APMA0650 or APMA1650, which is the right course?

There is a big difference. 

APMA0650

Essential Statistics is an introductory first course intended for students from a variety of disciplines.

The goals of APMA0650 are to that provide the framework for statistical reasoning and statistical judgment and to provide an understanding of the connections between probability and statistics, the basic principles of estimation and confidence intervals, the philosophy and technical definition of hypothesis testing, and a handful of the most commonly used methods of estimation and hypothesis testing.  The course emphasizes an understanding of statistical principles but is light on mathematical foundations and derivations.

The coverage of APMA0650 is not substantially different from a high-school course in AP Statistics. (Students who have mastered high-school AP Statistics are advised not to take APMA0650.) The level is somewhat more technical than the usual AP Statistics course, but accessible to any Brown student who is comfortable with high-school algebra. APMA0650 can be taken by non-science majors. It is not a pre-requisite for APMA1650, but students who have taken APMA0650 can go on to take APMA1650 for a more comprehensive and mathematical treatment.

APMA1650

In contrast, APMA1650 (Statistical Inference I) is a rigorous first course in probability and mathematical statistics with applications.  The course includes a self-contained introduction to probability theory, and statistical methods are derived from their probabilistic roots.  Applications of these principles to textbook problems and active research problems at Brown are also an essential component of the course. The course provides the background and is a prerequisite for many other courses concerning probability and statistics in the Division of Applied Mathematics, and other departments. The successful student will be in a good position to understand new and more complex statistical methods, to critically evaluate the strengths, weaknesses and the appropriateness of existing statistical methodologies. They will also have the foundation to learn to apply these principles in new settings.