by
lmh » Mon Aug 06, 2012 9:45 am
the square-root thing is nothing to do with chromatography, it's just statistics in action, and it's only half the story.
There are two sorts of error when sampling across a peak, errors in the y-direction and errors in the x-direction (area of a peak is, of course, affected by both). It's easiest to think about errors in the x-direction for a hypothetical square-wave shaped peak. If we managed to synchronise our detector such that it always started to measure just as the peak began, we'd have an uncertainty on the peak's width, because the peak may have stopped anywhere between our last non-baseline peak, and the first base-line peak. This error is uniformly distributed, not normally distributed. If you were to decrease the time between points linearly, you would see the error increase linearly to a point, instantly drop, and then increase linearly again, but each saw-tooth would be a little smaller than the one before.
This average size of this error is linearly related to the sampling time. Half the sampling time means half the error.
The Finnigan square-root on S:N ratio is, I suspect, another thing altogether: it's looking at the y-error. Each point is an estimate of the height of the peak (at that time). The more estimates you have, the better your knowledge of the peak height. If each measurement has a standard deviation (which is constant, and is "noise"), then our best estimate of the peak's height ("signal") is the standard error of lots of measurements, and the standard error decreases with the square root of the number of measurements we make (SE = sd/sqrt(n)), so 1/100th the scanning time means 100 times as many points, which means a 10-times better SE estimate of the y-measurement of the peak.
Another way to look at it, is that there isn't much difference between calculating a peak-area and calculating a mean of a load of measurements. A mean is the sum of lots of values divided by n. An area is a weighted sum of lots of values multiplied by their spacing. The error of a mean is therefore influenced by the same things as the error in an area, and we can apply everything we know about standard errors to the calculation of peak areas.