# Standard Deviation greater than mean implies no normal distribution?

I understand that the mean $\mu \pm \sigma$ gives a better approximation of the measurements. But how is it related to the normal distribution? Is it because since $\sigma > \mu$ so the normal curve won't have a maximum at the middle anymore?

Anyway, the question is as follows, if it helps in any way:

According to the Environmental Protection Agency, chlorofoam, which in its gaseous form is suspected to be a cancer-causing agent, is present in small quantities in all the country's 240,000 public water sources. If the mean and standard deviation of the amounts of chlorofoam present in water sources are 34 and 53 micrograms per liter ($\mu g / L$) respectively, explain why chlorofoam amounts do not have a normal distribution.

Thanks in advance!

## 2 Answers

If standard deviation is much larger than mean for a random variable $X$ with normal distribution, it implies that $\mathbb P(X<0)$ is large. But if $X$ is a positive variable, it is just not possible.

Note that any positive random variable can not have a normal distribution because for any normal distribution $\mathbb P(X<0)>0$. But we say that as an approximation.

Let $X$ be the concentration in a randomly selected source. If $X$ has normal distribution, then the probability that $X\lt 0$ is the same as the probability that $$Z\lt -\frac{34}{53},$$ where $Z$ is standard normal. This is definitely non-zero, in fact it is about $0.26$. However, a negative concentration is impossible.

This means that a normal distribution model gives an extremely poor fit to reality. The fact that a sample of $200000$ all had a positive concentration is not particularly relevant.

• Thank you and also thanks @Xoff. Both makes sense and helpful as now I don't know which one I should pick as an accepted answer. – PandaMan Jan 16 '14 at 7:56
• You are welcome. Toss a coin and choose Xoff. – André Nicolas Jan 16 '14 at 8:04