U-Pb Geochronology Practcal: Background Basc Concepts: accuracy: measure of the dfference between an expermental measurement and the true value precson: measure of the reproducblty of the expermental result random error: unpredctable fluctuatons n observatons that yeld results that dffer from experment to experment systematc error: errors that cause the measurement to dffer from the true values wth reproducble dscrepancy Sample versus Populaton A sample s a group of data whch s a subset of an nfnte populaton. populaton dstrbuton: the true dstrbuton of values of a gven quantty sample dstrbuton: the measured dstrbuton of those values; n the lmt of nfnte measurements, the sample dstrbuton approaches the parent dstrbuton How do we descrbe the sample versus populaton dstrbutons? Often wth concepts lke the mean, the varance, and the standard devaton. Parametrc statstcs The mode s the most probable value of x The medan (x / ) s the value for whch half the observatons are greater and half less The mean (x) s the average value of a quantty x x = n n = x µ = lm N N x
The devaton (d ) of a measurement x from the mean µ of the parent dstrbuton s the dfference between x and µ d = x x d = x µ The average devaton, α, s the (n the lmt of nfnte measurements) the average of the absolute values of the devatons from the mean µ but the absolute value s computatonally nconvenent α = x x n a = lm N N x µ The varance, σ, s (n the lmt of nfnte measurements) the average of the squares of the devatons from the mean µ σ = n ( x x ) s = lm N N ( x µ ) The standard devaton, σ, s (n the lmt of nfnte measurements) the square root of the varance; the standard devaton estmates the lkelhood of a measurement fallng wthn a certan nterval about the mean: σ = σ s = s The standard error of the mean s a measure of how well the sample mean, x, estmates the populaton mean, µ t does not represent the same thng as the standard devaton! It s calculated from the standard devaton dvded by the square root of the number of measurements: σ mean = σ n Note that as n approaches nfnty, the standard error approaches zero, e.g. the sample mean must approach the populaton mean.
Dstrbutons Recall that a hstogram s a graph of the frequency of bnned measurements; the apparent shapes of dstrbutons llustrated n hstograms are hghly senstve to the choce of bn sze A probablty densty functon appears as the smooth curve over the columns of a hstogram, n the lmt of nfnte measurements. The shape of the probablty densty functon s ndependent of the choce of bn sze, and thus always looks the same unlke assocated hstograms. The most commonly observed (or assumed) dstrbuton n scence and engneerng s the Gaussan or normal dstrbuton, whch s the symmetrc bell-shaped curve arsng when the number of dfferent possble outcomes s nfnte but the probablty of each s fnte: $ P G = & P G = lm ' & N s ' + ) exp $ - & π (,- x x ' ) (. 0 / 0 (, * exp x µ (. ' * π ) -. & s ) / 0 Ths equaton yelds the unt normal dstrbuton; the ntegral under the curve s of unt area (=). It can be subsequently scaled to n measurements.
Confdence Intervals The probablty that a measurement wll dffer from x by some amount Δx s called the confdence nterval; t s smply the area under the unt probablty densty functon, bound by ±Δx: A G = x +Δx x Δx $ & ' + ) exp $ - & π (,- x x ' ) (. 0 dx / 0 For a Gaussan probablty functon, If Δx = σ, then A G = 0.6869 ~ 68 the σ confdence nterval If Δx = σ, then A G = 0.95449 ~ 95 the σ confdence nterval Schoene et al. (03) Note that n hgh-precson geochronology the tradton s to tabulate and llustrate analytcal data wth ther σ (95) confdence nterval uncertantes. By contrast, n stu geochronologcal methods (on probe, LA-ICPMS) tradtonally llustrate ther data wth σ (68) confdence nterval uncertantes, e.g.
