Mean square weighted deviation

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.

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Mean squared error — In statistics, the mean squared error (MSE) of an estimator is one of many ways to quantify the difference between values implied by a kernel density estimator and the true values of the quantity being estimated. MSE is a risk function,… …   Wikipedia

  • Root mean square deviation (bioinformatics) — The root mean square deviation (RMSD) is the measure of the average distance between the backbones of superimposed proteins. In the study of globular protein conformations, one customarily measures the similarity in three dimensional structure by …   Wikipedia

  • Mean — This article is about the statistical concept. For other uses, see Mean (disambiguation). In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). the expected value of a… …   Wikipedia

  • Weighted mean — The weighted mean is similar to an arithmetic mean (the most common type of average), where instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean… …   Wikipedia

  • mean — mean1 /meen/, v., meant, meaning. v.t. 1. to have in mind as one s purpose or intention; intend: I meant to compliment you on your work. 2. to intend for a particular purpose, destination, etc.: They were meant for each other. 3. to intend to… …   Universalium

  • Standard deviation — In probability and statistics, the standard deviation is a measure of the dispersion of a collection of values. It can apply to a probability distribution, a random variable, a population or a data set. The standard deviation is usually denoted… …   Wikipedia

  • Geometric mean — The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word average, except… …   Wikipedia

  • Median absolute deviation — In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample. For a… …   Wikipedia

  • Variance — In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value (mean). Whereas… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”