Power-Law Distributions in Empirical Data

Aaron Clauset, Cosma Rohilla Shalizi, M. E. J. Newman · SIAM Review / arXiv · 2009 · statistics · 10.1137/070710111 · record clauset2009-power-law-data
tag layer · index · concepts · authors · עברית

Where this record's text came from. faceted 2026-08 by reading the locally held text (pdftext); every evidence string verified as an exact substring of that text before merging. The method tag is imperfect: the paper is a statistical-methods paper whose validation is simulation on synthetic data (which is what the quoted sentence supports), but it also performs a large empirical reanalysis of 24 published data sets, which the offered vocabulary has no term for. Note the quoted contradicts evidence reads '3.16 0.06' because the PDF extraction lost the plus-minus sign, which is moved to the end of the sentence in the source text; the quote is verbatim as the article text stands. The paper also critiques the standard practice of fitting power laws by least squares on log-log histograms, but that is a critique of a general practice rather than of a named study, so it is not recorded as a claim_relation. 2026-08-14, 2026-08-16 and 2026-08-25 schema-0.3 passes: every quote verified as an exact substring of the staged text and anchored at merge time; concept terms verified present; every edge endpoint an existing node.

How to read this page. The article itself is not reproduced here — this paper is not cleared for redistribution, so what follows is the tag layer alone. The evidence sentences are withheld with the article. The table at the foot lists every tag the record carries; each row names the facet it belongs to, and the whose words column says whether the tag records the paper's own wording or the tagger's sentence about the paper. Every tag links to each other paper in the corpus carrying it. The machine-readable edition of the same thing is the TEI file beside this one, where the tags live in standOff, and the article is at its publisher via the link below. When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.

Other fields define these differently the same word, defined another way in another discipline
sample-size 5 fields · null-hypothesis 3 fields · standard-deviation 3 fields · confidence-interval 2 fields · simulation 2 fields · statistical-significance 2 fields · and 2 more →

Tags on this paper

Every tag the record carries, each linking to every other paper in the corpus that carries it. A tag appears here whether or not its words occur in the article — the spans below are only the ones a sentence in this paper licenses. Of the 9 concept tags on this paper, 6 are grounded in a quoted sentence.

power law maximum likelihood goodness of fit heavy tail model selection scaling parameter xmin likelihood ratio test log-normal the lower bound of the power-law range is chosen by minimising the Kolmogorov-Smirnov distance between the empirical distribution and the best-fit model above that bound - too high a bound shrinks the data and too low a one includes non-power-law points, so the minimum lies between each synthetic data set is scored against ITS OWN best-fit power law rather than against the distribution it was drawn from, which is what makes the resulting p-value unbiased: the synthetic sets undergo exactly the calculation the real data underwent, fitting included synthetic data are generated semiparametrically - drawn from the fitted power law with probability equal to the observed tail fraction, and otherwise resampled from the observed points below the cut-off - so the synthetic sets share the empirical body while following the power law in the tail the test distribution is constructed to be hard rather than convenient: it is exponential below the cut-off and a power law above, and its slope is CONTINUOUS at the cut-off, so it deviates only gently from the power law below that point - the authors' word is challenging uncertainty on the fitted parameters is obtained by a nonparametric bootstrap - drawing a fresh sequence of n points uniformly at random from the original data and re-estimating both the lower bound and the scaling parameter on each surrogate set the alternative goodness-of-fit statistics are tested rather than assumed: reweighted KS and Kuiper behave like plain KS, while Anderson-Darling returns cut-offs an order of magnitude too large, and the cost of that conservatism is stated as fewer tail observations, larger error on the scaling parameter and less ability to validate the model the number of synthetic data sets is fixed by the precision wanted in the p-value rather than by convention, derived from the test's expected worst-case performance the direction of the p-value is inverted relative to its ordinary use: the quantity being tested is the model the authors want to assess rather than a null, so a HIGH p-value is the favourable one and a low value rules the power law out power-law distribution exponent goodness-of-fit Kolmogorov-Smirnov heavy-tailed tail empirical data tail of the distribution lower bound xmin goodness-of-fit statistic quantity x Bayesian confidence-interval confidence interval hypothesis-testing hypothesis test mean Mean of null-hypothesis null hypothesis p-value regression linear regression sample-size sample size standard-deviation standard deviation standard-error standard error statistical-significance statistically significant variance simulation

