. But statistics is a different bird altogether. It explores the fuzziness of nature. It allows you to make meaningful statements even when you can't come up with exact solutions.

It allows you to make excrutiatingly precise determinations of totally random phenomena. You can determine accurately how many people will die in auto accidents over a long holiday weekend without having a clue as to the causes of those accidents, or who will be involved. One standard deviation, two? What does that actually mean? A system of differential equations can describe complex physical assemblies with many moving parts, but statistics can give you some idea of how many things have to happen in what order before you can reasonably believe something else will happen. That's a differnt kind of question altogether. Yes, it appears to be a discipline really suited to model mental decisions. The real world responds to analysis, but your decisions and choices are best based on statistics.

There is another branch of mathematics, much neglected now, called numerical analysis. It is used to model real equations which have no easily determined solutions with other, simpler equations which give you approximations good enough for engineering work. It is no longer used too much, because now we have computers that can be used to brute-force any calculation. But at one time, it was often critical to come up with easily-computed algorithms which simulated the formal equations of real and complex analysis and converged to acceptable solutions close enough to those given by the "real" equations. It was actually a branch of experimental mathematics, because you didn't always know if, or under what circumstances, a numerical solution would be as good as a formal one. You had to actually try it and find out!

Sometimes the numerical analysis algorithm would quickly converge, after just a few iterations, to the formal values. At other times it would approach it asymptotically, or oscillate around it. Other times, the algorithm would approach the correct value, but eventually start to veer away.

For example, you can solve a definite integral by integrating the function and solving for a specific interval. But some functions can't be integrated, or maybe you just don't know how, and you have to divide the area under the curve into little rectangular strips and add up their areas.
In my time, sometimes we even plotted the integrals out on graph paper, cut out the curves with scissors and actually weighed them on a Mettler balance to get the area under the curve (the definite integral).

The ancient Egyptians could not calculate pi, or use the Pythagorean theorem, but they had developed numerical and graphical solutions to get decent enough results to build their pyramids.

Its like you're exploring the mental space between the mathematics and the engineering. Today, the computer can be instructed to do the calculations, and you can try it in multiple ways so you can determine how the approximations converge, or if they do at all. But I remember when you could take courses in it, for those times when you just didn't have a good mathematical model of what you were studying.

Sorry. Sometimes I just ramble aimlessly.