Career switch to Big Data Analytics
Due to high demand, it is possible to start a career in data science without a formal degree. My experience is that having a degree is often a 'requirement' in job descriptions, but if the employer is desperate enough, then that won't matter. In general, it's harder to get into large corporations with formalized job application processes than smaller companies without them. "Knowing people" can get you a long way, in either case.

Regardless of your education, no matter how high demand is, you must have the skills to do the job.

You are correct in noting that advanced statistics and other mathematics are very hard to learn independently. It is a matter of how badly you want to make the career change. While some people do have 'natural talent' in mathematics, everybody does have to do the work to learn. Some may learn more quickly, but everybody has to take the time to learn. 

What it comes down to is your ability to show potential employers that you have a genuine interest in the field, and that you will be able to learn quickly on the job. The more knowledge you have, the more projects you can share in a portfolio, and the more work experience under your belt, the higher level jobs that will be available to you. You may have to start  in an entry level position first.

I could suggest ways to study mathematics independently, but that isn't part of your question. For now, just know that it's hard, but possible if you are determined to make a career change. Strike while the iron is hot (while demand is high).