Is the standard deviation of numbers \(x, y,\) and \(z,\) positive?

[M7MBA]

Is the standard deviation of numbers \(x, y,\) and \(z,\) positive?

(1) The average (arithmetic mean) of \(x, y,\) and \(z,\) is less than \(x.\)
(2) The median of \(x, y,\) and \(z,\) is greater than \(z.\)

Answer: D

Source: GMAT Club Tests

[deloitte247]

Statement 1 => The arithmetic mean of x, y, and z is less than x

$$Given\ that\ \left(\frac{x+y+z}{3}\right)<x$$
definitely there is at least one number that is less than x so all numbers are not equal and the standard deviation is not equal to 0 hence it is positive. SD > 0.
Statement 1 is SUFFICIENT

Statement 2 => The median of x, y, and z is greater than z
Given that the median is greater than z, then z must be the smallest number and it is different from the other two numbers so all numbers are not equal and the standard deviation is not equal to 0. It is positive
Statement 2 is SUFFICIENT

Since each statement alone is SUFFICIENT,
Answer = D