Smart Community Arts and Technologies Introduces the Consumption Formula ...
Invented by Smart Community owner, John G. Freeman, copyright 1992 Figure 1

What it does. The P function fits a curve to consumption. The letter P stands for either percentage or proportional, because the meaning of the formula is in its relationship to total consumption or conversion. Additional information regarding the consumption process can be determined once the P function has been applied. Those in the sales analysis part of business know how important being able to fit a curve to any process can be, as it opens the door to being able to make further analyses and predictions. 

How it is different than statistical formulas that normally would be used. Most formulas determine a linear trend. The more sophisticated formulas are polynomials that are only accurate within very limited ranges, not reliable for the whole consumption picture. The P function is so accurate that it provides two otherwise unavailable functions: One is that the entire consumption, from start to end, is accurately mapped (curve-fitted). The other is that the fundamental standard deviation used to build the curve could also be used as a corollary to link inconsistencies in the consumption factors with other consumption curves. For example, the corollation among car paint deterioration and inconsistencies in the paint properties (due to inaccurate mixing) could be made.

Applicability. This formula is useful whereever something is or could be completely converted into another form or category (i.e., consumed in its original state) by a quantifiable cause. Remarkably, the determination of the data used to apply the formula is simple. The largest power of this formula is in the fact that it can be used to predict the change of something that is comprised of inhomogeneous properties, such as found in the reasons for most real events. A factor of its applicability is its versatility, which I describe separately.

• In marketing, one of the factors in selling is the consumer's cyclic usage of any product. In other words, how quickly a consumer returns to make another purchase. The factors that determine the return rate for a product are inhomogeneous. There are delays and accelerations throughout all the factors that determine the purchase rate, two of which, for examples, are the consumption rate and the shelf exposure.  If a store wants to predict when any number of 100 items of a product would be sold, this formula would provide the reference curve, based on previous sales experience of that item. The quantifiable cause (sometimes called load) would be time itself and would be the x in the formula. 
• In simple strength quality prediction, this formula would fit a curve to breakage with respect to application frequency. With vehicle tires, the mileage would be the load and the number of dead tires at each mileage would provide the P values for the fitted curve. Ideally, the data collection would be complete if a tracking of all tires were kept.

Versatility. Sometimes, all that would be required to know in order to determine that this curve-fitting tool would be usable would be that your data's curve is a result of cumulative data. This formula will still work with such a simple understanding of your data. In that situation, this versatility allows the formula to be a tool to research further the nature of the data you have.

The components.

• P is the percentage (when factored by 100/max) or portion that some item would be consumed.
• x is the amount of the component that is causing the consumption. 
• d is the bin width between amounts of the consuming component.
• s is the standard deviation determined by the distribution of the difference in the number of events of consumption at each consumption strength interval (d).
• d is a variable that, in the rare event the bin width is not known, can be decided with a zeroing process. When bin widths are equal, such as with a normal independent variable number line (e.g., x = 1, 2, 3, etc.).
• d equals the bin width, d, which simplifies the applicability of the formula.
• µ_x is the mean of the difference frequencies, which I will explain.

How to Build the P Function.

Figure 2: Frequencies of Consumption (or Conversion) are cumulative and S -shaped

                                    x   x
                            x   x   x   x
                        x   x   x   x   x
                        x   x   x   x   x
                    x   x   x   x   x   x
                    x   x   x   x   x   x
                x   x   x   x   x   x   x
            x   x   x   x   x   x   x   x
    |___|___|___|___|___|___|___|___|___|_
    0   1   2   3   4   5   6   7   8   9
Diff:   0   1   1   2   2   1   0   1   0
                                            (Difference Frequencies, Bin_i-Bin_i-1 )

Calculate Standard Deviation

• Mean, µ_x , of the Difference Frequencies = (1(2)+1(3)+2(4)+2(5)+1(6)+0(7)+1(8)+0(9))/8 = 37/8 = 4.63
• Differences from the mean are 2.63, 1.63, 0.63, 0.63, -0.37, -0.37, -1.37, -3.37
• Variance, s² = (6.92 + 2.66 + 0.40 + 0.40 + 0.14 + 0.14 + 1.88 + 11.4)/8 = 23.9/8 = 3.00
• Standard Deviation, s, = v3.00 = 1.73.
• Accept that  d equals the bin width, d.
• The first summation term equals 8, the total possible events per load point.
• d, the bin width, equals 1.
• Replace the variables with the known values.
• Calculate for any load value (x, the independent variable).