Which of the following represents gain across the database of customers and prospects assuming that desired outcome is purchase?

Question

BlackTom AI · Accepted Answer

Gain is the proportion of all purchasers that a model captures when you target the top-ranked subset — mathematically, cumulative number of purchasers divided by total number of purchasers.

1. Relevant knowledge points and concepts
- A gain chart measures how many of the desired outcomes (here, purchasers) are captured by targeting a top fraction of the population. 
- Gain is a ratio that normalizes the captured purchasers by the total number of purchasers in the entire population, so it is independent of sample size.
- Symbols: let C = cumulative number of purchasers captured by the subset, T = total number of purchasers in the whole database, N = total number in sample, overall purchase rate r = T/N.

2. Step-by-step reasoning
- The intuitive quantity we want is “what fraction of all purchasers have we captured?” That is C/T.
- Option 2 explicitly states Cumulative number of purchasers/Total number of purchasers = C/T, which matches the definition.

3. Calculations and algebraic confirmation
- For option 3: Cumulative number of purchasers/(Total number in sample * Overall purchase rate) = C / (N * r). Since r = T/N, N * r = N * (T/N) = T. So option 3 simplifies to C/T, exactly the same as option 2.
- For option 1: (Cumulative percentage of purchasers) * (Overall purchase rate) = (C/T) * (T/N) = C/N. That equals the proportion of the total population that are purchasers within the targeted subset, not the fraction of all purchasers — so it’s C/N, not C/T.
- For option 4: Cumulative number of purchasers = C (unnormalized). That raw count doesn’t express gain as a fraction of all purchasers.

4. Why the correct answer is correct
- Gain by definition is the fraction of total purchasers captured, C/T. Option 2 gives this directly and thus correctly represents gain.

5. Why other options are incorrect
- Option 1 yields C/N (proportion of population), not C/T (proportion of purchasers).
- Option 3, though algebraically identical to option 2, is a more convoluted expression; option 2 is the direct, standard form.
- Option 4 is unnormalized and so doesn’t represent gain as a proportion.

6. Conclusion
- The correct representation of gain is Cumulative number of purchasers / Total number of purchasers, matching the provided answer.

Which of the following represents gain across the database of customers and prospects assuming that desired outcome is purchase?Single choice

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