Dispersion is the variability around the central tendency (i.e. if mean addresses returns, then dispersion addresses risk).
Essentially, it shows how far the units are from the central tendency. Absolute Dispersion is the amount of variability that is present, without comparison to any reference point or benchmark. They include (i) Range, (ii) Mean Absolute Deviation (“MAD”), (iii) Variance, and (iv) Standard Deviation. Population Variance is the arithmetic average of the squared deviations around the mean- Formula: (Sum of (Xi - µ)2) / N; where (i) µ is the population mean, (ii) N is the total number of observations, and (iii) Xi refers to the specific observation.
Population Standard Deviation is the square root of the “Population Variance”- Standard Deviation explains the distance from the mean from either direction, where i.e. if mean =5 and standard deviation = 2, interval of expected values can run from 3 to 7.
Sample Variance is the sample measure of the degree of dispersion of a distribution- Formula: (Sum of (Xi - Xbar)2) / n-1; where (i) Xbar is the sample mean, (ii) n-1 are the degrees of freedom (number of observations -1) to help improve statistical significance , and (iii) Xi refers to the specific observation.
- Sample Standard Deviation is the square root of the “Sample Variance”
Variance and standard deviation takes into account returns above or below the mean. We may sometimes want to concentrate only on the downside risk- Therefore, it is important to understand (i) semi-variance, (ii) semi-deviation, (iii) target semi-variance, and (iv) target semi-deviation. The formula is following the same logic for variance and standard deviation, only accounting for the target below the mean.
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