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How can the percent increase in Product $A$ ($52\%$) and Product $B$ ($75\%$) each be greater than the combined increase ($47\%$)?

In $2015$, I have $\$1.1$M revenue of Product $A$ with $1,000$ units sold for an average price of $\$1,100$ per unit. In $2016$, I have $\$1$M revenue of Product A with $600 $ units sold for an average of $\$1,667$ per unit. The increase in percent from $\$1,100$ per unit to $\$1,667$ per unit is $52\%$.

In $2015$, I have $\$0.6$M revenue of Product $B$ with $350$ units sold for an average price of $\$1,714$ per unit. In $2016$, I have $\$0.3$M revenue of Product $B$ with 100 units sold for an average of $\$3,000$ per unit. The increase in percent from $\$1,714$ per unit to $\$3,000$ per unit is $75\%$.

Combining the products, in 2015, I have $\$1.7$M revenue of both products with $1,350$ units sold for an average price of $\$1,259$ per unit. In $2016$, I have $\$1.3$M revenue of both products with $700$ units sold for an average of $\$1,857$ per unit. The increase in percent from $\$1,259$ per unit to $\$1,857$ per unit is $47\%$.

Restating the question, how can the percent increase in Product $A$ ($52\%$) and Product $B$ ($75\%$) each be greater than the combined increase ($47\%$)?

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    This is an example of 'Simpson's Paradox'. – 2017-01-12
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    The Wikipedia article on [Simpson’s paradox](https://en.wikipedia.org/wiki/Simpson's_paradox) is a decent starting point. – 2017-01-12

1 Answers 1

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To take an extreme case:

Suppose in year $1$ I sell $100$ units of $A$ for $\$1000$ each and $1$ unit of $B$ for $\$1$. That's $101$ units sold for $\$100001$ so the unit price is $\$990.1089109$.

Suppose in year $2$ I sell $1$ unit of $A$ for $\$1100$ and $100$ units of $B$ for $\$1.1$. That's $101$ units for $\$1210$ so a unit price of $\$11.98019802$.

Thus both products go up by $10\%$ but the unit price goes way down. Of course, that's because in year $1$ I sold lots of the expensive product and in year $2$ I sold lots of the cheap one. That's what you did in your problem...in the first year you sold about three times as much $A$ as $B$, while in the second year that jumped to six times as much.