So we have arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM). Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., Harmonic mea...
The mean is the number that minimizes the sum of squared deviations. Absolute mean deviation achieves point (1), and absolute median deviation achieves both points (1) and (3).
Context is everything here. Are these theoretical variances (moments of distributions), or sample variances? If they are sample variances, what is the relation between the samples? Do they come from the same population? If yes, do you have available the size of each sample? If the samples do not come from the same population, how do you justify averaging over the variances?
What do you mean by "the derivative at 1 SD is +- 1"? Derivative of what? If you mean of a density plot, then what distribution? The normal? Different distributions will have different derivatives at 1 SD from the mean.
What does it imply for standard deviation being more than twice the mean? Our data is timing data from event durations and so strictly positive. (Sometimes very small negatives show up due to clock
The mean you described (the arithmetic mean) is what people typically mean when they say mean and, yes, that is the same as average. The only ambiguity that can occur is when someone is using a different type of mean, such as the geometric mean or the harmonic mean, but I think it is implicit from your question that you were talking about the arithmetic mean.
The above calculations also demonstrate that there is no general order between the mean of the means and the overall mean. In other words, the hypotheses "mean of means is always greater/lesser than or equal to overall mean" are also invalid.
After calculating the "sum of absolute deviations" or the "square root of the sum of squared deviations", you average them to get the "mean deviation" and the "standard deviation" respectively. The mean deviation is rarely used.
Hence, the mean acts as the balancing point in a distribution. This visual allows an immediate understanding of the mean as it relates to the distribution of the data points. Other property of the mean that becomes readily apparent from this demonstration is the fact that the mean will always be between the min and the max values in the ...
I have represented standard deviation as "±SD" before in publications. But I like to have opinions on this. Is it appropriate to use the notation '±' with SD ? Or ...
