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So we have arithmetic mean (am), geometric mean (gm) and harmonic mean (hm) 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. Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., harmonic mea.

What is the Central Tendency in Statistics? (Mean, Mode, Median and

The mean is the number that minimizes the sum of squared deviations Remember that the sample mean $\bar x$ is itself a random variable Absolute mean deviation achieves point (1), and absolute median deviation achieves both points (1) and (3).

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?

If you mean of a density plot, then what distribution 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.

I'm struggling to understand the difference between the standard error and the standard deviation How are they different and why do you need to measure the standard. 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.