The problem with calculus as taught today in most schools is that most of the mathematical tricks for integrating functions that they hammer the students with are essentially useless in real-world problems so you might as well just start with numerical methods of approximation right at the beginning, once basic concepts like 'the integral of a function corresponds to the area under the curve of that function' are grasped. I know many math purists are dedicated to the pencil and paper approach, but it's not all that useful for non-math majors who will certainly be building or using computational models almost exclusively.
Another prerequisite for statistics should be linear algebra, because without a grasp of vector representations of data and matrix operations on that data, a lot of statistical concepts, e.g. multivariate statistics, won't be very understandable.
Fundamentally, trying to use statistics without this deeper understanding of the mathematics underlying much of it can lead to all kinds of issues, like choosing the wrong statistical method for a given problem.