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
When Mean deviation is divided by the average used in finding out the mean deviation itself, the resulting quantity is described as_____________.
.Match List - I with List - II.
List - I List - II (A) Coefficient of Dispersion (I) Standard deviation as a percentage of mean (B) Standard Deviation (II) Mean of squared deviations of individual scores from mean (C) Coefficient of variation (III) Variance in terms of mean (D) Variance (IV) Positive square root of variance - Two sets of data consisting of 10 and 20 observations have same mean 8 with standard deviations of 1 and 2, respectively. If the two data sets are combined, then the variance is
Two data set of sizes 9 and 6 have standard deviation 3 and 4 respectively and arithmetic means 3 respectively. The standard deviation of combined data set of size 15 is
Variance is independent of change of :
