Introduction
I'm just learning statistics so bare with me. I understand that the variance is $\sigma^2 = \frac{1}{n}\sum\limits_{i=1}^n (x_i-\mu)^2 $ and the standard deviation being the square root of that.
Since the variation coefficient is expressed as $\rho = \frac{\sigma}{\mu}$ what do I do in case the average mean $\mu = 0$?
When evaluating $\sigma$ all the values are being squared and become positive thus the only way $\sigma$ can be zero is if $x_1=x_2=x_3=…=x_n$ thus the variance is zero and the deviation is zero (because there is no difference between the elements obviously).
Situation 1
If I have a set of numbers {0, 0, 0, 0} $\mu = 0$, $\sigma = 0$ thus $\rho = \frac{0}{0}$. What do I say my variation coefficient is as this isn't defined? Straight forward guess is saying that its zero but I cant do that?
Situation 2
If I have the set {-1, 0, 1} $\sigma = 1$ and $\mu = 0$ thus $\rho = \frac{1}{0}$ Is this any different from Situation 1?
What do I do in these cases?
Best Answer
The population $coefficient\; of\; variation,$ $\sigma/\mu$ is defined only for a distributions that has a $positive$ mean $\mu.$ And the sample coefficient of variation $S/\bar X$ is defined only when the sample mean $\bar X$ is positive.
Indeed, the most common uses of the coefficient of variation (population or sample) are for populations with $only\; positive\; values.$ So the situations that concern you should never arise.
Additional information that may help you understand the CV.
The coefficient of variation (CV) has no units. This means that it doesn't matter whether weights of animals in a sample are measured in pounds or in kilograms, the CV will still be the same. (And there is no such thing as an animal with a zero or negative weight.)
The CV measures variability relative to the mean. If we are measuring adult elephants, the variance (and its square root, the standard deviation) will be huge because all of the weights are huge. If we are weighing ants, the variance and SD will be very small. However, the CV for ants may be much larger than the CV for elephants. It is not unusual for some individual ants to weigh 10 times what others weigh, but the same is not true for elephants.