To illustrate the robust nature of the median absolute deviation, consider the following hypothetical dataset of 250 payments.
This payment data does not exhibit the normal distribution pattern which is not uncommon, as discussed above. Of particular importance is the small number of payment observations in which the company took significantly longer than usual to pay the invoice. This dataset produces a median of 33 due to the large cluster of payments around 30 days and an average of 64 due to the influence of the invoices which were paid after a long number of days. The median absolute deviation for the payments is 4 days due to the tight grouping of payments around the median. However, the standard deviation for the payments is 76 days due to the influence of the long dated payments.
From a visual standpoint, it is apparent that the historical payments fell overwhelmingly between 30 and 35 days with a small number of payments falling outside of that period. Utilizing a range of one median absolute deviation on each side of the median would yield an ordinary course range between 29 and 37 days and would indicate that approximately 65 percent of the historical payments can be said to represent an ordinary range. It is important to note that this percentage is comparable to the 68 percent of the population captured by a one standard deviation range for the normal distribution presented in Figure 1.
However, utilizing one standard deviation on each side of the mean to determine the ordinary course of historical payments would yield a range between 0 and 140 days. This methodology would indicate that approximately 92 percent of the historical payments represent an ordinary range.
Now that the ordinary course ranges defined by the median absolute deviation and standard deviation methodologies have been examined, let us consider the effect of the results of each methodology when applied to the following hypothetical dataset consisting of 50 payments in the 90 day preference period preceding a bankruptcy filing.
A visual analysis of this data once again reveals that there are several payments clustered around 33 days, as was the case in the historical period. However, there is no large spike in the frequency of payments around this point as there was in Figure 3 and the remaining payments show that only 1 payment was made between 40 and 60 days with the remaining payments falling relatively evenly from 60 to 110 days.
Utilizing the standard deviation method discussed above, all of the payments in the preference would meet the definition of ordinary course since they fall within the range of zero to 140 days. If the median absolute deviation methodology is employed, all payments over 37 days would be considered outside of the ordinary course of business based on the payment history between the parties.
To understand how the magnitude of the difference between the ordinary course indications under the two different methods, it is often useful to express the standard deviation range in terms of median absolute deviations. In this case, payments of 60 days would represent a variance of 6.75 median absolute deviations from the historical median observation of 33. A payment of 100 days would represent a variance of 16.75 median absolute deviations and a payment of 140 days would represent a variance of 26.75 median absolute deviations. Therefore, the influence of outliers in the standard deviation methodology becomes apparent when expressed in terms of median absolute deviations.
Conclusion
While many sophisticated statistical analyses have been developed by financial experts to support their opinions of the ordinary course of dealings between parties to a preference dispute, the standard deviation methodology has often been utilized due to its simplicity and familiarity when presented in a legal proceeding. However, due to the sensitivity of the standard deviation calculation to outliers, the financial expert is often forced to subjectively remove these outliers, which can create the appearance of bias. If the outliers are not removed, as demonstrated above, the ordinary course indication produced by the standard deviation methodology can appear at odds with the data analyzed.
If a dataset is normally distributed as presented in Figure 1, the mean and median will be very similar to one another and the median absolute deviation will typically also produce similar results. However, due to several factors specific to common payment practices, payment data does not typically exhibit a normal distribution pattern. Therefore, the median absolute deviation methodology can be employed by financial experts to present an easily understandable approach to triers of fact that does not require the subjective removal of outliers.
The Pillowtex decision once again thrusts the subjective issue of the ordinary course of payments between the parties in the historical period into the spotlight. As is the case with all financial analyses, the most appropriate methodology will depend on the facts and circumstances of each project. By utilizing the median absolute deviation, a financial expert can eliminate the need to eliminate outlier data from a population of payment data and can also test the number of implied median absolute deviations represented in the ordinary course range indicated by standard deviation or other methodology.
