Reproducibility of the K-Ar Ages 

of Rocks and Minerals: 

An Empirical Approach 

Reproducibility of the K-Ar Ages 

of Rocks and Minerals: 

An Empirical Approach 

By R.W. Tabor, R.K. Mark, and R.H. Wilson 

Derivation of a model for estimating 

the statistical error in K-Ar ages 

U.S. GEOLOGICAL SURVEY BULLETIN 1654 
DEPARTMENT OF THE INTERIOR 

DONALD PAUL HODEL, Secretary 

U.S. GEOLOGICAL SURVEY 

Dallas l. Peck, Director 

UNITED STATES GOVERNMENT PRINTING OFFICE 1985 

For sale by the 

Distribution Branch, Text Products Section 

U.S. Geological Survey 

604 South Pickett St. 

Alexandria, VA 22304 

Library of Congress Cataloging in Publication Data 

Tabor, Rowland Whitney, 1932-

Reproducibility of the K-Ar ages of rocks and minerals: An 

empirical approach. 

(U.S. Geological Survey Bulletin 1654) 

Bibliography: p. 5 

Supt. of Docs. no.: 119.3:1654 

1. Potassium-argon dating-statistical methods. 2. Error 

analysis (mathematics). I. Mark, Robert K. II. Wilson, 

R. H. Ill. Title. IV. Series: United States. Geological 

Survey. Bulletin 1654. 

QE75.B9 no. 1654 

[QE508] 

557.3s 

[551.7'01] 

8~00068 
CONTENTS 

Abstract 1 

Introduction 1 

Deriving the model 2 

Discussion 3 

References cited 5 

FIGURES 

1. Plot of coefficient of variation in K-Ar ages as a function of average fraction of 

radiogenic 40 Ar in multiply analyzed samples 2 

2. Plot of medians of pooled data 3 

3. Plot showing model derived from regression equation 4 

III 

Reproducibility of the K-Ar Ages of Rocks and Minerals: 

An Empirical Approach 

By R.W. Tabor, R.K. Mark, and R.H. Wilson 

Abstract 

The error assigned to potassium-argon ages is 

important to geologists who need to place the ages 

within a stratigraphic or historical context. Errors in 

many single analyses reported in the past have been 

based on theoretical equations and commonly have 

been optimistically small. We have derived a model 

for predicting the reproducibility of single ages based 

on replicate age determinations on 362 separate 

samples of rocks and minerals analysed in the U.S. 

Geological Survey's K-Ar isotope laboratory in Menlo 

Park, Calif. This empirically derived model agrees 

well in form with the theoretical models but predicts 

larger errors than the theoretical model for samples 

contaminated by a significant fraction of atmospheric 

argon. 

INTRODUCTION 

The error assigned to potassium-argon ages-the 

"mysterious ±," as it was called by Dalrymple and 

Lanphere (1969, p. 100)-has always been difficult to 

assign and commonly is not fully appreciated or understood by the geologist who uses the ages. Generally, 

errors in K-Ar ages are expressed in years, a value 

calculated from the standard deviation expressed as a 

percent (the coefficient of variation or standard 

deviation/average), and represent an estimate of 

precision based on the formula of Cox and Dalrymple 

(1967): 

sdAge= { (sdK2o>2+(sdt)2+(1/£_)2(sd40/38)2 

(1) 

where sdAge is the standard deviation of the age 

determination, sdK2o is the standard deviation of the 

potassium analysis, sdt is the standard deviation of the 

tracer calibration, sd40138 and sd36138 are the 

~tfnd~~ deviations of the measured 40 Ar/38 Ar and 

Ar/ Ar ratios, respectively, and r is the fraction of 40 Ar that is radiogenic. -

As originally discussed by Lipson (1958, p. 144), 

the value of r is the overwhelming factor in 

determining the reported error because, as the proportion of contaminating atmospheric 40 Ar increases, 

accurate measurement of the radiogenic-40 Ar content 

becomes more difficult. K2o-content errors are 

generally small. A large variation in the K20 analyses 

probably reflects inhomogeneities in the sample, a 

condition that bodes ill for precision in the argon 

analysis. Other sources of error are generally even 

smaller (see below). 

