Question

What are mean deviation and standard deviation?

Answer 1

Mean deviation:

• The mean deviation is a statistical measure that is used to calculate the average deviation from the mean value of the given data set.
• Formula: $MD=\frac{1}{n}\sum _{i=1}^n\left|x_i-\mu \right|$

Standard deviation:

• Standard Deviation is the measure of dispersion. It means how far the data values are spread out from the mean value.
• Formula: $\sigma =\sqrt{\frac{\sum _{i=1}^n{\left(x_i-\mu \right)}^2}{n}}$

Example: Data set: $3,4,5,7,8,9$

The mean of the data set is,

$\mu =\frac{3+4+5+7+8+9}{6}$

$\Rightarrow$$\mu =\frac{36}{6}$

$\Rightarrow$ $\mu =6$

The mean deviation of the data set is,

$MD=\frac{\left|3-6\right|+\left|4-6\right|+\left|5-6\right|+\left|7-6\right|+\left|8-6\right|+\left|9-6\right|}{6}$

$\Rightarrow$$MD=\frac{\left|-3\right|+\left|-2\right|+\left|-1\right|+\left|1\right|+\left|2\right|+\left|3\right|}{6}$

$\Rightarrow$$MD=\frac{12}{6}$

$\Rightarrow$$MD=2$

The standard deviation of the data set is,

$\sigma =\sqrt{\frac{{\left(3-6\right)}^2+{\left(4-6\right)}^2+{\left(5-6\right)}^2+{\left(7-6\right)}^2+{\left(8-6\right)}^2+{\left(9-6\right)}^2}{6}}$

$\Rightarrow$$\sigma =\sqrt{\frac{{\left(-3\right)}^2+{\left(-2\right)}^2+{\left(-1\right)}^2+{\left(1\right)}^2+{\left(2\right)}^2+{\left(3\right)}^2}{6}}$

$\Rightarrow$ $\sigma =\sqrt{\frac{9+4+1+1+4+9}{6}}$

$\Rightarrow$ $\sigma =\sqrt{\frac{28}{6}}$

$\Rightarrow$ $\sigma =2.16$