Dear All, is there a sensible way of combining Mean and Variance into 1 measurement, so as to communicate both these aspects together? For example, showing a lot of spread and low mean in 1 value.
Basically, instead of conducting analyses on the Means per condition, and then analyses on the spread in the data per condition, in an ideal world I would conduct analyses on a combination of the two, which helps me to discern different levels of dispersion and different levels of the mean.
Thanks..
Dear Michael Puntiroli
if you look at the Gauß-Markov Theorem, then you will see, that under a normally distributed random variable the mean estimate and the variance estimate are independent of each other. That is, if you combine the two into one measure, you will basically lose information. The higher the correlation between the two, the less information is lost by combining them. E.g. in the case of a Bernoulli experiment the variance is a function of the expectation only.
In the context of composite indicators, losing information is accepted for obtaining easy interpretable results. How to construct such a composite indicator depends on the subject of research again.
Coefficient of variation (CV) is often used in practice as a measure of relative dispersion. CV=std/mean, provided that the value of mean is positive and is not zero. Please search for coefficient of variation (CV) on internet and you will find many papers that discuss the statistical property of CV.
Dear Jan
You can use coefficient of variation
Cv=(standard deviation /mean) *100
.
The aim of the question was
"For example, showing a lot of spread and low mean in 1 value. "
This cannot be achieved by CV as far as I know. It is not possible to see whether a high or low value of CV is due to high dispersion or high mean value estimates.
Thanks a lot to everyone, I appreciate it.
Yes, I think you are right Jan Pablo Burgard. I studied your replies and CV just seems a measure of dispersion/spread alone, even though it contains the mean in its calculation. What I mean by that is that I don't think one could make any reasonable conclusion about both mean and spread by examining the CV.
In my data I have many types of ecological behaviors and I would like to see whether certain psychological values (altruism, egoism, ect..) predict inconsistency between the behaviors (high/low variance). However, variance is not enough, because one could be consistently (low variance) non-ecological (low mean), just as its possible to be consistently ecological (high mean).
Hence my desire to have both variance and mean meaningfully combined.
I seem to have composite indicators, just as you mention Jan Pablo Burgard.
Having "low mean" implies that you can compare it to sth, so I assume you have many distributions, for which you want to devise a metric which tells you which distributions have low mean (compared to the others?) and high spread (or any other combination of high or low). If that is the case you could try binning the means into classes of "high" "low" and "middle", for example, using percentiles, doing the same with the std dev. and then combining the bins into 9 classes (high mean and high std dev, high mean and middle std dev ... etc). Of course this is not exactly what you asked for, but it might do what I assume you want to do...
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