Although Cox and Dalrymple (1967, p. 2605) 

stressed that their formula is not exact and breaks 

down entirely for values of r approaching zero, it has 

been widely applied either in its original form or in a 

modified form for special needs (for example, Mahood, 

1981, p. 220-243). 

We have derived an empirical relation between 

the coefficient of variation in age (cv Age) of multiple 

analyzed samples and !:. which suggests that the error 

assigned to many single age determinations may be too 

small. Our model can be used to estimate the 

precision in ages based on single argon analyses when 

there is no reason to otherwise believe that the age is 

in error. 

Acknowledgments 

Many geochronologists supplied multipleextraction data for this study. We thank Julie 

Donnelly-Nolan (108 analyses), Robert Fleck (140), 

Wendy Hillhouse (62), Marvin Lanphere (57), Martha 

Pernokas (31), and James Smith (15) for use of their 

unpublished data. Six analyses are from McKee and 

Silberman (1970), 39 from Bailey and others (1976), and 

M. A. Lanphere (written commun., 1983), 75 from 

Duffield and others (1980) and M. A. Lanphere and G. 

B. Dalrymple (written commun., 1983), and 18 from 

Smith and Diggles (1981). Tabor, with the help of 

William Gaum, Jean Hetherington, Steve Connelley, 

and Kathleen Ort, supplied 52 analyses; and Wilson, 

with the help of Nora Shew, William Gaum, and Paige 

Herzon, supplied 226. Special thanks go to Kathleen 

Ort, Fred Zankowsky, and Mitchell Swanson for tabulating and computing the data in figures 1 through 3. We 

have benefited greatly from discussions with our 

colleagues within and outside of the K-Ar isotope 

laboratory in Menlo Park, Calif., especially G. Brent 

Dalrymple, Kenneth Fox, Ronald Kistler, Marvin 

Lanphere, Robert Fleck, Edwin McKee, and John 

Tinsley. 

Introduction 1 
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Figure 1. Coefficient of variation in ::tge (cvAge) as a function of average fraction of radiogenic 40 Ar (r) in 

multiply analyzed samples. Crosses, two analyses; circles, three analyses; squares, more than .three 

analyses. A total of 42 points with cvAge>60 and !_<0.1 are not plotted but were used in the regressions. 

DERIVING THE MODEL 

For this study, we collected 829 replicate 

analyses of samples, including hornblende, biotite, 

feldspar, and whole rock, from workers in the K-Ar 

isotope laboratory of the U.S. Geological Survey in 

Menlo Park, Calif. The analyses were made between 

1969 and 1983, using the isotope-dilution method of 

Dalrymple and Lanphere (1969). Argon-38 tracers 

were made by both batch and bulb methods. 

We calculated the cvAge of 362 separate 

samples; 293 were analyzed twice, 50 three times, 13 

four times, 3 five times, 1 six times, 1 seven times, 

and 1 ten times. Figure 1 plots the cv Age versus mean 

!.· For 70 percent of these samples, the standard 

deviation from the mean of the r is less than 0.10, the 

maximum is 0.26, and the overall average is 0.05. 

Because we assumed that the . cv Age of a sample 

analyzed many times is more reliable than that of one 

analyzed only twice, we weigh ted the data to give 

more influence to those sets of analyses derived from 

more than two extractions: a rock analyzed twice is 

treated as two datums, one analyzed three times as 

three datums, and so on. As we show below, this 

weighting becomes less influential because we use 

pooled means to derive the final equations. 

We used the University of Pennsylvania Minitab 

program (Ryan and others, 1976) to investigate a 

various regression models for cv Age. The only 

variable that has a high correlation with cv Age is r. 

Surprisingly, cvK20 does not appear to have any 

significant influence. We found little correlation 

between cvK2 0 and cv Age even when we limited the 

data to analyses with greater than 80 percent 

radiogenic argon. 

