[1][2] It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean.
It has some desirable properties: it is zero for any symmetric distribution; it is unaffected by a scale shift; and it reveals either left- or right-skewness equally well.
In some statistical samples it has been shown to be less powerful[3] than the usual measures of skewness in detecting departures of the population from normality.
[5][6] This range is implied by the fact that the mean lies within one standard deviation of any median.
The bounds of this statistic ( ±1 ) were sharpened by Majindar[8] who showed that its absolute value is bounded by with and where X is a random variable with finite variance, E() is the expectation operator and Pr() is the probability of the event occurring.
[11] Another extension for a distribution with a finite mean has been published:[12] The bounds in this last pair of inequalities are attained when
[19] Assuming a symmetric underlying distribution, a modification of S was studied by Miao, Gel and Gastwirth who modified the standard deviation to create their statistic.
The following values for S are known: In 1895 Pearson first suggested measuring skewness by standardizing the difference between the mean and the mode,[29] giving where μ, θ and σ is the mean, mode and standard deviation of the distribution respectively.
Estimates of the population mode from the sample data may be difficult but the difference between the mean and the mode for many distributions is approximately three times the difference between the mean and the median[30] which suggested to Pearson a second skewness coefficient: where ν is the median of the distribution.
Bowley dropped the factor 3 from this formula in 1901 leading to the nonparametric skew statistic.
The relationship between the median, the mean and the mode was first noted by Pearson when he was investigating his type III distributions.
[31][32][33] Analyses have been made of some of the relationships between the mean, median, mode and standard deviation.
[34] and these relationships place some restrictions on the sign and magnitude of the nonparametric skew.
The mean (0.9) is to the left of the median (1) but the skew (0.906) as defined by the third standardized moment is positive.
Doodson in 1917 proved that the median lies between the mode and the mean for moderately skewed distributions with finite fourth moments.
Haldane required a number of conditions for this relationship to hold including the existence of an Edgeworth expansion and the uniqueness of both the median and the mode.
This result was confirmed by Hall under weaker conditions using characteristic functions.
[39] Hall also showed that for a distribution with regularly varying tails and exponent α that[clarification needed][38] Gauss showed in 1823 that for a unimodal distribution[40] and where ω is the root mean square deviation from the mode.
For a large class of unimodal distributions that are positively skewed the mode, median and mean fall in that order.
[41] Conversely for a large class of unimodal distributions that are negatively skewed the mean is less than the median which in turn is less than the mode.
[42] For a unimodal distribution the following bounds are known and are sharp:[43] where μ,ν and θ are the mean, median and mode respectively.
The middle bound limits the nonparametric skew of a unimodal distribution to approximately ±0.775.
In 1964 van Zwet proposed a series of axioms for ordering measures of skewness.
It has been suggested that random variates from distributions with a positive nonparametric skew will obey this law.
[47] This statistic is very similar to Bowley's coefficient of skewness[48] where Qi is the ith quartile of the distribution.
Replacing the denominator with the standard deviation we obtain the nonparametric skew.