"algo-rank-wilson"

Wilson Score Ranking

Overview

Wilson Score interval provides a lower confidence bound on the true proportion of positive ratings. Unlike simple averages, it penalizes items with few ratings, preventing a 5/5 review item (1 review) from outranking a 4.8/5 item (1000 reviews). Computes in O(1) per item.

When to Use

Trigger conditions:

• Ranking items by user ratings when review counts vary widely
• Building "top rated" or "best of" lists that are fair to well-reviewed items
• Sorting binary feedback (upvote/downvote) with confidence

When NOT to use:

• For continuous scores (use Bayesian average instead)
• When comparing items with similar sample sizes (simple average suffices)

Algorithm

IRON LAW: Never Rank by Simple Average When Sample Sizes Differ
A 5.0 average from 1 review is NOT better than 4.8 from 1000 reviews.
Wilson Score lower bound accounts for sample uncertainty:
Items with few ratings get a LOWER bound, properly reflecting our
uncertainty about their true quality.

Phase 1: Input Validation

Collect per item: number of positive ratings (p), total ratings (n). For star ratings, convert to binary (e.g., 4-5 stars = positive). Gate: n > 0 for all items, confidence level chosen (typically 95%, z=1.96).

Phase 2: Core Algorithm

1. Compute observed proportion: p̂ = positive / total
2. Wilson lower bound: (p̂ + z²/2n - z × √(p̂(1-p̂)/n + z²/4n²)) / (1 + z²/n)
3. Rank by Wilson lower bound descending (conservative estimate of true quality)

Phase 3: Verification

Check: items with many positive reviews rank above items with few reviews and same proportion. Items with very few reviews are appropriately penalized. Gate: Ranking intuitively correct on manual inspection.

Phase 4: Output

Return ranked items with scores and confidence intervals.

Output Format

{
  "rankings": [{"item": "Product_A", "wilson_lower": 0.89, "positive": 950, "total": 1000, "proportion": 0.95}],
  "metadata": {"confidence": 0.95, "z": 1.96, "items_ranked": 500}
}