Sum of Squares Deviation Property Explained
The question asks about the property of the sum of squares of deviations when calculated from the mean of a dataset.
Mathematical Property of Deviations
For any set of observations $x_1, x_2, \dots, x_n$, let $\bar{x}$ be their arithmetic mean.
The sum of the squares of deviations from the mean is given by:
$ S_{\bar{x}} = \sum_{i=1}^{n} (x_i - \bar{x})^2 $Now, consider the sum of the squares of deviations from any arbitrary value '$a$':
$ S_a = \sum_{i=1}^{n} (x_i - a)^2 $It can be shown mathematically that:
$ S_a = \sum_{i=1}^{n} (x_i - \bar{x})^2 + n(\bar{x} - a)^2 $Since $n$ (the number of observations) is positive and $(\bar{x} - a)^2$ is always non-negative (zero only if $a = \bar{x}$), the term $n(\bar{x} - a)^2$ is always greater than or equal to zero.
Therefore, $S_a \ge S_{\bar{x}}$. Equality holds only when $a = \bar{x}$.
Conclusion
This proves that the sum of the squares of deviations is minimized when the deviations are taken from the arithmetic mean.
Hence, the sum of squares of deviation taken from the Mean is always the Least.
Important Questions from Measures of Dispersion
If for a moderately symmetrical distribution mean deviation is 12, then the value of standard deviation is
Variance is independent of change of :
- Mean (M) and Standard Deviation (S) of different datasets are given in A-D below. Compute the coefficient of variation of each dataset and arrange in ascending order.
A. M = 60, S = 14
B. M = 70, S = 16
C. M = 80, S = 5
D. M = 90, S = 4
Choose the correct answer from the options given below: - Which of the following are methods of dispersion ?
A. Median
B. Mean
C. Mean deviation
D. Standard deviation
E. Range
Choose the correct answer from the options given below : - Mean (M) and coefficient of variation (CV expressed as a percentage) of different datasets are given in A-D below. Compute the standard deviation of each dataset and arrange in ascending order.
A. M = 60, CV = 70/3
B. M = 80, CV = 6.25
C. M = 70, CV = 160/7
D. M = 90, CV = 40/9
Choose the correct answer from the options given below:
