How to obtain Incremental standard deviations from a set of standard deviations?

Question

I have a data set containing three columns, first column represents number of trials, second column represents experimental values, and the third column represents corresponding standard deviation.

With each experiment there is an increment in my experimental values. To get the incremental values, I hold my first value as the reference value and subtract this reference value from each subsequent value and use them to create fourth column of these incremental values.

My problem begins right from here. How do I create a new set of incremental standard deviations for the incremental experimental values I got? My apology if the problem is not well defined but hopefully someone will eventually be able to help me out. Many thanks!

Below is my data set,

Trial   Mean    SD      Incr Mean   Incre SD
1      45.311   4.668   0   
2      56.682   2.234   11.371  
3      62.197   2.266   16.886  
4      70.550   4.751   25.239  
5      80.528   4.412   35.217  
6      87.453   4.542   42.142  
7      89.979   2.185   44.668  
8      96.859   3.476   51.548  

Answer 1

To be clear, for other readers, your incremental mean is actually the difference between trial 1 and the other trials.

Variances add directly when you subtract (or add) independent normal distributions. So you first want to convert that standard deviation to a variance by squaring it, and then you can add the variances, and then you can take the square root to turn it back into a standard deviation. Note when using this kind of Pythagorean combination, you are assuming that trial 1 is independent from the trials, so for example, you cannot do things like have some sample in both trials.

Logically this makes sense that your so called "incremental SD" will always be greater than the individual SDs, since the uncertainty of both distributions contributes towards the uncertainty of the difference.