×

# Can the standard deviation be greater than the mean?

Jan 12, 2015

In a perfect normal distribution it can be.

In the ideal normal distribution ALL values are theoretically possible, from $- \infty$ to $+ \infty$.
And then any standard deviation $\sigma$ is possible

In the real world we work with datasets, that can often be well descibed by a normal distribution.

Say you have a filling machine for kilo-bags of sugar. The actual weight of the bags can be described as a normal distribution with a mean $\mu$ of 1000 gram.
In this case a $\sigma$ of more than 1000 would be unthinkable, as this would mean that 18% of your bags would have a negative weight!
(a machine this unreliable would be unthinkable anyway!).

In theory: YES -- in practice: (almost) NEVER

BTW :
In a standardised normal distribution the mean $\mu$ is calculated back to $0$ and the standard deviation $\sigma$ is reduced to $1$, so there the answer is always YES.

Dec 25, 2016