= Character error rate
{title2=$\mathrm{CER}$}
The <character error rate> measures disagreement between recognized text and a reference transcription using <edit distance>. With unit-cost insertions $I$, deletions $D$ and substitutions $S$ in a minimum-cost alignment, and $N$ reference characters, the reference-length convention is
$$
\mathrm{CER}=\frac{I+D+S}{N},\qquad N>0.
$$
The alignment matters: counting mismatched positions without allowing edits can exaggerate the effect of one omitted character. This convention is used to evaluate <optical character recognition> against a specified transcription.<Quality Assurance in OCR-D by OCR-D>[Evaluation Metrics, Levenshtein Distance and Character Error Rate: reference-length formula and edit operations]{ref}
For reference `cat` and recognized `cut`, one substitution gives $\mathrm{CER}=1/3$. For reference `a` and recognized `aaaa`, three insertions give $\mathrm{CER}=3$, or $300\%$. The second result is valid: under this denominator, <character error rate> is not bounded by one. Clipping the result or using the longer string as denominator defines a different metric.<Quality Assurance in OCR-D by OCR-D>[Character Error Rate, discussion of rates beyond 100% and alternative normalizations]{ref}
Aggregation also changes the question. Suppose one one-character sample has one error and a hundred-character sample has none. The unweighted mean of their two rates is $50\%$, while pooling edits and reference lengths gives $1/101\approx0.99\%$. Reporting the aggregation rule prevents a short, difficult sample from being confused with a large fraction of the complete text.
The definition of character must be fixed too. <Unicode> code points and user-perceived <grapheme clusters> can give different lengths; <Unicode normalization>, spaces and punctuation can change both the alignment and its denominator. An empty reference needs an explicit convention because the displayed formula divides by zero. A score is meaningful only together with these evaluation choices.
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