I tried a group of numbers, four 1, four -1, a 24, a 25, the mean of these ten numbers is 4.9, the standard deviation is 9.8 (using Excel's STDEVP function). Another group of numbers, six 0, two 10, a 14, a 15, mean 4.9, standard deviation is 6.17.
As long as the increase in the gap between the data, the standard deviation must be expanded, but the expansion of the gap at the same time, the average can remain unchanged.
Also, if the numbers are non-negative, it is possible to realize a situation where the standard deviation is greater than the mean. For example, nine 1s, a 100, have a mean of 10.9 and a standard deviation of 29.7.
The problem may be related to a common phenomenon: the larger the mean, the larger the standard deviation often is. But this experience does not prevent the use of the standard deviation formula to calculate extreme values.
Additionally, values that are too different from one another do not usually converge in the same set of data in a social phenomenon - a "homogeneous aggregate" - and individual differences are too large to constitute a "homogeneous aggregate". "
This is the first time that we have seen a data set that is not homogeneous.