Relationship With Other Means
If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.
Let r be a non zero real number and let the rth power mean ( Mr ) of a series of real variables ( a1, a2, a3, ... ) be defined as
For r = -1, 1 and 2 we have the harmonic, the arithmetic and the quadratic means respectively. Define r = 0, -∞ and +∞ to be the geometric mean, the minimum of the variates and the maximum of the variates respectively. Then for any two real numbers s and t such that s < t we have
with equality only if all the ai are equal.
Let R be the quadratic mean (or root mean square). Then
Read more about this topic: Harmonic Mean
Famous quotes containing the words relationship with, relationship and/or means:
“Some [adolescent] girls are depressed because they have lost their warm, open relationship with their parents. They have loved and been loved by people whom they now must betray to fit into peer culture. Furthermore, they are discouraged by peers from expressing sadness at the loss of family relationships—even to say they are sad is to admit weakness and dependency.”
—Mary Pipher (20th century)
“When a mother quarrels with a daughter, she has a double dose of unhappiness—hers from the conflict, and empathy with her daughter’s from the conflict with her. Throughout her life a mother retains this special need to maintain a good relationship with her daughter.”
—Terri Apter (20th century)
““When I use a word,” Humpty Dumpty said, in a rather scornful tone, “it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master—that’s all.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)