Calculating Geometric Mean Annual Increase: Cable TV Subscribers 1990-2000

[PARAJON]

I have a question that I would like your assistance to see if I have the correct info:


In 1990 there were 9.19 million cable TV subscribers. By 2000 the number of subscribers increased to 54.87 million. What is the geometric mean annual increase for the period ?


Answer:

(9.19*54.87)^(1/2) = 22.46 million

I would appreciate your help in this matter to come up with the correct answer.

I also have another possible answer of 19.56%


Thank you!

 

[HallsofIvy]

Neither of those answers has any hope of being correct.

In the first case, "(9.19*54.87)^(1/2) = 22.46 million" you are averaging the number of viewers, not the annual increase in the number of viewers.
In the second case, you have a percentage and the problem asks for "annual increase" not percentage annual increase.

IF the problem had asked for "arithmetic" mean (or just "mean") it would be easy: the total increase from 1990 to 2000 is 54.87- 9.19= 45.68 million which would be a mean annual increase for those 10 years of 45.68/10= 4.569 million.

Letting a₁, a₂, ... , a_n, be the actual annual increase, you want a number a such that a₁*a₂*...*a₁₀= a¹⁰ AND a₁+ ...+ a₁₀= 45.68. Since sums and products don't combine nicely, I don't see any way of getting that.

 

[pig]

the 19.56% answer is the annual increase.

9.19 * x^10 = 54.87
x = (54.87/9.19)^(1/10)

which is 1.195644 ~ 19.56%.

22.46 is the geometric mean of 9.19 and 54.87 but i don't see how that is important here.

 

[mathman]

Take the ratio (R) of no. of 2000 over no. in 1990. The answer you want is R^1/10. I'll let you do the arithmetic.

 

[HallsofIvy]

"the 19.56% answer is the annual increase."

No, it's the annual percentage increase. The (arithmetic) mean annual increase is 4.569 million as I said. There's a big difference.

"Take the ratio (R) of no. of 2000 over no. in 1990. The answer you want is R^1/10. I'll let you do the arithmetic."
That is 1.1956= "1+r" where r is the annual percentage increase. I don't see how that could be called the "geometric mean annual increase".

 

[pig]

No, it's the annual percentage increase.

you are right, of course :)