Mean square weighted deviation

Mean square weighted deviation is used extensively in geochronology, the science of obtaining information about the time of formation of, for example, rocks, minerals, bones, corals, or charcoal, or the time at which particular processes took place in a rock mass, for example recrystallization and grain growth, or alteration associated with the emplacement of metalliferous ore deposits..

Often the geochronologist will determine a series of age measurements on a single sample, with the measured value $\ {x_i}$ having a weighting $\ {w_i}$ and an associated error $\sigma_{x_{i}}$ for each age determination. As regards weighting, one can either weight all of the measured ages equally, or weight them by the proportion of the sample that they represent. For example, if two thirds of the sample was used for the first measurement and one third for the second and final measurement then one might weight the first measurement twice that of the second.

The arithmetic mean of the age determinations is:

$\overline{x} = \frac{\sum_{i=1}^N x_i}{N}$

but this value can be misleading unless each determination of the age is of equal significance.

When each measured value can be assumed to have the same weighting, or significance, the biased and unbiased (or "population" and "sample", respectively) estimators of the variance are computed as follows:

$\sigma^2 = \frac{\sum_{i=1}^N (x_i - \overline{x})^2}{N} {\rm \ \ and\ \ } s^2 = \frac{N}{N-1}\cdot\sigma^2 = \frac{N}{N^2-N}\cdot\sum_{i=1}^N (x_i - \overline{x})^2.$

The standard deviation is the square root of the variance.

When individual determinations of an age are not of equal significance it is better to use a weighted mean to obtain an 'average' age, as follows:

$\overline{x}^{\,*} = \frac{\sum_{i=1}^N w_i x_i}{\sum_{i=1}^N w_i}$

The biased weighted estimator of variance can be shown to be:

$\sigma^2 = \frac{\sum_{i=1}^N w_i (x_i - \overline{x}^{\,*})^2}{\sum_{i=1}^N w_i}$

which can be computed on the fly as

$\sigma^2 = \frac{\sum_{i=1}^N w_i x_i^2 \cdot \sum_{i=1}^N w_i - (\sum_{i=1}^N w_i x_i)^2} {(\sum_{i=1}^N w_i)^2}$

The unbiased weighted estimator of the sample variance can be computed as follows:

$s^2 = \frac{\sum_{i=1}^N w_i}{{(\sum_{i=1}^N w_i})^2 - {\sum_{i=1}^N w_i^2} } \ . \ {\sum_{i=1}^N w_i (x_i - \overline{x}^{\,*})^2}$

Again the corresponding standard deviation is the square root of the variance.

The unbiased weighted estimator of the sample variance can also be computed on the fly as follows:

$s^2 = \frac{\sum_{i=1}^N w_i x_i^2 \cdot \sum_{i=1}^N w_i - (\sum_{i=1}^N w_i x_i)^2}{(\sum_{i=1}^N w_i)^2 - \sum_{i=1}^N w_i^2 }$

The unweighted mean square of the weighted deviations (unweighted MSWD) can then be computed, as follows:

MSWD $_u = \frac{1}{N-1} \ . \ \sum_{i=1}^N\frac{ (x_i - \overline{x})^2}{\sigma_{x_i}^2 }$

By analogy the weighted mean square of the weighted deviations (weighted MSWD) can be computed, as follows:

MSWD $_w = \frac{\sum_{i=1}^N w_i}{(\sum_{i=1}^N w_i)^2 - \sum_{i=1}^N w_i^2 } \ . \ \sum_{i=1}^N \frac{w_i . (x_i - \overline{x}^{\,*})^2}{(\sigma_{x_i})^2 }$

Notes and references

McDougall, I. and Harrison, T.M. 1988. Geochronology and Thermochronology by the ⁴⁰Ar/³⁹Ar Method. Oxford University Press.

Dickin, A.P. 1995. Radiogenic Isotope Geology. Cambridge University Press, Cambridge, UK, 1995, ISBN 0521431514, 0521598915

Examples of MSWD in current practical use can be found below

Lance P. Black, Sandra L. Kamo, Charlotte M. Allen, John N. Aleinikoff, Donald W. Davis, Russell J. Korsch, Chris Foudoulis 2003. TEMORA 1: a new zircon standard for Phanerozoic U–Pb geochronology. Chemical Geology 200, 155-170.

M.J. Streule, R.J. Phillips, M.P. Searle, D.J. Waters and M.S.A. Horstwood 2009. Evolution and chronology of the Pangong Metamorphic Complex adjacent to themodelling and U-Pb geochronology Karakoram Fault, Ladakh: constraints from thermobarometry, metamorphic modelling and U-Pb geochronology. Journal of the Geological Society 166, 919-932 doi:10.1144/0016-76492008-117

Discussions of the basic mathematical principles

Roger Powell, Janet Hergt, Jon Woodhead 2002. Improving isochron calculations with robust statistics and the bootstrap. Chemical Geology 185, 191-204.