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The document frequency ${\mathrm{df}}_t$ of term $t$ is the number of documents in a specified collection that contain it at least once. Repeating a term many times inside one document contributes only one to its document frequency.^[ref]

For a collection $D$, write the count using an indicator function:Here ${\mathrm{tf}}_{t,d}$ is the raw term frequency. When an inverted index has one document record per containing document, the number of those records in the term's postings list equals ${\mathrm{df}}_t$. A record's position list can still contain several occurrences.

Consider red red boat, boat red red and blue boat. Red occurs four times but has document frequency two. Boat occurs only three times but has document frequency three. Thus the term with more total occurrences need not appear in more documents. Blue has document frequency one.

Adding another hundred red occurrences to the first document increases its term frequency without changing red's document frequency. Adding a new document containing red increases the latter by one. The collection boundary and preprocessing rules are therefore part of the statistic's definition. In information retrieval, this distinction lets scoring use evidence about how widely a term is distributed, rather than confusing repetition in one document with coverage of many documents.

A phrase query asks for documents containing a specified sequence of indexed terms in the required order. For an exact phrase query, the occurrences occupy consecutive positions. A document containing every requested word can still fail because its words occur in a different order or with intervening terms.^[ref]

Let $P_d(t)$ be the set of positions of term $t$ in document $d$. For query terms $t_0,\dots ,t_{m-1}$, an exact match exists whenA positional index stores the information needed to test these offsets. Repeated query terms need their own offsets even though their positions come from the same postings list.^[ref]

Number words from one, without removing any. Document 1 is red red boat, document 2 is boat red red, and document 3 is blue boat. The exact phrase query red boat matches only document 1, at positions 2–3. Document 2 has exactly the same term frequencies as document 1, but its boat occurs before both red occurrences. Intersecting document identifiers for the two terms would retain it incorrectly.

Phrase semantics depend on the indexing pipeline. If some words are removed, preserving their position gaps can prevent a false claim of adjacency. A search allowing intervening words is a different matching condition; quotation marks alone do not specify every implementation's preprocessing choices.

A postings list is the collection of records associated with one term in an inverted index. Each record identifies a document containing the term; optional fields include its term frequency in that document and, for a positional index, occurrence positions. The number of distinct document records is the document frequency, which differs from the total number of occurrences.^[ref]

For documents 1 red red boat, 2 blue boat, and 3 red train, the document identifiers in the red postings list are $1,3$. Its document frequency is two, although red occurs three times. The boat postings list contains $1,2$. Their intersection is $1$, answering red AND boat.

With identifiers in increasing order, compare the current identifiers and advance whichever is smaller; when equal, emit that identifier and advance both. Each advancement consumes a record, so intersecting lists of lengths $m,n$ costs $O(m+n)$ comparisons in this simple merge algorithm. Ordering gives a direct route from stored evidence to a query result without reading every document's text.^[ref]

The raw term frequency ${\mathrm{tf}}_{t,d}$ counts occurrences of term $t$ in one document $d$, after the chosen indexing preprocessing. It records how often the term appears there. Document frequency instead counts how many documents contain the term.^[ref]

For indexed terms $w_1,\dots ,w_L$, the indicator function givesIn red red boat, the raw term frequency of red is two and that of boat is one. The proportions $2/3$ and $1/3$ are normalized counts; they are not the raw counts in this convention.

Reordering the document to boat red red preserves both term frequencies but changes which phrase queries match. Counts therefore compress away ordering information. A positional index can retain the occurrence positions as well as the counts.^[ref]

A retrieval score can use term frequency as evidence that a document discusses a query term, but repetition alone does not establish relevance. Nor must doubling a raw count double a score: a scoring rule can deliberately reduce the effect of repeated occurrences. Whenever a transformed or normalized quantity is called term frequency, its formula should be specified.