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First, let me apologize if this question has already been answered. I'm new here (first post!) and while I've been searching for similar questions, I haven't found any that match my problem. I also understand this is a general question so I wont mind if this gets closed for being as such, though hopefully some answers come through before that happens.

I'm looking for some direction that can help me "fill in the gaps" of my mathematics knowledge. The problem is, I'm finding it hard to describe what exactly these gaps are. For the most part, it's a general feeling that I'm not seeing things the way others who are more astute at math do. Or that there is some general foundation in quantitative thinking that I'm lacking. The best way I can describe my struggles is through a couple examples.

Let's take the area of a triangle. Up to a certain point in my life, I just memorized the formula $\frac{1}{2}bh$ without understanding why it is that way. It wasn't until I learned you're simply taking the area of a square and cutting it in half that it clicked. (I know, this is a gross simplification)

Another example comes from a stats class I'm taking now. The text gives us the standard deviation for a sample:

$$s = \sqrt{{\sum_{k=1}^{n}(x_{k}-\bar{x})^2}\over{n-1}}$$

I can see why we are summing all of the samples but why are we subtracting $\bar{x}$ from it, squaring it, dividing it by ${n-1}$, and taking the square root of all of it?

These illustrate a fundamental problem I have: there is a disconnect between seeing these symbols on a page and understanding what they are describing in real life. Sure, I can memorize how to compute the answers, but in the long run I want to truly understand what I'm doing. Not just because I find it interesting but because I need that understanding to be able to read more advanced topics. So my questions for ya'll are:

1) Are there recommendations on resources that can help with this?

2) Do you have any insight from your own experiences?

Rounak Sarkar
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MJCoate
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    The sample standard deviation is given by $$s = \sqrt{\frac{\sum_{k = 1}^n (x_k - \bar{x})^2}{n - 1}}.$$ This is slightly different than what you’ve written; look at where the $n - 1$ is in regards to the square root and where the square is placed in the sum. – shoteyes Jul 07 '21 at 05:35
  • Thanks! I've updated it. – MJCoate Jul 07 '21 at 06:04
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    I think you just have to make it part of your mission in life to grok math. Try to understand how to derive all the formulas you encounter. Try to imagine how someone might have thought of it or discovered it. Find good books to read and if you're unsatisfied with the way the material is presented, and you can't figure out a better presentation yourself, then look on math.stackexchange or other internet resources to try to find better explanations. I think it's a genuinely difficult process, because it takes a lot of time, and that's why not everyone is great at math. – littleO Jul 07 '21 at 06:09
  • @MJCoate Unfortunately, it’s still not correct. The $n - 1$ should be inside the square root, not outside. You can right-click on the math in my previous comment, click “Show Math As” and “TeX Commands” to see how I was able to type it. As a tip, \frac{numerator}{denominator} is easier to handle and preferable to {{numerator} \over {denominator}}. – shoteyes Jul 07 '21 at 06:09
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    Yeah, I screwed up as I'm editing this on my phone. It should be correct now. – MJCoate Jul 07 '21 at 06:11
  • It's a serious commitment. From my own experience - there are no shortcuts. If you want to really understand the theory, you have to put in the hours. I am a to-be-mathematician-in-some-(distant)-future, so I can justify this commitment to myself. Can you? – AlvinL Jul 07 '21 at 06:24
  • To some extent, I think it's important for your instructor and your text to explain in detail the meaning of this formula. It shouldn't be expected that you understand it at first glance. That said, it's also your responsibility to take the explanation given there and digest it yourself. – Jair Taylor Jul 07 '21 at 07:09
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    In this particular case, it's understandable that you don't understand the significance immediately. Why take the squares instead of the absolute value? I remember being perplexed by this as well on first seeing this definition. There are several reasons why this definition is useful, but one of them is that the standard deviation appears naturally in many important expressions such as the formula for a normal distribution (which in turn is important due the Central Limit Theorem.) – Jair Taylor Jul 07 '21 at 07:12

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Maybe not the specific question that you mentioned, I would highly recommend the videos of 3 Blue 1 Brown and Mathologer to get these intuitions that you are talking about.

Also, there are books that often give you these ideas to understand what symbols represent. For example, if you read Herstein's Group Theory, it gives you so much geometry behind things. I'm sure, similar books are available for statistics as well (although I'm not a stat student).

3 Blue 1 Brown - https://youtube.com/c/3blue1brown

Mathologer - https://youtube.com/c/Mathologer

Also, there are some things which doesn't have any specific geometry- they are just very useful algebraic quantities. For example, I asked our Linear Algebra professor about the geometry of the trace of a matrix. He said, it's easy to force some geometry using divergence or curl or something similar. But, the best thing is to just look at it as the sum of the diagonal entries.

Sayan Dutta
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  • Thanks! I'm not necessarily looking for things specific to stats, that was just an easy example to illustrate my difficulties. 3B1B is excellent though I wish he had more content. I've looked for Herstein's Group Theory but can't find it. Is that a book or the section in his Topics in Algebra? – MJCoate Jul 07 '21 at 19:35
  • @MJCoate No no, what I meant was I was reading group theory from Herstein's "Topics In Algebra". – Sayan Dutta Jul 07 '21 at 20:11