Figure 1 indicates that several points in each 

interval of r have a radically higher cv Age value than 

most of the data. We suspect that these points 

represent inhomogeneous rocks or minerals, or faulty 

analyses-clearly in the minority, although we have no 

objective reason to exclude them. Because these 

points unduly influence the averages in the pooled data 

2 Reproducibility of the K-Ar Ages of Rocks and Minera1s 
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Figure 2. Medians of pooled data. Numbers indicate number of analyses in each interval. 

sets, we used the medians, rather than the means, in 

the modeling. By using the medians, we have 

effectively decreased the influence of the outliers. 

We attempted to fit ~ mod~ in the for'!2 of 

equation 1 with terms (K20) , (1/!:) , and ({1-!:)/!J to 

our data. The model fits the pooled data well, as 

might be expected, but analysis of the regression 

indicates that only the term [(l-r)/r1 2 is significant. 

Although the importance of this term was clearly 

emphasized by Lipson (1958, p. 144) and Cox and 

Dalrymple (1967, p. 2605), we were unable to produce 

a satisfactory curve based on that term alone. Thus, 

we adopted an empirical approach and looked for the 

simplest equation to fit the data illustrated in figure 

1. We also tried constructing curves with selected 

data, such as less than 1 percent cvK20 and greater 

than 1 percent cvK2o, samples that yielded ages 

concordant with other phases of the same rock, and 

samples that did not. All these models gave approximately the same results. 

Because r (and, thus, cv Age) is not entirely 

independent of-the K2o content of the samples, we 

separated low-K20 samples-basalt, hornblende, 

plagioclase, and so on--from high-K2o samplesrhyolite, biotite, muscovite, K-feldspar, and so on. 

The derived model, though nearly congruent at large 

values of !J predicts somewhat better reproducibility 

in low K2o than in high-K 0 samples at low values of 

!..· For example, when !_=O.l, the difference in cv Age is 

about 2.5 percent, but considering the overall generalizations in the model, this difference is insignificant. 

As might be expected from examination of the 

pooled median values (fig. 2), the model derived from 

them has a good fit: 

cv Age=-2.42+3.38r - 0•

65• (2) 

DISCUSSION 

Our model (eq. 2; fig. 3) corresponds in general 

shape to the error curves constructed by Cox and 

Dalrymple (1967, p. 2606). We selected this model to 

have a low positive value (approx 1 percent) for cv Age 

when r=l. This remnant error can be viewed as the 

sum of errors unrelated to the uncertainty in 

measuring radiogenic argon versus atmospheric 

argon. The average cvK2o in our data set is 0.4 

Discussion 3 
0 

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Figure 3. Model derived from regression equation, cvAge=-2.43+3.38!,-o•65• 

percent (s.d., 0.3), and the errors in tracer calibration 

range from about 0.40 to 1.65 percent, although most 

are less than 1.0 percent. The quadrature sum is about 

1 percent. For comparison, Cox and Dalrymple (1967, 

p. 2605) estimated the error for K20 at 0.5 percent, 

for tracer calibration at 0.3 percent, and for measurement of 40 Ar/38 Ar ratio at 0.2 percent. According to 

Cox and Dalrymple (1967, p. 2605), the error in 

measurement of the 36 Ar/38 Ar ratio is about 1.5 to 2 

percent, but as little as 0.5 percent for small values of 

!:.· Our model most closely corresponds to Cox and 

Dalrymple's (1967, p. 2606) model that uses values of 

1.5 percent for the 36 Ar/38 Ar ratio, of 0.5 percent for 

the K2o content, and of 0.3 percent for tracer 

calibration. From our experience, cv36 Ar/38 Ar is 

rarely greater than 1.0 percent, except for r 

approaching 1. Because this factor is the squared 

coefficient of the all-important term (1-r)/r in 

equation 1, it leads to predictions of small error: -

The model in figure 3 predicts rather poor 

precision for samples with small !J though probably 

well within every geochronologist's experience. Our 

model predicts errors of the same magnitude, albeit 

somewhat more pessimistically than the estimates of 

error cited by Edwin McKee (written commun., 1982), 

based on a study of several hundred samples from the 

San Francisco volcanic field: "In general, the fraction 

of radiogenic 40 Ar is an index of the± figure. Samples 

with 20 percent or more 40 ArRad have a possible error 

of 5 percent or less of the calculated date, those with 

10 to 20 percent 40 ArRad about 10 percent, those with 

5-10 percent 40 ArRad about 20 percent, and those 

with less than 5 percent 40 ArRad as much as, or more 

than, 100 percent of the age." Dalrymple and 

4 Reproducibility of the K-Ar Ages of Rocks and Minerals 
Lanphere (1969, p. 119) suggest a preciSion of 2 to 5 

percent for samples with r greater than 30 percent. 

