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

Our previous post left off with calculating a quantile (or percentile) from a single-equation-two-unknown situation. We now continue that discussion with how we can provide a reasonable estimate from the unknowns, if we have available two additional pieces of information. Namely, the mean (M) and the standard deviation (SD).

First, we start with a presumption that skewness (S) can also be related to a mean (M), median (m), and standard deviation (SD) by the following (simplified) equation:

Eq. 7
S = 3(M-m)/SD

Of course, this equation provides only a rough estimate, and may not be true in all circumstances. However, as Pearson proposed, it provides a reasonable estimate of the skewness.

Insofar as the standard deviation for normalized curves is related to the quantiles or percentiles by a well-defined rule, a simple look-up of that relationship can provide the needed information to estimate a quantile or percentile from the mean (M), median (m), and standard deviation (SD).

So, arbitrarily defining standard deviation (SD) as a function:

Eq. 8
SD = f[Q]

or

Eq. 9
Q = g[SD]

Insofar as the f[] and g[] are inverse equations.

We see that the skewness (S) can now be defined in terms of f[Q] to be:

Eq. 10
S = 3(M-m)/f[Q]

From our previous post, skewness (S) was calculated to be:

Eq. 4
S = (U+L-2m)/(U-L)

Thus, if U and L can be calculated from Eq. 1, and be defined as:

Eq. 11
U = g[SD_U]

and

Eq. 12
L = g[SD_L]

The skewness equations can be re-written as:

Eq. 13
S = 3(M-m)/SD = (g[SD_U]+g[SD_L]-2m)/(g[SD_U]-g[SD_L])

This now provides an equation of skewness (S) in terms of the mean (M), median (m), and standard deviation (SD), without any need for quantiles or percentiles. Alternatively, knowing that the relationship between quantiles or percentiles is defined in terms of the standard deviation (SD), we can re-write Eq. 13 as:

Eq. 14
S = 3(M-m)/f[Q] = (U+L-2m)/(U-L)

which now defines skewness (S) in terms of mean (M), median (m), and percentiles or quantiles.

With these estimates, our next post is dedicated to providing example calculations of quantiles or percentiles using these equations.

Reporting for Traton News,

Ingrid Ingram.