Settlement Rank Size Index

Fit log size versus log rank for supplied comparable settlement sizes and report primacy ratio; not a settlement hierarchy diagnosis or a mapped population estimator.

Description

Rank–size analysis compares settlement sizes after ordering them largest first.

Rank–size analysis compares settlement sizes after ordering them largest first. A log–log slope and largest-to-second ratio describe different aspects of the supplied size distribution.^[1]

Fit log size versus log rank for supplied comparable settlement sizes and report primacy ratio; not a settlement hierarchy diagnosis or a mapped population estimator.

When to use Settlement Rank Size Index

Use this tool to fit log size versus log rank for supplied comparable settlement sizes and report primacy ratio; not a settlement hierarchy diagnosis or a mapped population estimator.

Prepare the artifact, site or survey measurements and any independently established calibration or model choices. Keep paired observations aligned and confirm units and coordinate conventions.

How to use Settlement Rank Size Index and understand its results

Prepare the artifact, site or survey measurements and any independently established calibration or model choices. Keep paired observations aligned and confirm units and coordinate conventions.
Enter settlementSizes using the field table and calculation rules, then run the tool.
Read sizeRanks, logSizeLogRankSlope, logFitRSquared, observedLargestToSecondRatio. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
settlementSizes	Settlement sizes	list of number	Required

Supply at least two positive comparable settlement sizes. Sort descending, assign ranks starting at one and fit log size against log rank by ordinary least squares. Return ranks, slope, R² and observed largest-to-second ratio. Equal sizes are valid and produce slope zero with the helper's constant-response R² of one.

The least-squares fit needs at least two aligned observations and a varying independent variable. Where R² is returned, the helper assigns 1 for a constant dependent variable.

ln(size) = intercept + slope ln(rank); primacy ratio = largest/second largest

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
sizeRanks	Size ranks	list of number
logSizeLogRankSlope	Log size log rank slope	number
logFitRSquared	Log fit r squared	number
observedLargestToSecondRatio	Observed largest to second ratio	number

json

{
  "input": {
    "settlementSizes": [
      100,
      50,
      25
    ]
  },
  "output": {
    "sizeRanks": [
      1,
      2,
      3
    ],
    "logSizeLogRankSlope": -1.2336619422521746,
    "logFitRSquared": 0.9776539585182622,
    "observedLargestToSecondRatio": 2
  }
}

Limitations and assumptions

Size units and settlement contemporaneity must be consistent. The tool does not map sites, infer population or diagnose a political hierarchy from the fit.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Population Estimate From Roofed Area: Screen occupied residential roofed-area population scenarios with building-specific use/occupancy and caller area-per-person bounds; not census counts or inferred contemporaneity. For a complementary task, use Seriation Frequency Ordering Tool: Produce deterministic type-frequency centroid seriation on a supplied type axis, input order breaking ties; not an optimal unimodal or independently dated ordering.

References

Settlement Rank Size Index — tool.io

Similar or alternative tools

Population Estimate From Roofed Area
Screen occupied residential roofed-area population scenarios with building-specific use/occupancy and caller area-per-person bounds; not census counts or inferred contemporaneity.
Seriation Frequency Ordering Tool
Produce deterministic type-frequency centroid seriation on a supplied type axis, input order breaking ties; not an optimal unimodal or independently dated ordering.