Precision, recall and F1 for a spam filter

Build the confusion matrix first: TP, FP, FN, TN.

Worked answer.

Confusion matrix (positive class = spam)

• TP (spam correctly flagged) = 60
• FP (legitimate emails wrongly flagged) = 80 − 60 = 20
• FN (spam missed) = 100 − 60 = 40
• TN (legitimate correctly passed) = 1,000 − 60 − 20 − 40 = 880

Check: 60 + 20 + 40 + 880 = 1,000 ✓, and actual spam TP + FN = 60 + 40 = 100 ✓.

Metrics

• Accuracy = (TP + TN) / total = (60 + 880) / 1,000 = 0.94 (94%)
• Precision = TP / (TP + FP) = 60 / 80 = 0.75 — of the emails flagged, 75% really were spam.
• Recall = TP / (TP + FN) = 60 / 100 = 0.60 — the filter caught 60% of all spam.
• F1 = 2 × P × R / (P + R) = 2 × 0.75 × 0.60 / (0.75 + 0.60) = 0.90 / 1.35 = 0.667 (to three decimals).

Why accuracy misleads
Only 10% of emails are spam. A useless model that marks everything as "not spam" would get 900/1,000 = 90% accuracy while catching zero spam (recall = 0). Our 94% hides that the model misses 40% of spam and wrongly flags 20 genuine emails.

Which matters more here?

• Spam filter: a false positive (an offer letter sent to spam) is usually costlier, so precision is prioritised.
• Disease screening: missing a patient is worse, so recall is prioritised.
• Raising the threshold usually increases precision and lowers recall.

Common follow-up: "Why use the harmonic mean for F1 instead of the arithmetic mean?" The harmonic mean is dragged down by the smaller value, so a model cannot score well by being excellent on one metric and poor on the other. Here the arithmetic mean would be 0.675, slightly above the F1 of 0.667; the gap grows when precision and recall differ more.