Gurney equations

Mean square weighted deviation is a statistical method used extensively in geochronology also known as the reduced chi-squared.

The Mean Square Weighted Deviation (MSWD) is a measure of goodness of fit that takes into account the relative importance of both the integral and external reproducibility. In general when:

MSWD = 1 if the age data fit a univariate normal distribution in t (for the arithmetic mean age) or log(t) (for the geometric mean age) space, or if the compositional data fit a bivariate normal distribution in [log(U/He),log(Th/He)]-space (for the central age).

MSWD < 1 if the observed scatter is less than that predicted by the analytical uncertainties. In this case, the data are said to be "underdispersed", indicating that the analytical uncertainties were overestimated.

MSWD > 1 if the observed scatter exceeds that predicted by the analytical uncertainties. In this case, the data are said to be "overdispersed". This situation is the rule rather than the exception in (U-Th)/He geochronology, indicating an incomplete understanding of the isotope system. Several reasons have been proposed to explain the overdispersion of (U-Th)/He data, including unevenly distributed U-Th distributions and radiation damage.

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 $σ_{x_{i}}$ 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:

σ^{2} = \frac{\sum_{i = 1}^{N} (x_{i} - \overline{x})^{2}}{N} a n d s^{2} = \frac{N}{N - 1} \cdot σ^{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:

\overset{*}{\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:

σ^{2} = \frac{\sum_{i = 1}^{N} w_{i} (x_{i} - \overset{*}{\overline{x}})^{2}}{\sum_{i = 1}^{N} w_{i}}

which can be computed on the fly as

σ^{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} - \overset{*}{\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}}{σ_{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} - \overset{*}{\overline{x}})^{2}}{(σ_{x_{i}})^{2}}

Notes and references

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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 0-521-43151-4, ISBN 0-521-59891-5

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 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.

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.

Gurney equations

Notes and references

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