1995issue C021
Constructing mean-deviation histograms and price quantiles
Editorial interpretation: before any forecast is discussed, rebuild three checkable objects from ordered prices, a mean as location, a mean-deviation scale, and a bin-wise empirical quantile ladder. The archive facts walk that construction on two short lists a reader can recompute by hand.
- In the eight-price example the mean is 4.125, obtained by dividing the sum 33 by the observation count 8.
- Each mean deviation in that example is the observation minus 4.125; the signed gaps run from -3.123 to 2.875, and the summarized scale is 1.763.
- A second list of 14 prices is counted into seven integer bins from 1 to 7, with frequencies 1, 2, 2, 2, 3, 2, and 2.
- Dividing each running frequency total by 14 produces an empirical cumulative probability of 0.5 at bin 4 and 1 at bin 7.
Three objects before any forecast
Editorial interpretation: the lesson is the baseline itself. Before any forecast is discussed, rebuild three objects from ordered prices and keep them checkable by hand: a location, a mean-deviation scale, and a bin-wise empirical quantile ladder.
The archive facts walk that construction on two short lists. They do not present a forecast.
The mean as a location
The mean is the sum of the ordered observations divided by how many observations are in the list. In the eight-price example the mean is 4.125, obtained by dividing the sum 33 by the observation count 8. That single figure is the location later gaps are measured from.
Mean deviation as a scale
Mean deviation is the signed gap between each ordered observation and the series mean, then summarized by averaging the squared gaps and taking the square root. Each mean deviation in that example is the observation minus 4.125, and the signed gaps run from -3.123 to 2.875.
The eight squared mean deviations sum to 24.875. Dividing by 8 yields 3.109, and the square root of that quotient is 1.763. That last figure is the scale of the same list.
A histogram of integer prices
A price-change histogram is a count of how often each value in a defined integer range appears in an ordered price list. A second list of 14 prices is turned into a histogram by counting occurrences in seven integer bins that cover the observed range from 1 to 7.
A bin is one slot on the histogram axis that collects every observation equal to a single value in the observed range. Those seven bins have frequencies 1, 2, 2, 2, 3, 2, and 2, which together equal the 14 observations.
Empirical probabilities on the finished histogram
Quantile analysis is reading an empirical probability from the running share of observations at or below each bin of a finished histogram. The empirical cumulative probability at a bin is defined as that bin’s cumulative frequency divided by the number of prices. That running frequency, divided by the number of observations, is the empirical share of the sample at or below that bin.
Dividing each running frequency total by 14 produces an empirical cumulative probability of 0.5 at bin 4 and 1 at bin 7. Those two marks let a reader recompute the quantile ladder on this histogram by hand.
Where the construction stops
Editorial interpretation: once the mean, the mean-deviation scale, and the bin-wise cumulative probabilities can be recomputed from the lists, the construction is complete. The archive facts stop there.