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$$ \sigma^2 = \frac{\sum( I_i - I_{av} )}{n -1 } $$
where $\sigma^2$ is the variance, $\sigma$ is the standard deviation, $n$ is the size of the replicated spectra set with i being the corresponding index, $I_i$ is the peak intensity for spectrum $i$ , and $I_{av}$ is the mean over all spectra i .e. the sum of all peak intensities divided by $n$.
$$\sigma_{vol}^2 = \sigma_i^2 * N $$where $\sigma_{''σ<sub>vol}$ </sub>'' is the standard deviation of the volume, $\sigma_{''σ<sub>i}$ </sub>'' is the standard deviation of a single point assumed to be equal to the RMSD of the baseplane noise, <br>and $''N$ '' is the total number of points used in the summation integration method. For a box integration method, this converts to the<br>
$$<math>\sigma_{vol} = \sigma_i * \sqrt{(n*m)}$$</math>
Also {{note |there are a number of programs, for example peakint (http://hugin.ethz.ch/wuthrich/software/xeasy/xeasy_m15.html) that not use all points within the box. And if the number N can not be determined, this category of error analysis is not possible.}} {{note|Also for non-point summation methods, for example when line shape fitting is used to determine peak volumes, the equations above cannot be used. <br> Hence again this category of error analysis cannot be used. <br>This is the case for one of the three integration methods used by Sparky (http://www.cgl.ucsf.edu/home/sparky/manual/peaks.html#Integration). <br>And if fancy techniques are used, for example as Cara does to deconvolute overlapping peaks (http://www.cara.ethz.ch/Wiki/Integration), this again makes this error analysis impossible.}}
A problem often arises when the combination of several series of measurements performed under similar conditions is desired to achieve an improved estimate of the imprecision of the process. If it can be assumed that all the series are of the same precision although their means may differ, the pooled standard deviations $s_p$from $k$ series of measurements can be calculated as$$s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2+...+(n_k-1)s_k^2}{n_1+n_2+...+n_k-k}}$$The suffices $1, 2, ..., k$ refer to the different series of measurements. In this case it is assumed that there exists a single underlying standard deviation $\sigma$ of which the pooled standard deviation $s_p$ is a better estimate than the individual calculated standard deviations $s_1, s_2, ..., s_k$, For the special case where $k$ sets of duplicate measurements are available, the above equation reduces to$$s_p=\sqrt{\frac{\sum(x_{i1}-x_{i2})^2}{2k}}$$Results from various series of measurements can be combined in the following way to give a pooled relative standard deviation $s_{r,p}$:$$s_p=\sqrt{\frac{\sum(n_i-1)s_{r,i}^2}{\sum n_i -1}}$$ == See also ==[[Category:Analysistechniques]]
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== Intensity Spectrum error analysis ==
[http://www.nmr-relax.com/manual/spectrum_error_analysis.html See the manual]
=== Peak heights with partially replicated spectra ===
When spectra are replicated, the variance for a single spin at a single replicated spectra set is calculated by the formula
<math>\sigma^2 = \frac{\sum( I_i - I_{av} )}{n -1 }</math> where σ<sup>2</sup> is the variance, σ is the standard deviation, ''n'' is the size of the replicated spectra set with ''i'' being the corresponding index, ''I<sub>i</sub>'' is the peak intensity for spectrum ''i'' , and ''I<sub>av</sub>'' is the mean over all spectra i.e. the sum of all peak intensities divided by ''n''. As the value of $''n$ '' in the above equation is '''always very low''' since normally only a couple of spectra are collected per replicated spectra set, '''the variance of all spins''' is '''averaged''' for a '''single replicated spectra set'''. <br>
Although this results in all spins having the same error, the accuracy of the error estimate is significantly improved.
Then the error is simply given by the sum of variances:
<math>
\sigma_{vol}^2 = \sigma_i^2 * N
</math>
Nicholson, Kay, Baldisseri, Arango, Young, Bax, and Torchia (1992) Biochemistry, 31: 5253-5263 equation:
where $''n$ '' and $''m$ '' are the dimensions of the box.<br>Note that a number of programs, for example peakint (http://hugin.ethz.ch/wuthrich/software/xeasy/xeasy_m15.html) does not use all points within the box.<br>And if the number N can not be determined, this category of error analysis is not possible.
=== Peak volumes with partially replicated spectra ===
With all spectra replicated and again using any integration methodology, the intensity errors can be calculated as described in the `Peak heights with all spectra replicated' section above.
== IUPAC - pooled standard deviation See also ==# [http://goldbook.iupac.org/P04758.html IUPAC :pooled standard deviation[RMSD]]