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The following joke reads "Be greater than average".

$$ B > \frac{1}{n} \sum\limits_{i=1}^{n}x_i $$

But as a new and n00b mathematician, I find the syntax difficult to understand and I have a question about it.

If the formula for average is $$ \text{Average} = \frac{\text{Sum of the terms}}{\text{Number of terms}} $$

Then I can see how the equation in the joke might be:

$$ \frac{ \sum\limits_{i=1}^{n}x_i }{n} $$

How (or perhaps a better a question would be why) does the $$ \frac{1}{n} $$

get moved to the side of that?

I really appreciate you helping me with my dumb question. Thanks.

Thecoder001
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Jimmyt1988
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    How can I improve my question? (I note that I have a -1). I'm happy to make any changes. Thanks in advance. I understand the question may appear below par on the "does not show research effort" but that's actually why I'm here. I would love to know the name of or deeply understand how the 1/n is required or important instead of my version of it. I guess I just never got taught it – Jimmyt1988 Sep 29 '21 at 23:16
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    It means the same thing, just as $\frac{1}{2}(3 + 5)$ means the same as $\frac{3 + 5}{2}$. – Theo Bendit Sep 29 '21 at 23:20
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    It's just a notational difference. $\frac{1}{n} A$ and $\frac{A}{n}$ mean the same thing (namely, $A$ divided by $n$, or equivalently times $n^{-1}$). – Connor Harris Sep 29 '21 at 23:21
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    I don't understand downvotes to this question. – Adam Rubinson Sep 29 '21 at 23:24
  • Ah! So. Eg. 1/5 = 0.2. then 5+10+20+200+100 (5 numbers) = 335. therefore. 0.2*335 = 67. Now if i try that the other way... 335/5 = 67. So, is there a name for this notation? Or a reason it has become this rather than my version? Thanks in advance! – Jimmyt1988 Sep 29 '21 at 23:24
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    This notation is neat : compare how $\frac{\sum_{i=1}^n x_i}{n}$ and $\frac 1n \sum_{i=1}^n x_i$ look. The size of the font of the summation on the first one is really tiny. The other comments make the point that these expressions are the same. – Sarvesh Ravichandran Iyer Sep 29 '21 at 23:26
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    I rate that. Yes, I think you're right, having it on one line does look a bit more sexy... I leave that to philosophers, psychologists and genealogists to explain why. – Jimmyt1988 Sep 29 '21 at 23:27
  • Last question which ties along with this... Is it just a matter of thinking about the equality between them... Or is there a rule or set of rules I should investigate that give me some pattern to remember or recall next time. The rule of "making things look sexy". - Perhaps it's a bit too obvious to even want to answer this one. – Jimmyt1988 Sep 29 '21 at 23:31

1 Answers1

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$\frac{1}{n} a = \frac{a}{n} = a \frac{1}{n}.$

Now consider when $a=\sum\limits_{i=1}^{n}x_i.$

Adam Rubinson
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  • Ahhh, I love that additional equality clause in your first line. I suppose it's just what it is. Actually, you writing it like that has helped me to just think about it in a different, more... explicit "who cares" kind of way. Thank you! – Jimmyt1988 Sep 29 '21 at 23:27