Say we add 5 to each element. We sum them up and divide by the number of elements (compute mean). Well we added 5n to that total sum and are dividing by n. So if the mean was 10 before, now it’s 15. (We had 10 datapoints added too 100, but we added 50, dividing by 10 we get 15).
Now every single data point is just as close to the mean as it was before. The mean shifted with 5, but so did the datapoints. Remember, variance is the sum of squared errors divided by n, or n-1 for sample. Well, the sum of squared errors did not change. So our estimate of variance remains the same as well as our estimate of standard deviation.
This is without assuming normality. (ie through the equation of mean and standard deviation themselves). In general expected values shift with constants, and variances remain stable.
