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I know this is a broad, perhaps off-topic question but please bear with me.
I graduated 4 years ago with a degree in Mathematics (but didn't take many Statistics courses). It was only after a few years of working in a non-related field that I discovered my passion in Data Science. I decided to quit my day job to focus on learning all the necessary skills needed to be a data scientist. It became very apparent however that there are many concepts to learn and it is not going to take a short time to catch-up with the competence of the typical professional data scientist that has a PHD in statistics. Data science an interdisciplinary field requiring strong competence in Computer Science, Stats, Math, Economics, Psychology and so forth, but despite this, I have started (6 months) immersing myself in lots of content such as conferences, lectures, PDFs, CV questions and answers to attempt to bridge this gap, and I am incrementally learning more new stuff.\

I guess my question is, what is the best way to approach a career in this field? Should I take the approach of getting an entry-level job as a Data scientist and "learn on the job", or rather dedicate a year (the length of a Master's degree) to immersing myself in high quality content, then applying for a job? I read many answers on CV, and find myself admired by the level of competence that many reputable users have here, and thinking that there is still a long way to go.

Any advice — greatly appreciated.

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A few comments:

  • I think that you should first make a precise and realistic list of your goals, especially if you're going to learn on your own. Why? Because data science is a vast domain, nobody knows everything, not even the top contributors on CrossValidated or here on DSSE. Additionally most of them have a degree related to data science and years of experience, so you're unlikely to reach this level in one year. Since the main goal is to have a job, you could start by identifying more precisely which kind of job you want and which skills you need for these jobs.
  • In general I would say that the best way is to actually obtain a formal degree. It offers you guaranteed recognition, reasonably well structured curriculum, easy access to professors and fellow students who can help you learn more efficiently.
  • Don't focus only on statistics. With a maths background, you should probably focus at least as much on your programming skills. In particular you should practice as much as possible on hands-on projects. If you learn on your own these are the skills that you'll need to demonstrate the most to companies in order to get the job that you want.
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