# Testing whether the variance of a data set is significant

hope someone can help me out with some stats for a data set I have- I originally thought it was easy but I think I'm over complicating everything in my head.

Ok so for example- I'm looking at the body length of dogs. I've taken measurements of 20 individuals of each breed and I've done this for 10 breeds. I now want to see if the measurements I've taken for each breed are a good representative of the actual breed- so I think I want to see if the variance for each breed is significant and then decide overall if all these variances are significant so that I can tell when I compare the breeds if I've actually got means that are representing the breed.

Plotted coefficient of variation against the breeds mean as suggested:

Still confused about how (or if its possible) I can tell if the variance within the breeds is significant. Would performing a stats test on the coefficients of Variance work?

• It is not entirely clear what you mean with "variancve for each breed is significant" How did you obtain the data? random sample (from which population) or otherwise? Do you have external data specifying what is tha variability for each breed? ... – kjetil b halvorsen Mar 1 '17 at 13:35
• Sorry struggling with how to express what I want- I'll try and explain it better. Ok so all together I've taken 30 different breeds of dog and taken measurements of individuals from each breed. For 20 of these breeds I've only used 5 individuals but for 10 I've used 20 individuals to ensure reliability- that the measurements I'm getting from these smaller groups are an actual representation of the population of the breed as a whole. So now I want to see using the samples of 20 if the measurements taken are all roughly the same (so low variance) or if they are really spread (high variance). – Coconutter Mar 1 '17 at 14:01
• But then I want some way of telling if this variance is significantly too big and so the results are too far spread that the mean I'm using for comparison isn't actually a good representation of the data. – Coconutter Mar 1 '17 at 14:01
• It still is somewhat unclear what you want. Maybe this breeds are of very different size? then maybe the variance could be modeled as some function of the mean, if so, you could see if some empirical variances were outliers compared to such a relationship? – kjetil b halvorsen Mar 1 '17 at 14:27
• Yeah I believe that would be right actually- I need to see if the mean for each breed is a good representation of the overall measurements I'm getting for each breed. Would this be done through a chi-square test? – Coconutter Mar 1 '17 at 15:03

I now want to see if the measurements I've taken for each breed are a good representative of the actual breed

This is impossible based on the sample itself. I understand what you want but it can't be done without knowing something in addition to the sample at hand, something about the entire population (i.e. breed in this case).

For instance, you know what is the average weight of dogs in the population ( for entire breed), then you look at your 10 dogs and their weight is somehow not representative, then you have an issue with the sample. You probably can't use the length measurements either.

Another example, suppose you know that the variance of length in population is $\sigma^2=100$. You measure the variance in 10 dog sample and it's $S^2=1000$. If we assume that the variance of sample variance $S^2$ is $\sigma^4/n=100^2/10=1000$, then the standard deviation is approximately $\sqrt{1000}\approx 33$, hence $S^2=1000$ is too high relative to $\sigma^2=100$. Note, we established that the sample variance was too high using external information, in this case the population variance

• Ahhh ok thank you. How about just of my sample then? Is there anyway I can just statistically tell if variation of multiple categories is significantly too high? In short is the variation of breed variances significant? Thank you for your help. – Coconutter Mar 1 '17 at 20:09
• What is too high? Too high for what? – Aksakal Mar 1 '17 at 20:21
• The variance within the breeds. Is there no way I can statistically show the measurements I have taken within each group are all very close together and that this applies across all the groups? Like if I had a set of data on say poodles and I had two outliers either side the variance would be high so the mean I have may not be a good indication of what I would get if I took the measurements of another poodle. I believe the data I have doesn't vary much within groups so the mean is a good indication of what I would get it I had a larger sample- is there no way I could show this statistically? – Coconutter Mar 1 '17 at 20:36
• @Coconutter, there's no way around this: you can't say whether the sample is representative of the population based solely on the sample data. You need to know something about the population that is not implied by the sample. Otherwise you have a logical trap. – Aksakal Mar 1 '17 at 20:40
• Ahhh ok- well thank you for your help. Back to the drawing board it is. – Coconutter Mar 1 '17 at 20:49