Explanation of sums

Question

How would you explain this notation in plain English? I am having issues getting my head around the meaning of the summation symbol.

$$ x = \frac1n\sum_{i=1}^n x_i\;. $$

Answer 1

$\sum_{i=1}^nx_i$ is simply $x_1+x_2+x_3+\ldots+x_n$, the sum of the $n$ numbers $x_1,\ldots,x_n$. Now you’re multiplying that by $\frac1n$, or, equivalently, dividing it by $n$, so you’re just taking the average (arithmetic mean):

$$\frac1n\sum_{i=1}^nx_i=\frac{x_1+x_2+\ldots+x_n}n\;.$$

If, say, $n=4$, $x_1=3,x_2=5,x_3=1$, and $x_4=5$, then

$$\frac14\sum_{i=1}^nx_i=\frac{3+5+1+5}4=\frac{14}4=3.5\;.$$

Answer 2

$\displaystyle x = 1/n\sum\limits_{i=1}^n xi $ means $\displaystyle x = \frac1n \left( x_1 + x_2 + x_3 + \ldots + x_{n-1} + x_n \right)$.

We see here that the variable $i$ takes on all the integer values from $1$ to $n$ since those are the lower and upper limits on the sum.

Let's say if $n = 3$ then $\displaystyle x = 1/3\sum\limits_{i=1}^3 xi = \frac13 \left( x_1 + x_2 + x_3 \right)$

Answer 3

The symbol $\sum$ denotes the greek letter sigma (or "s") in caps, and the integral symbol $\int$ comes from a long "s", for the Latin summa (sum, or total). Both thus represent "sums" of elements. The first one is generally reserved for a finite number of elements, or an infinite one, but discrete enough (imagine you could sum the integers $1$, $2$, $3$, $\dots$).

For $\sum$, by convention, a letter below denotes an index, here $i$, that can change but is unique for each of the elements you are summing. Generally, calling the index $i$, $k$ or whatever does not changes: it is a dummy variable. What is important though is how you construct your index. Suppose you want to sum $3$, $7$ and $8$, you build a set $S=\{3,7,8\}$ with your elements. Writing: $$\sum_{s\in S} s$$ means that you will sum all the elements of $S$. For convenience, you can use indices. $S$ has three elements: one generally indexes it with consecutive integers, often starting from $0$ ($s_0=3$, $s_1=7$, $s_2=8$) or $1$ ($s_1=3$, $s_2=7$, $s_3=8$). Then, below the sum, you can put the set of indices $i$ instead of the elements $s\in S$: $$\sum_{i\in \{0,1,2\}} s_i \;\text{or }\; \sum_{i\in \{1,2,3\}} s_i \,.$$ You can imagine this can become quite long to write all the indices if you sum one hundred terms. So, by conventions, supposing consecutive integers, one often write the starting index below, and the last above the $\sum$, and everybody knows that you are summing all symbols in between, as in your case:

$$\sum_{i=1}^3 s_i $$ and then you can play, for instance summing all the terms but the first: $$\sum_{i=2}^3 s_i \,.$$