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...
When studying two independent samples means, we are told we are looking at the "difference of two means". This means we take the mean from population 1 ($\\bar y_1$) and subtract from it the mean from
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).
One takes the pairwise difference of each point of data [ the mean of the differences ] and the other takes mean A and subtracts it from mean B [ the difference of the means ]. While the differences can be calculated to come out the same, the confidence intervals for each are different. I am confused as to which formula to use for which situation.
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 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.
Remember that the sample mean $\bar x$ is itself a random variable. So the first formula tells you the standard deviation of the random variable $\bar x$ in terms of the standard deviation of the original distribution and the sample size.
