August 27, 2012

The Square Root Rule

I probably get more questions about the square root rule than anything else I talk about, aside from Gliebers Dresses.

The theory behind the square root rule is that demand from advertising channels does not increase in a linear manner.  Folks who execute email frequency testing know this better than anybody else.  Demand from email campaigns, on a weekly basis, looks something like this:

Pay close attention to the sales generated by about one email campaign per week ($0.18), and the total sales generated by about five email campaigns per week ($0.40).

This relationship is approximated by the square root rule.  If you know that you generate $0.40 in email demand per week by sending five campaigns per week, you can guesstimate what might happen if you send other combinations of campaigns.

  • 1 campaign per week = (1/5)^(0.5) * $0.40 = $0.18.
  • 2 campaigns per week = (2/5)^(0.5) * $0.40 = $0.25.
  • 3 campaigns per week = (3/5)^(0.5) * $0.40 = $0.31.
  • 4 campaigns per week = (4/5)^(0.5) * $0.40 = $0.36.
  • 5 campaigns per week = (5/5)^(0.5) * $0.40 = $0.40.
Now we can easily calculate the incremental impact of each contact.
  • 1 campaign per week = (1/5)^(0.5) * $0.40 = $0.18.
  • 2 campaigns per week = (2/5)^(0.5) * $0.40 = $0.07.
  • 3 campaigns per week = (3/5)^(0.5) * $0.40 = $0.06.
  • 4 campaigns per week = (4/5)^(0.5) * $0.40 = $0.05.
  • 5 campaigns per week = (5/5)^(0.5) * $0.40 = $0.04.
Clearly, the "right" answer is to actually test the right number of campaigns.  Unfortunately, so few people execute frequency tests that few people even understand the concept of incrementality.

So if your business leaders won't allow you to test the optimal contact frequency because of some odd excuse like "we can't afford to lose sales", then use the square root rule as a proxy for testing.

By the way, the square root rule works really well on estimating the budget for non-branded paid search terms.

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