How can I group numerical data into naturally forming "brackets"? (e.g. income)

Question

The following describes what I'm trying to accomplish, but it's possible an alternative problem statement can describe my goal:

I want to

1. divide the following numbers into groups where the variances of the numbers within each group are not too large, and the differences between the averages of groups are not too small

2. compare the distribution obtained in the end with the "perfect" ones and see how "different" it is from being perfect.

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Layman's explanation of goal

I'm trying to calculate income distribution, and determine the "income brackets" each population is in. The income bracket is supposed to be self-adjusting based on the input data.

My goal is to ultimately measure or calculate the difference between the income brackets. I assume there will be many brackets, and want to see how far "apart" each tier is.

Here is a sample of hourly income for a sample set of a population of 20, and a total income of 3587:

Population= 10                   pop=2   population=5              population =3
10, 11,13,14,14,14,14,14,15,20,  40,50  ,90,91,92,93,94      999,999,900 

How can I use mathematical concepts to group, sort and analyze data that acts like income distribution over a given population?

At the end of the calculation, I want to determine tiered income distribution, where a perfect distribution would look (something) like this

(each person makes $10 more per hour than the previous; total is 3587)
89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 219, 229, 239, 249, 259, 269, 279

or this:

(evenly distributed groups of people make the same per hour) 
(gaps between income groups is consistent and not "too far")
(income total is 3587)
99 99 99   129 129 129   159 159 159   199 199 199  229 229 229  269 269 269 

Question

How should I analyze the population groups, and measure the gap in a way that will tell me how much is needed to make it more like the last two model sets listed above?

Answer 1

Cluster analysis with a single variable makes perfect sense whenever there is some dimension along which values can be arranged. This could be a measurement scale, time or space.

Given ordered data on some measurement scale, there might be interest in looking for relative breaks within a frequency distribution (antimodes, in one terminology).

Note of caution: However, breaks defining bins that are, or that might seem, arbitrary are widely shunned in several areas of statistical science, and there is widespread and marked preference for binning with equal intervals, and very often for avoiding binning altogether when possible. This is partly a matter of taste, partly one of convention: practices have shifted as it becomes easier to store datasets in their entirety.

A time series could be divided into spells, epochs, periods, whatever, ideally with relatively small differences within subseries and relatively large differences between subseries. The same problem arises for space whenever a single spatial dimension (horizontal or vertical) is to be subdivided. In geological and other sciences, this is often studied under the heading of zonation.

Note that any formal clustering should always be accompanied by appropriate plotting of the data (for example, using a dot or quantile or line plot), which indeed may make clear either that breaks are obvious (so that formal clustering is merely decorative) or that convincing breaks do not exist (so that formal clustering may be pointless).

Consider a toy example of values ordered by magnitude:

    14 15 16 23 24 25 56 57 58 

where it is evident that a three-group clustering

    14 15 16 | 23 24 25 | 56 57 58 

is sensible. Whether the ordering is on the values themselves, or in time or in space, the data can always be laid out in one dimension, which gives special structure to the problem. Thus, although more general clustering methods can be used, that special structure ideally should be exploited. $k$ groups devised for $n$ values are defined by placing $k - 1$ markers (in the example above, $k - 1 = 2$); there are $n - 1$ possible places to put them. There are thus $n - 1 \choose k - 1$ possible clusterings. However, if $k$ is free to vary, then the total number of possible clusterings is $2^{n - 1}$, as each value can be in the same group as each neighbour, or not. For even modest $n$, that is a large number.

The problem can be made precise (Fisher 1958; Hartigan 1975) by placing markers to minimise, for a given number of groups, the

$$\text{sum over groups of variability around group centres}.$$

A sum of squared deviations from group means will spring to mind as the most obvious possibility. Sum of absolute deviations from group medians, and other measures, might well be entertained.

Hartigan (1975) showed how a dynamic programming approach makes such computation straightforward and presented Fortran code. A Stata implementation (Cox 2007) is group1d to be installed from SSC.

Cox, N.J. 2007. GROUP1D: Stata module for grouping or clustering in one dimension. http://ideas.repec.org/c/boc/bocode/s456844.html

Fisher, W.D. 1958. On grouping for maximum homogeneity. Journal, American Statistical Association 53: 789-98.

Hartigan, J.A. 1975. Clustering algorithms. New York: John Wiley. Ch.6.

Postscript This approach seems to match the first part of the specific question. I have pitched it generally because I think the formulation is of some general interest (and because it was easy for me to recycle part of the documentation of Cox 2007). But if the specific goal is to compare an income distribution with a reference uniform distribution, I don't see that binning has any part to play at all. That is a standard problem in economics for which Lorenz curves and inequality measures are the starting points. In essence, you can compare quantile to quantile or percent point to percent point.

Answer 2

Take a look for Jenks Natural Break:

https://en.wikipedia.org/wiki/Jenks_natural_breaks_optimization

I think it's what you need, and there are implementations in many languages.