How is standard deviation different from average deviation?

Question

Standard deviation is used as a measure of deviation or distribution in a sample or a population.Similarly mean is used as a measure of something in a group(say average marks of a group of students).How is standard deviation different from average deviation?

Answer 1

Let $(x_i)$ be a data set. Denote by the average deviation $D$. Then

$$ D = \frac{1}{n} \sum_{i=1}^n (x_i - \mu) $$

and

$$ \sigma = \sqrt{ \frac{1}{n} \sum_{i = 1}^n (x_i - \mu) ^2 } $$

Certainly two different quantities. Variance (the square of standard deviation) measure the average squared deviation - i.e., the average positive distance to the mean.