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I would like to write an equation to sum the values of each number in a vector divided by the length of the vector. What is the proper way to write this formula?

As an example, consider vector v:

v = [1, 2, 3, 4]

2.5 = 1/4 + 2/4 + 3/4 + 4/4

Is this the proper way to write this?

$$\sum_{i=1}^N \frac{i}{length}$$

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As far as I have seen in many calculus book from R.A Silverman to Adams; $|V|$ is what you want. –  Babak S. Nov 29 '12 at 18:42
    
Thanks. I've seen this, but in my actual use case I am also using absolute values in my equations, so it would be very confusing to use this notation. –  user1728853 Nov 29 '12 at 18:43
1  
$|v|$ is often used to denote $\lVert v\rVert_2$, the $2$-norm or Euclidean length of $v$, whereas it seems you want its length in the sense of the number of its components -- I'm not aware of any notation for that. –  joriki Nov 29 '12 at 18:45
    
And when you present $V$ as a vector in your context firstly; there won't happen any confusing point for example with numbers,... –  Babak S. Nov 29 '12 at 18:47

3 Answers 3

up vote 1 down vote accepted

I would be tempted to use the notation $\overline{{\bf v}}$, as is done with the arithmetic mean $\overline{x}.$ If your vector is given by ${\bf v} = (v_1,\ldots,v_n)$ and you are calculating

$$\frac{v_1+\cdots+v_n}{n}$$

then you are calculating the arithmetic mean of the data set whose datum are the components of the vector. It seems perfectly reasonable to represent data sets by vectors and to write $\overline{{\bf v}}.$

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You could use $N$, since you are already using it in the summation bound. Another notation to consider depending on the context is $dim(v)$.

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Do you mean $\dim V$ where $v \in V$? –  Fly by Night Nov 29 '12 at 18:58
    
Yes, that's more accurate, although I think $dim(v)$ is clear enough, depending on context. –  mdgeorge Nov 29 '12 at 19:48
    
The symbol $v$ denotes a vector. Then $\dim v$ denotes the "dimension of a vector". IMHO, the only way to reconcile that is to take it to mean the dimension of the span of that vector. –  Fly by Night Nov 29 '12 at 19:52
    
Yes fbn, you are absolutely correct and I hereby recant my advocacy for considering $dim~v$ as a reasonable notation. –  mdgeorge Nov 30 '12 at 4:41

As noted, this is the arithmetic mean of the elements in $v$. You can indicate this in a number of ways such as:

$$\sum_{x\in v} \frac{x}{|v|} = \sum_{i=1}^{|v|} \frac{v_i}{|v|}$$

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Thanks. Yes, I'm aware it's the mean. I choose a simplified example to depict what I am trying to convey. My actual use case is more complex. Thus, I was looking for the clearest way to convey vector length. –  user1728853 Nov 29 '12 at 20:38

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