Average of all data vs. average of group averages [duplicate]

Question

Assume I operate a chain of 20 hot dog stands. I am asked to figure out the average % of customers who buy a hot dog after visiting a stand.

Method 1: Divide total number of hot dogs sold by all 20 stands by the total number of people who visited one of my stands.

Method 2: Follow process for method 1 for each individual stand. Then take the average of the 20 individual averages.

Are there any inherent benefits/drawbacks of method 1 vs. method 2? Any implicit assumptions in either approach?

Answer 1

Let us create some fictional data, before we attempt to answer your question:

> hot.dog.stands <- data.frame(stands = c("A", "B", "C"), hot.dogs.sold = c(5000, 5000, 1), prospective.customers = c(1, 5000, 5000))
> hot.dog.stands
  stands hot.dogs.sold prospective.customers
1      A          5000                     1
2      B          5000                  5000
3      C             1                  5000
> method.1 <- sum(hot.dog.stands$hot.dogs.sold) / sum(hot.dog.stands$prospective.customers)
> method.2 <- sum(hot.dog.stands$hot.dogs.sold / hot.dog.stands$prospective.customers) / 3
> 
> method.1
[1] 1
> method.2
[1] 1667

Now, according to method 1, you have sold an average of one hot dog to one prospective buyer, so your hot dog stands appear to have sold hot dogs to 100% of prospective buyers.

But that cannot be true, because at stand C there were 4999 people who didn't buy a hot dog at all.

Then, according to method 2, stand A has sold hot dogs to 166,700% of prospective buyers. That seems like an insane success. Yet when you look at the data from the individual stands, you will immediately realize that you could have had the same success without stand C! In fact, if you close that stand, you will save the pay for that lazy salesperson.

And you can also see that the improbable success of stand A comes from that single prospective buyer who bought 5000 hot dogs for his colleagues when he told them he would retire after winning the lottery. That sale is unique and this person is not coming back, so your data tells you to close both stand A and C (or replace the salesperson or advertise more or whatever).

This data was of course made up, and it was made up in a highly improbable manner, but I hope that my fictional example serves to illustrate why you should never deduce too much from a mean but should always rather look at some more useful representation of the data.

What representation will be most useful, will depend on the information that you seek. If all you want to know is how well you are doing on average, then method 1 appears to be more representative of your overall success.

If you want to know how the stands are doing, then looking at the data for each individual stand in turn will give you the most valuable information.