Precision, Recall & F1

three core classification metrics derived from the confusion matrix. Precision $P = \frac{TP}{TP + FP}$ (of the samples predicted positive, how many are truly positive — “quality over quantity”); Recall $R = \frac{TP}{TP + FN}$ (of the truly positive samples, how many were found — “better safe than sorry”); $F1 = \frac{2PR}{P + R}$ is their harmonic mean, summarizing both in one number; more generally $F_\beta = \frac{(1+\beta^2)PR}{\beta^2 P + R}$, where $\beta > 1$ weights recall more (e.g. $F_2$) and $\beta < 1$ weights precision more. The harmonic mean punishes imbalance: $P = 1, R = 0.1$ gives $F1 \approx 0.18$, far below the arithmetic mean of 0.55.

almost guaranteed in interviews around imbalanced data and funnel-style products (search, recsys, risk control); the PR curve traces P vs R as the threshold moves, and the random-classifier baseline is $P = \rho$ where $\rho = \frac{TP+FN}{N}$ is the positive rate (e.g. baseline $P = 0.01$ at a 1% positive rate); be ready for “why F1 instead of accuracy” and “PR curve vs ROC curve”.

decouples evaluation from the majority class — P looks only at the predicted-positive column and R only at the truly-positive row, neither diluted by the huge negative population; the harmonic mean exposes models with one high and one low score (always predicting positive yields $P = \rho \approx 0$), which the arithmetic mean would hide. Fβ tunes the trade-off by business cost: screening with fatal misses uses $\beta = 2$ to favor recall, ad placement wasting budget uses $\beta = 0.5$ to favor precision.

Hand computation: a model has $P = 0.9, R = 0.5$; compute F1 and explain why it is below the arithmetic mean.

Why does F1 use the harmonic mean instead of the arithmetic mean? What is F1 when $P = 1, R = 0.1$?

Meaning of β in $F_\beta$: what do $F_2$ and $F_{0.5}$ emphasize, and which fits medical screening vs ad placement?

When to choose the PR curve over ROC: why is PR more sensitive to class imbalance, and what does the baseline $P = \rho$ mean?

How to derive the PR curve from model scores, and how to pick the operating threshold (F1 maximization / cost minimization)?