Standard deviation has been accepted universally as the ideal measure of dispersion of data.
However, standard deviation, though regarded as ideal measure of dispersion, is not based on the deviations between all pairs of observations.
A measure of dispersion can, in the true sense, be regarded as the proper measure of dispersion if the measure is based on the deviations between all pairs of data.
If a measure of dispersion is defined on the basis of the deviations between all pairs of observations then logically it should be a proper measure of dispersion.
The point in this case is whether it will be better than standard deviation as a measure of dispersion.
Therefore, the question is
" If a measure of dispersion is defined on the basis of the deviations between all pairs of observations, will it be superior to standard deviation ? "
Dhritikesh,
a very interesting question--
I calculated the standard deviation and the average deviation for three datasets:
(3, 6, 7) (3, 8, 9) (3, 10, 11). In each case the average deviation was greater than the standard deviation.
As for the relative superiority, of the two measures, the principal advantage of using the standard deviation is that it permits parametric tests, e.g., differences among means.
A search of Google Scholar for information related to average deviation among all pairs of data was unsuccessful. What is required to answer your question definitively is to compare Type 1 and Type II errors based on computing a parametric test versus a nonparametric test that incorporates average pairwise deviation as the measure of variability. As far as I know, no such test has been developed.
I hope this helps.
Don P.
@ Donald J. Polzella,
It is true that the principal advantage of using the standard deviation is that it permits parametric tests.
However, if standard deviation is not superior to a measure of dispersion based on the deviations between all pairs of observations then any test based on standard deviation will not yield correct findings/results.
Thus there is necessity of a measure of dispersion which is based on the deviations between all pairs of observations.
@ Sandra Brathwaite ,
I have not been able to understand the meaning of truly 'non- question'.
May I request to describe the meaning of truly 'non- question'.
@ Donald J. Polzella,
you have calculated the standard deviation and the average deviation for three data sets: (3, 6, 7) (3, 8, 9) (3, 10, 11)
and found that in each case the average deviation was greater than the standard deviation.
My point is
" Can this be a ground for the standard deviation to be superior to a measure of dispersion based on the deviations between all pairs of observations ? '
Also, logically it is difficult to accept the convenience of standard deviation in parametric test as a basic principle/criterion so that it can be superior to a measure of dispersion based on the deviations between all pairs of observations.
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