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Geometric Mean Question

  1. Apr 7, 2004 #1
    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!!!!
     
  2. jcsd
  3. Apr 7, 2004 #2

    HallsofIvy

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    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, ... , an, 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.
     
  4. Apr 7, 2004 #3

    pig

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    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.
     
  5. Apr 7, 2004 #4

    mathman

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    Take the ratio (R) of no. of 2000 over no. in 1990. The answer you want is R1/10. I'll let you do the arithmetic.
     
  6. Apr 8, 2004 #5

    HallsofIvy

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    "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 R1/10. 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".
     
  7. Apr 9, 2004 #6

    pig

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    you are right, of course :)
     
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