A search result can be accurate and incomplete
Imagine searching a document collection for reports about bicycle commuting. A search tool returns ten documents, and nine are relevant. The results look excellent. But suppose the collection contains 100 relevant documents. The tool found only nine of them. Its precision is 90%, while its recall is 9%.
Those figures describe the same performance from different directions. Someone hoping to find one useful report may be satisfied with that result. Someone reviewing every available report would face a different problem. This is why the word “accurate,” without a description of the task, can leave out the thing a reader most needs to know.
The formulas, with the denominators made visible
In a binary classification task, a true positive is a case correctly identified as positive. A false positive is incorrectly flagged. A false negative is a positive case that was missed. “Positive” names the class being counted; it does not mean desirable or beneficial.
| Metric | Calculation | Question answered |
|---|---|---|
| Precision | True positives divided by all positive predictions | Of the flagged cases, how many belonged here? |
| Recall | True positives divided by all actual positive cases | Of the cases that belonged here, how many were found? |
| Accuracy | All correct predictions divided by all evaluated cases | How often was the assigned label correct overall? |
Recall is also called sensitivity in many diagnostic settings. Precision corresponds to positive predictive value when the positive class and evaluation setup match. These connections make terminology easier to translate, but they do not make a document-search test equivalent to a clinical validation study.
A complete example with 1,000 documents
Suppose a fictional archive contains 100 relevant documents and 900 irrelevant ones. A tool returns 120 documents: 80 relevant and 40 irrelevant. It misses the other 20 relevant documents and correctly leaves out 860 irrelevant ones.
Precision is 80 divided by 120, or about 66.7%. Recall is 80 divided by 100, or 80%. Accuracy is 940 divided by 1,000, or 94%. The high accuracy does not contradict the fact that one third of the returned documents are irrelevant. Most items in the archive were irrelevant to begin with, and correctly excluding them contributes heavily to overall accuracy.
For comparison, a tool that returns nothing would correctly label all 900 irrelevant documents as irrelevant. It would have 90% accuracy and zero recall. Its precision would be undefined because it made no positive predictions. Software sometimes substitutes a configured value in this situation, so the displayed number needs its accompanying convention.
Why a threshold changes the picture
Many systems produce scores before assigning labels. A threshold determines which scores become positive predictions. Lowering it includes more cases. For a fixed set of scores, recall cannot decrease when more cases are included, but precision can move in either direction depending on which cases enter the group. The familiar trade-off is a common pattern, not a guarantee at every step.
Different thresholds may suit different uses. A broad search followed by human review can tolerate some irrelevant results. A notification that interrupts someone repeatedly may place more weight on avoiding false alarms. Neither preference implies that the people using the system care less about accuracy. They are dealing with different consequences and different opportunities to correct errors.
What an F1 score leaves out
F1 combines precision and recall using their harmonic mean. For the fictional archive above, it is about 72.7%. That can make comparisons more compact, but the combined number hides which type of error contributed more. It also does not use true negatives directly.
Two systems with similar F1 scores can create quite different workloads. One might produce a short, reliable list that misses many documents; another might produce a longer list requiring substantial review. Reporting the component metrics preserves that distinction. In multiclass tasks, averaging choices matter too: a macro average gives classes equal weight, while other averaging methods can emphasize more common classes.
A score belongs to an evaluation, not just a model
Precision depends on the mix of cases being evaluated. A score from a deliberately balanced test set may not describe the experience of searching an archive where relevant documents are rare. Labels also require judgment: if reviewers disagree about relevance, the evaluation contains that uncertainty.
The useful claim is therefore specific: this system, with this threshold, found these cases in this collection under this definition of relevance. That is more informative than an isolated percentage, and it makes comparisons possible without pretending that one metric captures every purpose.