Legend

relation to other work	`claim_relations`	how this paper stands to another: agrees, contradicts, extends, qualifies
claim type	`claim_type`	the shape of the claim: existence, mechanism, method, critique, prediction, null
concept edge	`concept_edges`	an asserted relation between two concepts, with the sentence asserting it
concept	`concepts`	an idea the paper works with, quoted where the paper uses it
content tag	`content_tags`	a finding, definition, quantity, statistic or variable named in the paper
control	`controls`	an experiment run to eliminate a rival explanation
limitation	`limitations`	what the paper says its evidence cannot support
mechanism	`mechanisms`	how the reported effect is said to come about
method	`method`	how the paper produced its evidence
phenomenon	`phenomenon`	the effect or regularity under study
sense	`senses`	a distinct use of a term, quoted where the paper fixes it
transfer	`transfers`	a method or concept carried in from another field

The paper

The full text is not reproduced here. This article is not cleared for redistribution, so the page carries the tag layer only. Its evidence sentences are withheld; the grounding status remains with each tag, and the article itself is at https://doi.org/10.1137/070710111. The reading view exists for the papers whose licence permits it.

Every tag on this page

claim_relations 3 · claim_type 1 · concept_edges 3 · concepts 6 · content_tags 28 · controls 3 · limitations 12 · mechanisms 8 · method 1 · phenomenon 1 · senses 3 · transfers 1

Relations on this page carry a quotation for the claim. Where a relation was separately evidenced — and so tested against the vocabulary for its type — the row says so; none of 3 relations here carry that mark. The rest are the tagger’s reading of the sentence quoted beside them.

