# How do the Averages Work?

I am trying to figure out the average items sold per customer for the year 2019

I have multiple customers per day, who each have a random number of items. Sometimes a customer makes more than one purchase a day - about 6% of the time. In that case - all sales for that customer are considered to be 1 sale.

I have calculated this in 2 fashions - The 1st:

1. Total # Items Sold ( 27,427,131 ) / Total # customers ( 6,556,133 ) = 4.18
2. AVG(Items Sold Per Customer who make a purchase Per Day) = 4.21

The second way I did using an SQL function on an MS SQL Server The first way I did by getting the total # from the same SQL Database

With > 1 million customers - I can't replicate the 2nd method in Excel - and so I can't step it through and confirm the data.

I don't quite understand why the 2 avgs are different - but I suspected they would be and would like to use the # I think is more accurate - the 2nd one: 4.21 My boss has asked me to provide the formula used, and wants me to "prove" my answer. Not an unreasonable request, but I'm a little lost in explaining why I am getting 2 different averages. How do I explain this?

OR - and this is a real possibility - The numbers SHOULD be exactly the same and I am doing something wrong in one of my steps to calculate this.

You're running into weighted averages on unequal sample sizes. The second formula is incorrect.

Suppose there's only the following two days:

1. 2 items sold, 1 customer
2. 2 items sold, 2 customers

It's obvious the first formula generates the correct result ($$4/3$$, 4 items/3 customers), the second is $$\mathrm{AVG}(2,1)$$ which is not $$4/3$$.

• Change it to 2/1 and 2/2 or whatever ($\mathrm{avg}(2,1)\neq4/3$). The point is avg(rate per sample) is fundamentally not correct to calculate a global rate. Jun 4, 2020 at 9:59