Benford's Law (The Law of Anomalous Numbers)In a previous post we discussed the significance of price levels. Prior highs and lows are often revisited, sometimes more than once and act as resistance and support. Like a magnet these major and minor highs and lows appear to attract and repel price over time. With this information we drew trendlines creating channels in order to anticipate future price levels.
To view a growing price chart over a long period of time is impractical using an Arithmetic scale for price. For the most part, all analysis is done using a Logarithmic scale instead. This allows us to view the percent change uniformly. To understand the drawback of an arithmetic chart, consider how it may look for prices to change from 2 to 10 then 10 to 50. The move from 2 to 10 is only a difference of 8 units, whereas, 10 to 50 is 40 units. The percent change is the same, as is the rate of change assuming the same amount of time eclipsed. Logarithmic charts allow for a better gauge on momentum, but our understanding of numbers as they relate to each other may still be incorrect.
The Law of Anomalous Numbers or Benford's Law states that given a data set that does not have built in constraints or parameters to influence the output of data, the leading digit will follow a power law distribution. The larger the data set and orders of magnitude covered, the greater the conformity to this distribution. We may have assumed that the most likely outcome of leading digits in a data set would be uniform, but this would be incorrect.
As a percentage the expected distribution of leading digits using numbers in base 10 is as follows:
P(1) = 30.1%
P(2) = 17.6%
P(3) = 12.5%
P(4) = 9.7%
P(5) = 7.9%
P(6) = 6.7%
P(7) = 5.8%
P(8) = 5.1%
P(9) = 4.6%
It should be noted that the shape of this distribution holds regardless of the base of the number system used. Using base 10, the median leading digit of the distribution is 3.16. This means that half of all data points should fall between 1.00 and 3.16 while the other half fall between 3.16 and 9.99. Without this understanding, we might have otherwise expected 5.5 to be the median leading digit as in the case of a uniform distribution.
The chart above shows equal spacing between price levels on a logarithmic chart. They can be found by taking the square root of 10 (= 3.16), then taking the square root of 3.16 (= 1.78) then cubing 1.78 (= 5.62). Midpoints play a significant role in my analysis and is the basis for using these numbers.
1.0 black
1.78 light blue
3.16 red
5.62 dark blue
This organizes the expected distribution into quarters. Over time the actual distribution of leading digits observed should gravitate towards the distribution of Benford's Law.
While I have had my doubts about the validity of using these numbers as price levels, I couldn't keep this to myself given its apparent relevance.