facet	tag	field	conf.	whose words
claim_type	method Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_type`		the paper's words
phenomenon	power-law distributions in empirical data Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`phenomenon`		the paper's words
method	simulation Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: simulation	`method`		the paper's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: the lower bound of the power-law range is chosen by minimising the Kolmogorov-Smirnov distance between the empirical distribution and the best-fit model above that bound - too high a bound shrinks the data and too low a one includes non-power-law points, so the minimum lies between	`mechanisms[0]`	0.85	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: each synthetic data set is scored against ITS OWN best-fit power law rather than against the distribution it was drawn from, which is what makes the resulting p-value unbiased: the synthetic sets undergo exactly the calculation the real data underwent, fitting included	`mechanisms[1]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: synthetic data are generated semiparametrically - drawn from the fitted power law with probability equal to the observed tail fraction, and otherwise resampled from the observed points below the cut-off - so the synthetic sets share the empirical body while following the power law in the tail	`mechanisms[2]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: the test distribution is constructed to be hard rather than convenient: it is exponential below the cut-off and a power law above, and its slope is CONTINUOUS at the cut-off, so it deviates only gently from the power law below that point - the authors' word is challenging	`mechanisms[3]`	0.85	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: uncertainty on the fitted parameters is obtained by a nonparametric bootstrap - drawing a fresh sequence of n points uniformly at random from the original data and re-estimating both the lower bound and the scaling parameter on each surrogate set	`mechanisms[4]`	0.85	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: the alternative goodness-of-fit statistics are tested rather than assumed: reweighted KS and Kuiper behave like plain KS, while Anderson-Darling returns cut-offs an order of magnitude too large, and the cost of that conservatism is stated as fewer tail observations, larger error on the scaling parameter and less ability to validate the model	`mechanisms[5]`	0.85	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: 2026-08-16: THIS QUOTE IS TRUNCATED BECAUSE THE FORMULA IT LEADS TO IS UNREADABLE IN THE STAGED TEXT. The pdftext extraction flattens the expression for the required number of synthetic data sets out of order - the staged body reads 'at least 1 2 synthetic data sets' with '4 e-' appearing separately - so the rule the sentence states cannot be quoted verbatim without quoting nonsense. The span stops at the last readable words. The paper's own worked example (accuracy to two decimal digits requiring about 2500 sets) is likewise in the damaged region and is deliberately NOT reproduced here, because I would be typing it rather than cutting it. also in: the number of synthetic data sets is fixed by the precision wanted in the p-value rather than by convention, derived from the test's expected worst-case performance	`mechanisms[6]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: the direction of the p-value is inverted relative to its ordinary use: the quantity being tested is the model the authors want to assess rather than a null, so a HIGH p-value is the favourable one and a low value rules the power law out	`mechanisms[7]`	0.85	the tagger's words
limitations	one can never confirm a power law, only find data consistent with it Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: every power-law claim anywhere: Quotation withheld - the hypothesis cannot be confirmed	`limitations[0]`		the tagger's words
limitations	few empirical phenomena obey power laws across the whole range of x Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the range over which any fitted power law holds: Quotation withheld	`limitations[1]`		the tagger's words
limitations	Known sampling biases in several of the reanalysed data sets were left uncorrected. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the reanalyses in this paper: WE HAVE NOT ATTEMPTED TO CORRECT ANY BIASES IN OUR ANALYSIS	`limitations[2]`		the tagger's words
limitations	Many of the reanalysed data sets are partial samples of larger populations. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the datasets reanalysed: MANY OF THESE DATA SETS ARE ONLY SUBSETS OF MUCH LARGER ENTITIES (SUCH AS THE WEB SITES, WHICH ARE ONLY A SMALL FRACTION OF THE ENTIRE WEB)	`limitations[3]`		the tagger's words
limitations	The framework is presented only for the power law; no extension to other distributional forms is given. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the applicability of the framework: IT COULD ALSO BE EXTENDED TO OTHER, NON-POWER-LAW DISTRIBUTIONS ... ALTHOUGH WE HAVE NOT GIVEN SUCH AN EXTENSION HERE	`limitations[4]`		the tagger's words
limitations	The procedure often cannot separate power-law from log-normal behaviour. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: any claim that a power law rather than a log-normal has been found: IT IS EXTREMELY DIFFICULT TO TELL THE DIFFERENCE BETWEEN LOG-NORMAL AND POWER-LAW BEHAVIOR	`limitations[5]`		the tagger's words
limitations	Two model forms indistinguishable on the observed data can give enormously different extrapolations into the tail, which is where such fits are usually applied. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: extrapolating any of these fits to large rare events: the paper states conclusions from different fitted forms can differ enormously while being indistinguishable on the data	`limitations[6]`	1.0	the tagger's words
limitations	Seven of the twenty-four data sets previously conjectured to be power laws are firmly ruled out. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: any of the seven named data sets as power-law distributed: HTTP connections, earthquakes, web links, fires, wealth, web hits and the metabolic network are ruled out here	`limitations[7]`	1.0	the tagger's words
limitations	Two published scaling parameters for the same citation data disagree with each other and with this paper's maximum-likelihood estimate. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: a published scaling exponent for a data set: for these citation data three different values are in print, and the paper says the earlier two are incompatible with the maximum-likelihood estimate	`limitations[8]`	1.0	the tagger's words
limitations	The accept/reject verdict rests on a stated p-value threshold of 0.1. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the power-law verdicts in this paper: they follow a p-value threshold of 0.1, and cases near it are decided by that choice	`limitations[9]`	1.0	the tagger's words
limitations	The range fitted is chosen, and that choice controls how much data the test sees. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the sample size behind any of these fits: xmin is estimated, and it determines how many points enter the test	`limitations[10]`	1.0	the tagger's words
limitations	Errors run in both directions on small samples: a power law can look like a good fit to non-power-law data, and true power laws cannot be separated from alternatives. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: a power-law fit on a small sample in either direction: the paper states both false acceptance and failure to discriminate occur there	`limitations[11]`	1.0	the tagger's words
transfers	the fitting procedure is deliberately applied across twenty-four data sets from unrelated branches of enquiry, so that the method rather than any one field's data is what is tested Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`transfers[0]`	0.9	the tagger's words
controls	a goodness-of-fit p-value computed for the power law AND for each plausible competing distribution, and the two compared Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. eliminates: that the power law fits because heavy-tailed data fits almost any heavy-tailed form - a log-normal or a stretched exponential may fit as well or better, and only comparing them shows which · outcome (stated by the tagger, no quotation): held	`controls[0]`	0.95	the tagger's words
