TITLE: What is the purpose of the standard deviation?
QUESTION [6 upvotes]: I don't have any knowledge of statistics beyond high school common sense. Why is the standard deviation usually seen in combinatorics textbooks, and why is the standard deviation defined intentionally? What is its purpose?
Thanks in advance. I have checked in Wikipedia and many other websites to see what this is, but they are not very concise and clear.

REPLY [3 votes]: Given a sample $x = \{x_1,x_2,\ldots,x_n\}$ the sample standard deviation is defined as
$$
   \operatorname{sd}(x) = \sqrt{\frac{1}{n} \sum_{k=1}^n(x_k - \bar{x})^2 }
$$
where $\bar{x} = \frac{1}{n} \sum_{k=1}^n$ denotes the sample mean.
Several properties of the standard deviation are worth noting. 

It does not change if all the values in the sample are shifted by the same amount, $x_k \mapsto x_k + a$. 
If all the values in the sample are multiplied by a common factor, i.e. $x_k \mapsto \lambda x_k$, the standard variation also gets multiplied by its absolute value, i.e. $\operatorname{sd}(\lambda \cdot x) \mapsto | \lambda | \cdot \operatorname{sd}( x)$.
Standard deviation is bounded from above by the maximal deviation from the mean: $\operatorname{sd}(x) \leqslant \sup_k | x_k - \bar{x} |$. In particular standard deviation of the sample with identical elements is zero.

Because of these properties, standard deviation is a measure of scale of the data, which indicates extent the sample elements deviate from the mean, hence the name.