F-Measure

F-Measure

The F-Measure is the weighted harmonic mean of precision and recall. It provides a single score that balances the trade-off between the two metrics.

F-Measure Formula

$F_{\beta }=\frac{(1+{\beta }^2)\cdot P\cdot R}{({\beta }^2\cdot P)+R}$

where:

• $P$ — Precision
• $R$ — Recall
• $\beta$ — Parameter indicating the relative importance of recall vs precision.

The F1 Score

The most common version is the F1 Score, where $\beta =1$ (equal weight to precision and recall). $F_1=\frac{2PR}{P+R}=\frac{2}{\frac{1}{P}+\frac{1}{R}}$

Why Harmonic Mean?

Unlike the arithmetic mean, the harmonic mean is sensitive to very low values. If either Precision or Recall is 0, the F1 score becomes 0. It penalizes extreme imbalances, forcing the system to perform well on both.

The Beta Parameter ($\beta$)

• $\beta =1$: Equal weight.
• $\beta >1$: Weights Recall higher than Precision (e.g., $F_2$). Used when missing a relevant document is more costly than a false alarm.
• $\beta <1$: Weights Precision higher than Recall (e.g., $F_{0.5}$). Used when false alarms are very costly.

Connections

• Components: Precision and Recall
• Context: Used throughout Machine Learning and Evaluation in IR.

Appears In

• IR-L04 - Evaluation