Q: What does it mean when standard deviation is higher than the mean?

My answer: It depends on the nature of the data.

If all of the values are positive, then it indicates that there is quite a bit of spread, and the ratio of sd/mean is the coefficient of variation. This can be useful to compare the degree of “spreadoutedness” of two distributions with different means.

But if some of the data are negative, then the comparison of sd to mean stops having any meaning.

E.g. if there are 3 variables X, Y and Z

X: 1, 2, 3, 4, 5

Y: 10, 20, 30, 40, 50

Z: -2, -1, 0, 1, 2

Intuitively, these all seem to have the same spread in comparison to their size.

X has mean = 3, sd = 1.58, CV = 0.53

Y has mean = 30, sd = 15.81, CV = 0.53

Z has mean = 0, sd = 1.58, CV = infinite