Why is standard deviation a good measure of central tendency
Answers
1) The square of the standard deviation is called the variance.
2) The variance is exactly the average value of the squared distance away from the middle (i.e. the mean) --wherever that middle might be.
3) This means that the variance is measuring something like how "far" away from the middle you "typically" are (where "far" means the square of the distance rather than the distance itself and "typically" means on average).
4) The standard deviation is the square root of the variance so it is also a measure of how "far" you "typically" are from the middle (where "far" now means the usual distance, but "typically" doesn't exactly mean on average anymore).
So the standard deviation measures how "spread out" a distribution is irrespective of where the center is. It is not trying to measure where the center is (like the mean, median, and mode do).
An important property of the standard deviation of a distribution is that if you translate the distribution (e.g. shift a density function horizontally in either direction), the standard deviation doesn't change. Obviously the middle of the distribution changes by exactly as much as the distribution is shifted.