# Growth Rates

Hi, this contracted company that I am working with has this data. Basically they collect the number of transactions each day. Then they graph these transactions out by Week, always monday - sunday.

The data is then stored in a table where the x axis is the date of the monday the week started and the y axis is the average of the transactions for that week. They like it that way because it smooths out the weekends.

The problem is they are reporting a growth rate. Since the last couple weeks of the year are generally slower because of holidays, they appear to pick the second or third last full week of the year, and the second or third last week of the previous year, divide, and then come up with a rate.

Given the data, and the structure that they keep it, is there a better way?

I was thinking that they could average the weekly averages of each year. But I remember an old college professor always used to warn against averaging averages.

Any input is appreciated.

HallsofIvy
Homework Helper
You cannot "average averages" in the "unweighted" sense of "add them all and divide by their number.

For an easy example, suppose all 10 of the first set are "2":
{2, 2, 2, 2, 2, 2, 2, 2, 2, 2}.
The average is, of course, 2.

Now suppose all 8 of the second are "3":
{3, 3, 3, 3, 3, 3, 3, 3}.
The average is, of course, 3.

Suppose, finally, that all 8 of the final set are "4":
{4, 4, 4, 4, 4, 4, 4, 4}
The average is, of course, 4,
If you just naively average those three numbers, you get (2+ 3+ 4)/3= 9/3= 3.

But in fact, you have a total of 26 numbers:
{2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4}
which add to 76 so their average is 76/26= 36/13= 2.92, approximately, not quite 3.

Of course, I didn't actually add those numbers to get 76, I argued that the 10 "2"s will add to 20, the 8 "3"s to 24, and the 8 "4" to 32 so the total is 20+ 24+ 32= 76.

That's why we could use, instead, a "weighted" average. Instead of keeping all the numbers to get a grand average, keep the average of each set and the number of entries in each set. Here, our first average was 2 and there were 10 numbers, the second was 3 and there were 8 numbers, the last average was 4 and there were 8 numbers. We can get the total of each by multiplying the average of each set by the number of terms in the set and add those:
$$\frac{2(10)+ 3(8)+ 4(8)}{10+ 8+ 8}= \frac{20+ 24+ 32}{26}= \frac{76}{26}= 36/13$$

Notice that we could also write that as
$$\frac{10}{26}(2)+ \frac{8}{26}(3)+ \frac{8}{26}(4)= \frac{5}{13}(2)+ \frac{4}{13}(3)+ \frac{4}{13}(4)$$
"weighting" each average by the fraction of total numbers that was based on.

Actually, I don't see the problem with the company's current method of comparing the number of transactions in a given week with the number a transactions at the same period a year ago. That naturally allows for seasonal fluctuations and public holidays, etc.

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