# Measures of Dispersion - Variance and Standard Deviation

#### description

- Variance and Standard Deviation for raw data:
- Variance and Standard Deviation for ungrouped frequency distribution:
- Variance and Standard Deviation for grouped frequency distribution :

#### notes

Let x_1, x_2, x_3, ..., x_n be n observations and x be their mean. Then
(x_1 - bar x)^ 2 + (x_2 - bar x) ^2 + ... + (x_n - bar x)^ 2
If this sum is  zero, then each (x_i - bar x)has to be zero. This implies that there is no dispersion at all as all observations are equal to the mean  bar x .
If $\displaystyle\sum_{i=1}^{n} (x_i - \bar{x})^2$  is small , this indicates that the observations x_1, x_2, x_3,...,x_n are close to the mean x and therefore, there is a lower degree of dispersion. On the contrary, if this sum is large, there is a higher degree of dispersion of the observations from the mean  bar x .

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