There are several weaknesses in the assumptions 

behind our error model. A primary one is that each 

sample analyzed is from a separate experiment with a 

separate set of circumstances; many procedures in the 

laboratory can increase the value of r, and they have 

little to do with the sample itself. Data in our set 

were not collected under uniform procedures to 

maximize r. Although averaging r makes the 

statistical approach a little neater, the -averages only 

crudely reflect the actual air contamination of the 

dated rna terial and, as such, can only crudely predict 

the reproducibility. As a result, the pooled data in the 

intervals of r close to zero (fig. 1) are of poor quality 

in comparison with those in the other intervals. For a 

few samples, we used average r values that included 

negative !. values because they are mathematically 

viable, even if physically meaningless. There seems to 

be no reliable way to predict the error in this region 

where !. approaches zero except in a gross statistical 

way. Thus, the errors predicted by the equation and 

the curves are probably unreliable at very low values 

of r. The cv Age of very young rocks probably will not 

be-accurately predicted by our model. With 

considerable care, reproducible ages can be obtained 

from very young rocks, as indicated throughout the 

literature and by the large number of data points in 

the lower left-hand corner of figure 1. In estimating 

precision, there is no substitute for multiple runs; and 

for estimating accuracy, no statistics can replace good 

stratigraphic control and numerous ages determined by 

various methods. 

All the data in our set came from one laboratory 

where procedures for extraction and spectometry are 

more or less uniform. Thus, our model might not be 

appropriate for data from other laboratories, although 

we expect that for a run-of-the-mill single analysis, 

this model will give a reasonable estimate of 

reproducibility. 

REFERENCES CITED 

Bailey, R.A., Dalrymple, G.B., and Lanphere, M.A., 

197 6, Volcanism, structure, and geochronology of 

Long Valley Caldera, Mono County, California: 

Journal of Geophysical Research, v. 81, no. 5, p. 

72.5-744. 

Cox, Allan, and Dalrymple, G.B., 1967, Statistical 

analysis of geomagnetic reversal data and the 

precision of potassium-argon dating: Journal of 

Geophysical Research, v. 72, no. 10, p. 2603-

2614. 

Dalrymple, G.B., and Lanphere, M.A., 1969, 

Potassium-argon dating: Principles, techniques, 

and applications to geochronology: San 

Francisco, W.H. Freeman and Co., 258 p. 

Duffield, W.A., Bacon, C.R., and Dalrymple, G.B., 

1980, Late Cenozoic volcanism, geochronology, 

and structure of the Coso Range, Inyo County, 

California: Journal of Geophysical Research, v. 

85, no. B5, p. 2381-2404. 

Lipson, J .I., 1958, Potassium-argon dating of 

sedimentary rocks: Geological Society of 

American Bulletin, v. 69, no. 2, p. 137-149. 

Mahood, G.A., 1980, The geological and chemical 

evolution of a Late Pleistocene rhyolitic 

center: The Sierra La Primavera, J alisco, 

Mexico: Berkeley, University of California' 

Ph.D. thesis, 257 p. 

McKee, E.H., and Silberman, M.L., 1970, 

Geochronology of Tertiary igneous rocks in 

central Nevada: Geological Society of America 

Bulletin, v. 81, no. 8, p. 2317-2327. 

Ryan, T.A., Joiner, B.L., and Ryan, B.F., 1976, 

Minita· A student handbook: North Scituate, 

Mass., Duxbury, 341 p. 

Smith, J.G. and Diggles, M.J., 1981, Potassium-argon 

determinations in the Ketchikan-Prince Rupert 

quadrangles, southeastern Alaska: U.S. 

Geological Survey Open-File Report 78-73-N, 16 

P• 

References Cited 5 


GPO 587-044/10025