You can get the mean and standard deviation of any set of numbers and come up with a Gaussian fitting them.
What does it mean if you do this with another distribution, such as a uniform distribution?
For instance, if i generate 1,000,000 uniform random numbers from 0 to 100, i get a mean of about 50 and a standard deviation of about 28.8.
Gaussians have the 68-95-99.7 rule (https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule) which say that 68% of the values are going to be +/- 1 standard deviation from the mean.
Using that logic, that says that 68% of the values in the uniform distribution are going to be between 50-28.8 and 50+28.8 aka between 21.2 and 78.8. those values have a range of 57.6 which is definitely not accurate for a uniform distribution. 68% of the samples should be +/-34 from the mean of 50.
When i plot the normalized histogram (pdf) of these two distributions, this is what I get.
So, does taking the standard deviation of a uniform distribution make sense? Can it tell me anything in particular other than a very general sense of how much dispersion the values have? It doesn't feel very useful when the standard deviation itself has to be interpreted differently based on what sort of distribution the data itself has. This doesn't seem to let you compare two pieces of data that may have different distributions.
Am i missing something?
Thanks!