Two types of averages for ratios of two things

Question

It was brought to my attention in this video that, where you have some attribute of your data that is the ratio of two other attributes, there are two ways of generating the mean "average" of this ratio.

Taking the example from the beginning of the video, suppose that $x_i$ is the total sales (in \$) for a given line item, and $k_i$ is the quantity sold:

Item ID	Sales	Qty sold
Accessory #1	$x_1$	$k_1$
Accessory #2	$x_2$	$k_2$
Accessory #3	$x_3$	$k_3$

For each individual line item, you might define "Sales per unit" as $u_i=\frac{x_i}{k_i}$. However, as the video points out, there are two ways one might then define the average Sales per unit for an Accessory. Using the video's terminology:

$$\text{AGG}=\frac{x_1+x_2+x_3}{k_1+k_2+k_3}$$

$$\text{AVG}=\frac{1}{3}\left(\frac{x_1}{k_1}+\frac{x_2}{k_2}+\frac{x_3}{k_3}\right)$$

So $\text{AVG}$ is just $\bar{u}$, whereas $\text{AGG}$ creates a composite Accessory item $\left(X, K\right)$, whose sales and quantity are the sum of the individual line items', and calculates $U=\frac{X}{K}$ for that composite item.

Plugging in example numbers for the $x_i$ and $k_i$ suggests that $\text{AGG}$ and $\text{AVG}$ do not in general give the same value. My question is: what are the qualitative differences in what $\text{AGG}$ and $\text{AVG}$ are doing? Which provides the better view of what average sales per unit for an Accessory are?

Answer 1

The simple average $$\text{AVG}=\frac{1}{3}\left(\frac{x_1}{k_1}+\frac{x_2}{k_2}+\frac{x_3}{k_3}\right)$$ is the mean selling price of the three types of accessories, whereas the weighted average $$\text{AGG}=\frac{x_1+x_2+x_3}{k_1+k_2+k_3}$$ is the mean selling price of the units of accessories, accounting for the unequal sales volumes among the three types.

Answer 2

AGG gives a weighted average, so that if a particular accessory (e.g. Accessory #1) is outselling all of the other accessories combined, then more weight is given to the ratio of $u_1 = \frac{x_1}{k_1}.$

The research is targeting the generic customer who comes into the store to buy $n$ items, where there is a narrow range between the typical maximum and minimum values of $n$. The store wants to know (in general) what the profit from that person will be. This can be useful when there is little variance between the ratio of how many items of accessory $i$ were sold versus how many items of accessory $j$.

There is a presumption here that the store can predict its overall gross sales by the amount of foot traffic inside the store. A 24 hour store might use such research to consider whether to have the store closed between $12$ midnight and $6$ am.

AVG presumes that it is irrelevant what the nature of the accessories are, or which accessories sell the most. The idea is that AVG is geared to a customer who only wants to spend a certain dollar range per item, no matter what he buys. So, AVG, charts typical price per accessory, giving equal weight among accessories, regardless of whether one accessory outsells another.