Editing Weighted arithmetic mean (section)

==== Bootstrapping validation ====

It has been shown, by Gatz et al. (1995), that in comparison to [[bootstrapping (statistics)|bootstrapping]] methods, the following (variance estimation of ratio-mean using [[Taylor series]] linearization) is a reasonable estimation for the square of the standard error of the mean (when used in the context of measuring chemical constituents):<ref>{{cite journal |last1=Gatz |first1=Donald F. |last2=Smith |first2=Luther |title=The standard error of a weighted mean concentration—I. Bootstrapping vs other methods |journal=Atmospheric Environment |date=June 1995 |volume=29 |issue=11 |pages=1185–1193 |doi=10.1016/1352-2310(94)00210-C|bibcode=1995AtmEn..29.1185G }} - [https://www.cs.tufts.edu/~nr/cs257/archive/donald-gatz/weighted-standard-error.pdf pdf link]</ref>{{rp|1186}}

:<math>
\widehat{\sigma_{\bar{x}_w}^2} = \frac{n}{(n-1)(n \bar{w} )^2}  \left[\sum (w_i x_i - \bar{w} \bar{x}_w)^2 -
2 \bar{x}_w \sum (w_i - \bar{w})(w_i x_i - \bar{w} \bar{x}_w)
+ \bar{x}_w^2 \sum (w_i - \bar{w})^2 \right]
</math>

where <math>\bar{w} = \frac{\sum w_i}{n}</math>. Further simplification leads to

:<math>\widehat{\sigma_{\bar{x}}^2} = \frac{n}{(n-1)(n \bar{w} )^2}  \sum w_i^2(x_i - \bar{x}_w)^2</math>

Gatz et al. mention that the above formulation was published by Endlich et al. (1988) when treating the weighted mean as a combination of a weighted total estimator divided by an estimator of the population size,<ref>{{Cite journal| doi = 10.1175/1520-0450(1988)027<1322:SAOPCM>2.0.CO;2| volume = 27| issue = 12| pages = 1322–1333| last1 = Endlich| first1 = R. M.| last2 = Eymon| first2 = B. P.| last3 = Ferek| first3 = R. J.| last4 = Valdes| first4 = A. D.| last5 = Maxwell| first5 = C.| title = Statistical Analysis of Precipitation Chemistry Measurements over the Eastern United States. Part I: Seasonal and Regional Patterns and Correlations| journal = Journal of Applied Meteorology and Climatology| date = 1988-12-01 | doi-access = free| bibcode = 1988JApMe..27.1322E}}</ref> based on the formulation published by Cochran (1977), as an approximation to the ratio mean. However, Endlich et al. didn't seem to publish this derivation in their paper (even though they mention they used it), and Cochran's book includes a slightly different formulation.<ref name = "Cochran1977">Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Nashville, TN: John Wiley & Sons. {{ISBN|978-0-471-16240-7}}</ref>{{rp|155}} Still, it's almost identical to the formulations described in previous sections.