the banana chart
How often do people buy a category? A negative binomial model of purchase frequency.
What is m?
m is the average number of purchases per person in the period, counting everyone, including the people who buy nothing. Raising m moves the whole distribution to the right.
What is r?
r controls how unevenly purchases are spread. A small r means a few heavy buyers make most of the purchases, and the tail of the chart is long. A large r means purchases are more evenly spread, with fewer extreme buyers.
What is penetration?
Penetration is the share of people who buy the brand at least once. The brand's P(0) is 1 − penetration. The people who do buy the brand keep the category's shape, scaled to fit the penetration, and the green line shows that.
In plain English
Learn how it works
The formula, in words
The chart shows the chance that a random person makes exactly k purchases, for k = 0, 1, 2, and so on.
Read it in three pieces:
Γ(r + k) / (Γ(r) · k!) counts the ways the purchases can be spread out. Γ is the gamma function, a version of the factorial that also works for decimals. For whole numbers, Γ(n) = (n − 1)!.
(r / (r + m))^r is the chance of zero purchases, P(0).
(m / (r + m))^k adjusts for each purchase beyond zero. Larger m makes higher k more likely.
The math, with your current numbers
Glossary
- Mean (m)
- Average purchases per person, including people with zero purchases.
- Shape (r)
- How uneven purchases are. Smaller r means a longer tail of heavy buyers.
- k
- The number of purchases a person makes in the period.
- P(k)
- The share of people who make exactly k purchases.
- Cumulative probability
- The share of people who make k or fewer purchases. It climbs from P(0) toward 100%.
- Truncation (K)
- The chart stops at the first k where the cumulative probability reaches 99.95%. Beyond that, the remaining share is too small to matter.
- Penetration
- The share of people who buy a given brand at least once.