Measures of Dispersion - Variance and Standard Deviation



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


 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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