TRATON STATISTICS: Math Lesson for the Day - Simple Calculations of Quantiles, Part I

In a prior post, we reported on statistical analysis and specifically how one calculates skewness (S) when one knows the median (m), the first quartile (Q1), and the third quartile (Q3). In this article, we move forward and provide a simple equation for calculating a particular quantile (%) in view of the median (m) and the skewness (S) of a distribution.

Previously, we provided a simplified equation for skewness (S) as being:

Eq. 1
S = 3(M-m)/SD

where:

Eq. 2
SD = 0.7413(Q3-Q1)

because of a well-defined relationship between the standard deviation and the percentiles.

This, however, can be more generalized to calculate skewness (S) in view of particular quantile numbers. So, instead of the left and right quartiles — i.e., the 25th and 75th percentiles — one can just as plausibly use the 20th and 80th percentiles, the 10th and 90th percentiles, or more generally the 100p^th and 100(1 – p)^th percentiles F^-1(p) and F^-1(1 – p).

So, designating:

Eq. 3
F^-1(p) = L

and

Eq. 4
F^-1(1 – p) = U

the revised equation reduces to:

Eq. 5
S = 3(M-m)/(k(U-L))

where k represents the conversion value between the percentiles (or quantiles) and the standard deviation (SD).

Thus, one can see that the quartile (Q1, Q3) equation can be derived from the quantile equation, since:

Eq. 6
L = Q1

and

Eq. 7
U = Q3

thereby reducing the general equation to the quartile equation by simple substitution.

Eq. 8
S = 3(M-m)/0.7413(Q3-Q1)

Intuitively, it may seem that with knowledge of S, M, m, and SD, one can calculate any percentile or quantile of L and U.

A detailed explanation of this calculation is provided in the next article by my colleague Morley Moore.

Reporting for Traton News,

Ingrid Ingram.