Constructng Probablty Functons (and Hstograms) Two approaches: ) Assume that the total sample dstrbuton can be modeled as a specfc probablty densty functon. Then, the area under a normalzed probablty densty functon s made equal to the area of the equvalent hstogram. For example: a) Calculate the unt Gaussan probablty functon from the approprate eqn. b) Normalze the probablty functon (Pg) by: y(x ) = P G (x ) nδx ) Assume that each measurement has a normally dstrbuted uncertanty wth a known varance. Then you can assgn a unt Gaussan curve to each measurement, and then sum all these up along the x-axs. Agan the area under the probablty densty functon s equal to the area under the hstogram. The beneft of ths approach s that t can handle non-unform varances for each observaton. Weghted Mean and Varance Followng from the dea that the ndvdual observatons wthn a sample may have non-unform varance, t s common to weght each observaton by the nverse of ts varance when calculatng the mean of the sample dstrbuton: (x /σ ) x = (/σ ) σ x = (/ ) The standard error of the weghted mean decreases wth the number of measurements.
Statstcal evaluaton of the best-ft soluton How do we evaluate how well a best-ft soluton, n ths case the weghted mean of a normal dstrbuton, descrbes our data dstrbuton? We can defne our goodness-of-ft parameter, χ, as the sum of the squares of the devatons of our observatons from our model mean value, weghted by the varances of those observatons: χ = n = " $ # x x In geochronology, we have a tradton of normalzng the χ statstc by the degrees of freedom of the system to derve the Mean Squared Weghted Devaton or MSWD (e.g. Wendt and Carl, 99). Ths statstc quantfes the extent to whch data scatter from the best-ft soluton beyond stated uncertantes. ' & MSWD = χ f = χ n ƒ s the degrees of freedom of the model, e.g. the number of expermental parameters (measurements) number of model parameters (unknowns) The expectaton (mean) value of the MSWD s, n other words: ) MSWD = scatter of data about the model s accurately descrbed by analytcal uncertanty ) MSWD >> scatter of data about the model s beyond analytcal uncertantes; the model s a poor ft to the data or the expermental error s underestmated 3) MSWD << scatter of data about the model s much less than the analytcal uncertanty; expermental error s probably overestmated Wendt and Carl (99) establshed the frequency dstrbuton of the MSWD about the mean value of, as a smplfed functon of ƒ: σ MSWD = f = n
References and further readng Bevngton, P.R., and Robnson, D.K., 003, Data Reducton and Error Analyss for the Physcal Scences, 3 rd Ed.: McGraw-Hll, New York, 30 p. Ludwg, K., 000, Decay constant errors n U Pb concorda-ntercept ages: Chemcal Geology, v. 66, no. 3-4, p. 35 38. Ludwg, K., 998, On the treatment of concordant uranum-lead ages: Geochmca et Cosmochmca Acta, v. 6, no. 4, p. 665 676. Ludwg, K.R., 003, Isoplot 3.00: a geochronologcal toolkt for Mcrosoft Excel: Berkeley Geochronology Center Specal Publcaton, v. 4, p. 7. McLean, N.M., Bowrng, J.F., and Bowrng, S.A., 0, An algorthm for U-Pb sotope dluton data reducton and uncertanty propagaton: Geochemstry Geophyscs Geosystems, v., no. null, p. Q0AA8, do: 0.09/00GC003478. Schmtz, M.D., and Schoene, B., 007, Dervaton of sotope ratos, errors, and error correlatons for U-Pb geochronology usng 05Pb- 35U-( 33U)-spked sotope dluton thermal onzaton mass spectrometrc data: Geochemstry Geophyscs Geosystems, v. 8, no. 8, p. 0, do: 0.09/006GC0049. Schoene, B., Condon, D.J., Morgan, L., and McLean, N., 03, Precson and Accuracy n Geochronology: Elements, v. 9, no., p. 9 4, do: 0.3/gselements.9..9. Wendt, I., and Carl, C., 99, The statstcal dstrbuton of the mean squared weghted devaton: Chemcal geology. Isotope geoscence secton, v. 86, no. 4, p. 75 85.