This is definitely a very basic question but it is something which is in my head for some time now as I struggle to get the real meaning or sense of the standard deviation. How to apply it is not difficult, I guess it is quite self-explained. But how to interprete it?
The standard deviation looks as following
and it is defined as
$ s = \sqrt{ \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x} )^2}$
So finally it is meant such that ~68 % of the data falls within one standard deviation, and ~95 % within two standard deviations.
My questions are:
- How many standard deviations will cover all data? Infinite?
- When you want to detect outliers, it is quite common to define that data within 3-5 standard deviations will be inliers and the rest outliers. So due to the definition, there will always be outliers (because 3-5 sd do not cover all data). What is the sense of that?
- For online/streaming data the standard deviation is probably subject to change so it is hard to apply anyways sensefully.
- When values are measured it is also common to write it such as Y = 50 +/- 1 std. What does this mean?
- Why is the normal distribution finally so important?