First of all, Statistics is a branch of Mathematics. To pursue your career with Statistics, it is very important to choose Mathematics/ Statistics just after Class 12th. I would personally suggest you not to go with JEE and all such things.

The common thing to become a statistician is to complete Bachelor's, Master and Post Graduation in Statistics.  The degree awarded for Bachelor's in Statistics is B.Stat (Bachelor of Statistics). After that the second step is to pursue Master's i.e., M.Stat with Statistics.

Now, after that the main question is Where to pursue these courses/ degrees, What are the reputed colleges which offer these degrees/ courses. The answer is you are not required to study in a foreign university to pursue such courses. There are some Institutes ( very few ) in India which offer these degrees/ courses. The most reputed of all is "Indian Statistical Institute, ISI". This institute has many branches in India, Main in Kolkata, one in Bangalore, one in Delhi, etc.. Apart from this, there are some other colleges also such as "Loyola College", "St. Xavier's College", etc.. These institutes also offer these degrees.

You can check the admission process for all such Institutes by visiting their official websites one by one. However, ISI has a very few seats for B.Stat programme i.e., about 50 seats in total (i.e., only 50 candidates are selected out of all the candidates appearing for its Entrance exam).

After you complete your Graduation and Post Graduation in Statistics, you are called a Statistician. After that, you may be selected by a Government agency to work for them, or some other Reputed agencies to work for them.

For any other query regarding this, do let me know through comment section.