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The F1 score is the harmonic mean of precision and recall for a specified positive class. It combines correctness and completeness with equal weight.
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Formula
dimensionless, usually between 0 and 1
| Symbol | Meaning | Unit |
|---|---|---|
| Correctly classified positive examples | count | |
| Incorrectly classified positive examples | count | |
| Missed positive examples | count | |
| Precision | TP/(TP+FP), share of predicted positives that are correct | dimensionless |
| Recall | TP/(TP+FN), share of actual positives found | dimensionless |
F_1 = \frac{2\,TP}{2\,TP + FP + FN}From precision = TP/(TP+FP) and recall = TP/(TP+FN), F1 = 2PR/(P+R) = 2TP/(2TP+FP+FN). The harmonic mean penalizes one very low value more than the arithmetic mean would. TN does not enter the formula. For multiple classes, the averaging method (such as macro, micro or weighted) must also be specified; these methods answer different questions. If no positive case is predicted or present, a convention for division by zero is needed.
From precision = TP/(TP+FP) and recall = TP/(TP+FN), F1 = 2PR/(P+R) = 2TP/(2TP+FP+FN). The harmonic mean penalizes one very low value more than the arithmetic mean would. TN does not enter the formula. For multiple classes, the averaging method (such as macro, micro or weighted) must also be specified; these methods answer different questions. If no positive case is predicted or present, a convention for division by zero is needed.
The F1 score is the harmonic mean of precision and recall for a specified positive class. It combines correctness and completeness with equal weight.
Defining relation
The equation states the defining relation within its stated domain.
Use the example for F1 score to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
The F1 score is the harmonic mean of precision and recall for a specified positive class.
It combines correctness and completeness with equal weight.
The relation applies only under its stated assumptions.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: From precision = TP/(TP+FP) and recall = TP/(TP+FN), F1 = 2PR/(P+R) = 2TP/(2TP+FP+FN). The harmonic mean penalizes one very low value more than the arithmetic mean would. TN does not.
Formula cluster
The F1 score is the harmonic mean of precision and recall for a specified positive class. It combines correctness and completeness with equal weight.
With TP=8, FP=2 and FN=4, precision=8/10=0.8 and recall=8/12≈0.667. Therefore F1=16/(16+2+4)=16/22≈0.727. A large share of true negatives does not directly change this value.
Compare classifiers, especially when positive detections matter and class frequencies are uneven; always report the class and averaging method.
Here is how to work through a typical F₁ score (F₁ = 2TP / (2TP + FP + FN)) task step by step:
Task
Use the example for F1 score to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
The F1 score is the harmonic mean of precision and recall for a specified positive class.
It combines correctness and completeness with equal weight.
The relation applies only under its stated assumptions.
The relation F1 = \frac{2\,TP}{2\,TP + FP + FN} summarizes The F1 score is the harmonic mean of precision and recall for a specified positive class. It combines correctness and completeness with equal weight. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example With TP=8, FP=2 and FN=4, precision=8/10=0.8 and recall=8/12≈0.667. Therefore F1=16/(16+2+4)=16/22≈0.727. A large share of true negatives does not directly change this value., symbols are connected to concrete values and the result is interpreted in context. Validity limit: From precision = TP/(TP+FP) and recall = TP/(TP+FN), F1 = 2PR/(P+R) = 2TP/(2TP+FP+FN). The harmonic mean penalizes one very low value more than the arithmetic mean would. TN does not Source: scikit-learn, User Guide: “Classification metrics”, F-measure and averaging.
First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example With TP=8, FP=2 and FN=4, precision=8/10=0.8 and recall=8/12≈0.667. Therefore F1=16/(16+2+4)=16/22≈0.727. A large share of true negatives does not directly change this value. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: From precision = TP/(TP+FP) and recall = TP/(TP+FN), F1 = 2PR/(P+R) = 2TP/(2TP+FP+FN). The harmonic mean penalizes one very low value more than the arithmetic mean would. TN does not Source: scikit-learn, User Guide: “Classification metrics”, F-measure and averaging.
The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: From precision = TP/(TP+FP) and recall = TP/(TP+FN), F1 = 2PR/(P+R) = 2TP/(2TP+FP+FN). The harmonic mean penalizes one very low value more than the arithmetic mean would. TN does not Source: scikit-learn, User Guide: “Classification metrics”, F-measure and averaging.
The stated units are: dimensionless, usually between 0 and 1. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: From precision = TP/(TP+FP) and recall = TP/(TP+FN), F1 = 2PR/(P+R) = 2TP/(2TP+FP+FN). The harmonic mean penalizes one very low value more than the arithmetic mean would. TN does not Source: scikit-learn, User Guide: “Classification metrics”, F-measure and averaging.
Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source scikit-learn, User Guide: “Classification metrics”, F-measure and averaging. supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: From precision = TP/(TP+FP) and recall = TP/(TP+FN), F1 = 2PR/(P+R) = 2TP/(2TP+FP+FN). The harmonic mean penalizes one very low value more than the arithmetic mean would. TN does not Source: scikit-learn, User Guide: “Classification metrics”, F-measure and averaging.
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