Expert: Josh - 2/1/2010

Question
Hi Josh

I am confused by what I would term a "rates" question.
I am trying to work out the kWh electricity used / person / $ revenue for a business with multiple locations. The problem is that when I work out the kWh/p/$ for two locations the result kWh/p/$ for the business does not seem logical to me. See the simplified example below.

Location A
electricity = 50 kWh
people = 10
revenue = $5
= 1 kWh/person/$

Location B
electricity = 20 kWh
people = 10
revenue = $5
= 0.4 kWh/person/$

Location C
electricity = 30 kWh
people = 10
revenue = $5
= 0.6 kWh/person/$

Total Business
electricity = 100 kWh
people = 30
revenue = $15
= 0.2 kWh/person/$

I would expect the rate for the total business would be somewhere between the highest and lowest location.

Is the problem that all three parameters are inconsistent, compared to when you do a similar calculation for litres/person/day for a water system and the number of days is not a sum but a consistent amount?

In which case do I need to keep one of the parameters I use consistent, if so which one and how? Or add another (which seems to me even more confusing)? Or am I wrong in thinking the overall rate for the total business should fall within the range of the locations?

Thank you for your help.

Answer
Hi Erin,

Generally speaking, the average of individual sample ratios is not the same as the ensemble average obtained from aggregates (more explanation to follow). However, in the special case where the value of "people" and "revenue" at each location are identical, the average calculations simplify significantly. It turns out that your intuition (that the rate for the total business being bounded by the highest and lowest figures at individual locations) is a correct one, provided some adjustment is made to account for the "double division" used to compute the ratio. In brief, I think the individual figures (for location 1, 2 and 3) need to be further divided by 3 to make sense; if we are to compare them with the rate obtained for total business.

The following elaborates on the comments above. The sketch is not very polished, as I do not have time to enter all the numbers to make everything as clear as it can be.

Let "f", "e", "p" and "r" be the efficiency measure, energy use, people count and revenue, respectively. Also, let f(n)=e(n)/p(n)/r(n) for location n=1,2,3. For example, f(2)=20/10/5.

The average of individual ratios F/3 = [f(1)+f(2)+f(3)]/3
is GENERALLY NOT EQUAL TO the ensemble average E/P/R,
where aggregates E=e(1)+e(2)+e(3), P=p(1)+p(2)+p(3), R=r(1)+r(2)+r(3).

This is not surprising, considering that
[K/N + L/P + M/Q]/3 DIFFERS from (K+L+M)/(N+P+Q) in general.

What the ensemble average (E/P/R) is saying is that the statistical measure comes from dividing the aggregate sum of "energy use" at all locations, by the total workforce "head count", by the sum of "revenue" contributed at each location.

Let's look more closely
-----------------------
If we consider only e(n)/p(n), then, we have 50/10 = 5kWh/person (at location 1), 20/10 = 2kWh/person (at location 2) and 30/10 = 3kWh/person (at location 3). These average to [5+2+3]/3=3.33kWh/person, exactly in agreement with (50+20+30)[kWh]/(10+10+10)[person] = 3.33kWh/person. Why does [e(1)/p(1) + e(2)/p(2) + e(3)/p(3)]/3 equal [e(1)+e(2)+e(3)]/[p(1)+p(2)+p(3)] in this case, you may ask? This only works out because the people count at each location are identical, i.e., p(1)=p(2)=p(3) (the fractions share a common denominator).

As the measure of interest "e(n)/p(n)/r(n)" involves two divisions, we can merge e(n)/p(n) together and consider what happens if we continue examining say Y/r(n).
So, forming fractions again, this time we can let the numerators be A=e(1)/p(1), B=e(2)/p(2), C=e(3)/p(3). We just determined that (A+B+C)/3=3.33kWh/person from above. The ultimate measure (in terms of kWh/person/$) can be expressed as [(A+B+C)/3]/[r(1)+r(2)+r(3)] on the condition that previously p(1)=p(2)=p(3). The question now is, does [(A/3)/r(1)+(B/3)/r(2)+(C/3)/r(3)]/3 = [(A+B+C)/3]/[r(1)+r(2)+r(3)]?  We see that [(A+B+C)/3]/[r(1)+r(2)+r(3)]=3.33/[5+5+5] = 2/9; this is identical to [(A/3)/r(1)+(B/3)/r(2)+(C/3)/r(3)]/3 = (0.3333+0.1333+0.2)/3 = 2/9 when r(1)=r(2)=r(3). So, the total business energy efficiency figure 0.22kWh/person/$ is the true average of {f(1)/p(1),f(2)/p(2),f(3)/p(3)}/3 = {0.3333,0.1333,0.2} when viewed in the right light. i.e., 0.22 [kWh/person/$] is upper-bounded by 0.3333 (the maximum value in the sample) and lower-bounded by 0.1333 (the minimum value in the sample).