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Which of the following sentences is mathematically correct:

  1. The harmonic mean of a set of numbers is directly proportional to the sum of numbers.
  2. The arithmetic mean of a set of numbers is directly proportional to the sum of the numbers.

In my opinion, the first sentence is incorrect because, for example, if the sum of the numbers increases twice the harmonic mean does not increase twice. Whereas the second sentence is true.

Am I right? Thanks in advance

Ami
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    You need to give an explicit counterexample for one case and a proof for the other. – copper.hat Apr 14 '20 at 22:34
  • Harmonic mean of (10,10) = 10; harmonic mean of (10,30) = 15 and not 20. 2. Arithmetic mean of (x,y) = (x+y)/2; Arithmetic mean of (2x, 2y) = (2x+2y)/2= x+y
    • – Ami Apr 14 '20 at 22:51
  • is good. For 2., just note that ${x_1+\cdots+x_n \over n} = {1 \over n} \sum_{k=1}^n x_k$. The constant of proportionality is ${1 \over n}$.
    • – copper.hat Apr 14 '20 at 23:08