Maximum Scattering

Age 16 to 18

Challenge Level Yellow star

a) The data obtained from a given experiment is a pair of numbers

$a$ and

$b$ , where

$a\geq 0$ and

$b\geq 0$ . It is known that

$a$ and

$b$ have mean

$1$ ; what is the largest value that the standard deviation can be?

(b) The data obtained from a given experiment is a triple of numbers

$x$ ,

$y$ and

$z$ , where each is non-negative. It is known that the mean of

$x$ ,

$y$ and

$z$ is

$1$ ; what is the largest value that the standard deviation can be?

$t_1,\ldots,t_n$ , where each is non-negative. It is known that the mean of the

$t_j$ is

$1$ . Show that the standard deviation may be as large as

$\sqrt{n-1}$ .

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