At Chegg, we test the relevance of our search engine using customer data. We extract anonymous information about queries and clicks, then use that to automatically test improvements to search. When our search engine provides results that better match what our customers are choosing, we call that an improvement.
The most basic measurement of search quality is clickthrough rate. For each search page that is shown, how often is at least one search result clicked on? This fits well with overall website conversion, showing how search works in getting a visitor to a successful result.
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