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How does analysis really make you better - what's the purpose?

Well, honestly, analysis can help you avoid your mistakes. Most of the times I'm analysing the game by an engine. If I don't understand the moves it recommends, I just quickly go through it's recommendations to try and understand what I missed and why did I miss it. Then I try to think differently in my games. My advice to you - Be careful, know your openings, think about how and why did you make a mistake, what pattern did you miss and how can you avoid missing it and finding them in your future games. Blitz games won't improve your chess skill at all. Blitz games are there for those that are, like me, struggling to play fast and want to be able to play good chess at a faster speed than usual. It's very important to know how to analyse your games. First try to do that yourself and if you can't find solutions that seem logical to you, ask friends or your coach (if you've got it). You can use an engine as well. You can't play good without knowledge. To possess knowledge, one must study the game. To study the game, there are books, coaches and online courses. You can't expect to play good without a lot of study. You study so you wouldn't make mistakes you made earlier. Try to think like a Grandmaster. Try to find out what are they thinking about, why are they thinking about things that they think about, and how can you implement that into your play. I hope this may have given you some insights.
Several good answers here. My ideas in short as addendum.

1. First diagnosis, then treatment.

You must know what you train. You might do this by "intuition" or by summarizing your weaknesses through data before. Empirical knowledge from psychological research teaches us, that in often happening cases statistical decisions mostly are superior to clinical, aka human knowledge based, decisions. (Don't forget: You need human input to create the statistical system.) Analysis is one way to deliver data in case you have an categorisation system für mistakes. This might be tactical motives, openings, material distribution, ....

2. Don't analyse anything beyond 10 minute games.

Better play 15/15 games at least, if you want to analyse. There is one exception: If you want to collect material about your automatic blundering 5 minute games or less are acceptable. Even in otb tournament games you you find enough material for this.

3. Build up good habits

It's better to train 4 days 15 minutes per day than 1 day one hour per week. Having a regular training schedule helps.

4. Focus yourself

First check, what's going wrong. Then focus on the worst mistake for some weeks. If you don't like the theme of the worst mistake, let it be and take the first theme you like. B.e. Studying openings is something is disliked completly. So every attempt to train openings in the past failed except once. And as this was an isolated success it went lost over time. Studying tactics and endings was fun for me. So I could build up training habits over this. Today I'm 65 and slowly studying openings is useful and fun. Allow yourself to change! But focus on a goal till it's time to set the next goal.

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