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

Official Solution:


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

The standard deviation of a list shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas a high standard deviation indicates that the data are spread out over a large range of values. So, basically we can say that it in a sense measures the distance and the distance cannot be negative, which means that the standard deviation of any list is greater than or equal to zero: \(SD\geq0\).

Next, the standard deviation of a list is zero if and only the set consists of identical numbers (or which is the same if the list consists of only one number).

So, the question basically asks whether \(x=y=z\)., If \(x=y=z\), then the standard deviation will be zero and if not, then the standard deviation will be positive.

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

If \(x=y=z\) were true, then the average of \(x\), \(y\), and \(z\) would be \(x\) and since we are told that the average is less than \(x\), then \(x=y=z\) is not true and thus the standard deviation is positive. Sufficient.

(2) The median of \(x\), \(y\), and \(z\), is greater than \(z\)

If \(x=y=z\) were true, then the median of \(x\), \(y\), and \(z\) would be \(z\) and since we are told that the median is greater than \(z\), then \(x=y=z\) is not true and thus the standard deviation is positive. Sufficient.


Answer: D