Consumer Panel Analytics & Brand Growth Science
Ehrenberg NBD Brand Penetration & Double Jeopardy Formula
Calculate Andrew Ehrenberg’s Negative Binomial Distribution (NBD) Non-Buyer Share P_0 = (1 + m/k)^(-k), Household Penetration b = 1 - P_0, Average Purchase Frequency per Buyer w = m/b, and Duplication of Purchase D*b_Y.
Diagnose brand growth realism using Ehrenberg’s NBD and Double Jeopardy Law!
Brand managers regularly pitch unrealistic annual plans promising to double "loyalty frequency w" among existing buyers without expanding household penetration b! Andrew Ehrenberg’s Negative Binomial Distribution (NBD) consumer-panel model (Gamma-Poisson purchasing across households with mean purchase rate m and heterogeneity shape k > 0) proves that Non-Buyer Share is P_0 = (1 + m / k) ^ (-k), Brand Penetration is b = 1 - P_0, and Purchase Frequency per Active Buyer is w = m / b! Crucially, NBD mathematically generates William McPhee’s Double Jeopardy Law: smaller brands with lower mean sales m suffer TWICE—vastly lower household penetration b AND noticeably lower repeat purchase frequency w! Alongside it, Goodhardt & Ehrenberg’s Duplication of Purchase Law predicts the fraction of Brand X buyers who also buy Rival Brand Y as b_Y|X = D * b_Y!
Verify on a Market-Leader CPG Brand—Mean_Household_Purchases m = 1.00 unit/qtr (B2), Gamma_Heterogeneity_Shape k = 2.00 (B3), Rival_Brand_Penetration b_Y = 0.30 (30.0%, B4), and Category_Duplication_Coeff D = 1.25 (B5): Non-Buyer Share P_0 is (1 + 0.5)^(-2) = 4/9 = 0.4444 (44.44% in E2); Brand Penetration b is 1 - 4/9 = 5/9 = 0.5556 (55.56% in E3); Average Purchase Frequency per Buyer w is 1.00 / (5/9) = 1.8000 purchases/buyer (E4); and Cross-Brand Duplication with Rival Y is 1.25 * 0.30 = 0.3750 (37.50% in E5)!
| Ehrenberg NBD Panel Metric (m=1.00/HH, k=2.0, Rival b_Y=30%, D=1.25) | Formula | Verified Output |
|---|---|---|
| Zero-Purchase Non-Buyer Share P_0 | (1 + m / k) ^ (-k) | 0.4444 (0.8264 @ m=0.20 Small Brand) |
| Household Penetration b | 1 - (1 + m / k) ^ (-k) | 0.5556 (0.1736 @ m=0.20 Small Brand) |
| Purchase Frequency per Buyer w | m / b (Double Jeopardy!) | 1.8000 (1.1524 @ m=0.20 Small Brand!) |
| Duplication of Purchase b_Y|X | MIN(1, D * b_Y) | 0.3750 (37.50% Shared Buyers) |
Ehrenberg NBD Buyer Frequency & Penetration Formula
Validate strictly positive mean purchase rate m > 0, Gamma dispersion k > 0, rival penetration 0 < b_Y < 1, and duplication D > 0!
=LET(m_hh, B2, k_sh, B3, b_rv, B4, dp_c, B5, IF(OR(m_hh <= 0, k_sh <= 0, b_rv <= 0, b_rv >= 1, dp_c <= 0), "Invalid NBD Panel Input", LET(p_zr, (1 + m_hh / k_sh) ^ (-k_sh), bn_p, 1 - p_zr, ROUND(m_hh / bn_p, 4))))Small-Brand Double Jeopardy Verification (Set m = 0.20)
Drop Mean_Household_Purchases B2 from 1.00 to 0.20 (niche challenger in the same category k = 2.0): Penetration E3 plunges from 55.56% to 17.36% AND Buyer Frequency E4 drops from 1.8000 to 1.1524!
=IF(ISTEXT(E4), E4, ROUND(MIN(1, B5 * B4), 4))Try the example
Use the original values or change them to test your understanding.