I was watching a TV commercial that claimed "x of y cars get A mpg" and knew the fleet average for y cars was B mpg. So I wondered what was the C mpg of the remaining cars?

My first attempt was to try and setup a series of equations that would solve for the B mpg of the remaining, ( y - x ), cars:

y - total number of cars
x - subset of total number of cars
A - MPG of x subset number of cars
C - MPG of all cars

(y - x ) * B + x * A = y * C
(y - x) * B = (y * C) - (x * A)
B = ( (y * C) - (x * A) ) / (y - x)

So let's test it, with "1 of 2 cars get 40 MPG" and "all cars average 30 MPG":

y = 2
x = 1
A = 40 mpg
C = 30 mpg fleet average

B = ( (y * C) - (x * A) ) / (y - x)
B = ( (2 * 30) - (1 * 40) / (2 - 1)
B = ( 60 - 40 ) / 1
B = 20 MPG -- a reasonable number

Ok, let's try it with numbers from the commercial:

y = 10
x = 9
A = 30 mpg
C = 20 mpg fleet average

B = ( (y * C) - (x * A) ) / (y - x)
B = ( (10 * 20) - (9 * 30) ) / (10 - 9)
B = ( (200) - (270) ) / (1)
B = -70 MPG -- an unreasonable number

Ok, let's try the problem from another direction. If the fleet average is 20 MPG and 1 out of 10 cars gets 0 MPG, what do the other 9 get?

(1 * 0) + (9 * B) = (10 * 20)
(9 * B) = 200
B = 200 / 9
B = 22.2 MPG -- best case, 9 of 10 cars get this MPG

You don't suppose something was missing from that commercial?

Bob Wilson

 

I'm assuming the commercial said 9 out of 10 cars get 30 mpg. I'm also assuming you know that company has a fleet average of 20 mpg, and that number did not come from the commercial.

I'm guessing you're not comparing apples and apples. Your number is probably a weighted City/Hwy mileage. They're average is probably just the highway mileage. So the fleet's highway mileage averge is probably a bit higher than 20 mpg.

 

Good points. I had uncritically taken a posting here that claimed one manufacturer was getting a fleet average of 20 MPG and another was getting 23 MPG. So I decided to find a source, "SUMMARY OF FUEL ECONOMY PERFORMANCE", March 2004, U.S. Department of Transportation, NHTSA, NVS-220.

http://www.nhtsa.dot.gov/cars/rules/cafe/CAFEData.htm

This report gives MPG data by vendor for domestic passenger, light truck and imports along with the total number of vehicles in each group. So I did the math and found:

A - 23.7 MPG for all vehicles
B - 24.8 MPG for all vehicles

This brings into question a past posting that vendor A was only getting a fleet average of "20 MPG" and vendor B "23 MPG."

One other curious observation was the MPG for the different classes of vehicles were identical! It was only the larger proportion of light trucks that impacted vendor A's fleet average.

This may cure the problem of the "9 out of 10" commerical. The necessary qualifications were missing from the narration although it might have been printed in pharacutical script (aka., 5 point, obscura.) If we take out the light trucks and use 27.5 MPG for the domestic and imported passenger vehicles, the 30 MPG is achievable with the remaining 1 of 10 vehicles:

5 MPG

This is achievable whereas the negative MPG was not. Lesson's learned, treat all unsourced Internet postings with suspicion.

Bob Wilson

 

Another lesson.

What do you want to prove? Given the raw data it's possible to prove anything with statistics. I bet someone could provide statistics showing smoking cuts down on the incidence of lung cancer.

 

Fortunately, no statistics were involved, just algerbra and fleet averages reported by the vendors, a metric.

My goal was to fill in the blank:
"9 out of 10 cars get 30 MPG" so
"1 out of 10 cars get <blank> MPG"

<blank> = 5, given the vendor reported fleet averages.

So now it reads:
"9 out of 10 cars get 30 MPG" so
"1 out of 10 cars get 5 MPG"

Nothing else needed nor supported by what is known.

Bob Wilson