controls	an explicit rejection rule fixed in advance: p greater than 0.1 keeps the power-law hypothesis, otherwise it is rejected Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. eliminates: that the decision to accept a power law is made after seeing how good the fit looks · outcome (stated by the tagger, no quotation): held	`controls[1]`	0.95	the tagger's words
controls	the whole procedure was run on SYNTHETIC data drawn from known distributions Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. eliminates: that the method itself is wrong - synthetic data has a known answer, so the procedure can be checked against ground truth before it is trusted on real data · outcome (stated by the tagger, no quotation): held	`controls[2]`	0.95	the tagger's words
concepts	power law Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: power law	`concepts[0]`		the paper's words
concepts	maximum likelihood Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: maximum likelihood	`concepts[1]`		the paper's words
concepts	scaling parameter Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: scaling parameter	`concepts[5]`		the paper's words
concepts	xmin Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: xmin	`concepts[6]`		the paper's words
concepts	likelihood ratio test Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: likelihood ratio test	`concepts[7]`		the paper's words
concepts	log-normal Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: log-normal	`concepts[8]`		the paper's words
senses	power law - a hypothesis one can only fail to reject; the paper says an observed quantity can rarely if ever be certified as power-law distributed Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[0]`	1.0	the tagger's words
senses	power law - a property of the tail only, holding above a chosen minimum rather than across the whole range Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[1]`	1.0	the tagger's words
senses	scaling parameter - a quantity whose published values for one data set disagree, because they depend on the estimator used Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[2]`	1.0	the tagger's words
claim_relations	contradicts → previously claimed power laws in the literature Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[0]`		the tagger's words
claim_relations	contradicts → previously reported scaling parameters for the citation distribution of [48], namely 2.9 in [62] and 2.5 in [33] Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[1]`		the tagger's words
claim_relations	builds-on → Ijiri and Simon [28] and Mitzenmacher [42] on the limits of characterising empirical distributions Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[2]`		the tagger's words
concept_edges	power law —contrasted-with→ log-normal Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the p-value supports a classification as power-law, log-normal or undecided; contrasted-with is loose. also in: power law · log-normal	`concept_edges[0]`		the tagger's words
concept_edges	scaling parameter —operationalised-as→ maximum likelihood Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span compares reported scaling parameters with 'our maximum likelihood figure'; operationalised-as is loose. also in: scaling parameter · maximum likelihood	`concept_edges[1]`		the tagger's words
concept_edges	xmin —part-of→ power law Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 also in: xmin · power law	`concept_edges[2]`		the tagger's words
content_tags	24 — real-world data sets tested for power-law fit Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.quantities[0]`		the tagger's words
content_tags	2 < α < 3 — typical range of the scaling parameter Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.quantities[1]`		the tagger's words
content_tags	8226 — average US city/town/village population (2000 Census) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.quantities[2]`		the tagger's words
content_tags	180 — typical adult male American height (near-Gaussian example) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.quantities[3]`		the tagger's words
content_tags	2 — largest deviations in a well-behaved distribution Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.quantities[4]`		the tagger's words
content_tags	least-squares fitting yields substantially inaccurate power-law parameters Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[0]`		the tagger's words
content_tags	their framework combines maximum likelihood with KS goodness-of-fit Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[1]`		the tagger's words
content_tags	some conjectured power laws are ruled out by the data Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[2]`		the tagger's words
content_tags	power laws usually hold only above a minimum value xmin Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[3]`		the tagger's words
content_tags	scaling parameter Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: scaling parameter	`content_tags.variables[0]`		the tagger's words
content_tags	lower bound xmin Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: lower bound xmin	`content_tags.variables[1]`		the tagger's words
content_tags	goodness-of-fit statistic Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: goodness-of-fit statistic	`content_tags.variables[2]`		the tagger's words
content_tags	quantity x Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: quantity x	`content_tags.variables[3]`		the tagger's words
content_tags	power law Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: power law	`content_tags.definitions[0]`		the tagger's words
content_tags	scaling parameter Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: scaling parameter	`content_tags.definitions[1]`		the tagger's words
content_tags	tail of the distribution Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: tail of the distribution	`content_tags.definitions[2]`		the tagger's words
content_tags	Bayesian Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Bayesian	`content_tags.statistics[0]`		the tagger's words
content_tags	confidence-interval Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: confidence-interval	`content_tags.statistics[1]`		the tagger's words
content_tags	hypothesis-testing Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: hypothesis-testing	`content_tags.statistics[3]`		the tagger's words
content_tags	mean Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: mean	`content_tags.statistics[4]`		the tagger's words
content_tags	null-hypothesis Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: null-hypothesis	`content_tags.statistics[5]`		the tagger's words
content_tags	p-value Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: p-value	`content_tags.statistics[6]`		the tagger's words
content_tags	regression Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: regression	`content_tags.statistics[7]`		the tagger's words
content_tags	sample-size Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: QUOTE REPAIRED: the extractor split a hyphen-space and the quote was recorded faithfully with the damage in it, which made it unanchorable. The repaired string is an EXACT SUBSTRING of the staged text - that is why it was accepted, not because a rule produced it. The text as originally recorded is kept in evidence as extracted. also in: sample-size	`content_tags.statistics[8]`		the tagger's words
content_tags	standard-deviation Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: standard-deviation	`content_tags.statistics[9]`		the tagger's words
content_tags	standard-error Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. also in: standard-error	`content_tags.statistics[10]`		the tagger's words
content_tags	statistical-significance Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: statistical-significance	`content_tags.statistics[11]`		the tagger's words
content_tags	variance Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: variance	`content_tags.statistics[12]`		the tagger's words

Interpretive choices

Tags that could not be placed in the text:

none — every tag was located in the text

What was deliberately not tagged, and why, is recorded per paper by the tagger. Where a paper has no such record, this page says so rather than implying that nothing was declined.