Estimating standard deviation from paired data

[wesbell]

I work advising chemical analysts as best I can on statistics.
There is one issue that I am really struggling with and would really appreciate some help.
The protocol used to determine the overall uncertainty of the methods is ot use a 'top down' approach, i.e. run standard materials of known value through the method and calculate the standard deviation from the obtained results. There are however some methods where this is not possible as no standard materials are available, so in this case samples are analysed in duplicate and the difference of each pair used to estimate the standard deviation.
My question is what is the best way to use this paired data to obtain a standard deviation.
I have been using a method given in an old stats book that uses the mean range divided by a factor (1.128), but one of the analysts has been using the square root of the sum of the squared differences that have been divided by 2*the number of pairs.
Which of these methods, if any, would be the most statistically correct option?

[Dragan]

I work advising chemical analysts as best I can on statistics.
There is one issue that I am really struggling with and would really appreciate some help.
The protocol used to determine the overall uncertainty of the methods is ot use a 'top down' approach, i.e. run standard materials of known value through the method and calculate the standard deviation from the obtained results. There are however some methods where this is not possible as no standard materials are available, so in this case samples are analysed in duplicate and the difference of each pair used to estimate the standard deviation.
My question is what is the best way to use this paired data to obtain a standard deviation.
I have been using a method given in an old stats book that uses the mean range divided by a factor (1.128), but one of the analysts has been using the square root of the sum of the squared differences that have been divided by 2*the number of pairs.
Which of these methods, if any, would be the most statistically correct option?

If we let the difference between the i-th first observation and the i-th second observation be:



where i = 1,..., N and where the mean of the difference scores is:

,

then (I wonder) why not just use the standard deviation (S) for the difference scores as: