📏 Harmonic Mean

Harmonic Mean#

The reciprocal-based mean underlying the F1 score.

Important

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What it is#

The harmonic mean is an average that leans toward the smaller of the values — the reciprocal of the average of reciprocals. For two numbers it is:

\[\text{HM} = \frac{2ab}{a + b}.\]

It is always ≤ the arithmetic mean, and equal only when the values match.

Its key property#

It penalizes imbalance. Averaging precision 0.95 and recall 0.20, the arithmetic mean gives a rosy 0.575, but the harmonic mean gives ~0.33 — correctly flagging that one component is poor. A high harmonic mean requires all inputs to be high.

Where it’s used#

That property is exactly why the F1-score uses it to combine precision and recall, and why harmonic means are the right average for rates and ratios (speeds, P/E ratios) rather than additive quantities.

Theme: Classification & Averaging Metrics · All terminology

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Mind map — connected ideas

F1-score · Mean · Average Precision (AP) · Precision (a.k.a. Positive Predictive Value, PPV) · Weighted Averaging · Macro Averaging

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More in Classification & Averaging Metrics

Accuracy · AUC (Area Under the Curve) · Average Precision (AP) · Binary Classification · Classification Probability · Discriminatory Power · F1-score · Gini Coefficient · Log Loss (also called Logarithmic Loss or Cross-Entropy Loss) · Macro AUC · Macro AUROC (Macro-Averaged AUROC) · Macro Averaging · Macro F1 · Macro Precision