The repeatability, reproducibility, and sources of error inherent in a given measurement are important considerations for potential users. To quantify errors arising from a single operator or multiple laboratories, most testing standards uses a oneway analysis of variance (ANOVA) based method, which utilizes a simple standard deviation across all measurements. However, this method does not allow users to quantify the sources of error and capacity (i.e., the precision to tolerance ratio). In this study, an innovative twoway ANOVAbased analysis method is selected to quantify the relative contributions of different sources of error and determine whether a measurement can be used to check conformance of a measured characteristic to engineering specifications. In this study, the standardized Atterberg limits tests, fallcone device Atterberg limits tests, and bar linear shrinkage tests widely used for determining the soil plasticity were selected for evaluation and demonstration. Comparisons between results of the various testing methods are presented, and the error sources contributing to the overall variations between tests are discussed. Based on the findings of this study, the authors suggest use of twoway ANOVAbased R&R analysis to quantify the sources of measurement error and capacity and also recommend using the fall cone device and ASTM standardized thread rolling device for determining liquid and plastic limits of soils, respectively.
The repeatability (i.e., the single operator or intralaboratory precision) and reproducibility (the interlaboratory precision) of a measurement are important characteristics which can be quantified to enable users to understand the variability of test results. The ASTM E691 standard practice on interlaboratory testing states “ASTM standard regulations require precision statements in all test methods in terms of repeatability and reproducibility” and specifies a oneway ANOVA (i.e., a simple standard deviation across all measurements) to quantify the singleoperator or multilaboratory errors [
To address these issues, the authors propose to use a twoway ANOVAbased R&R analysis to evaluate the repeatability, reproducibility, sources of error, and capacity of a measurement. In this study, several widely used Atterberg limits testing devices are employed for comparisons and demonstration of the statistical analysis. The ASTMstandardized Atterberg limits tests [
Swedish soil scientist Atterberg defined moisture content limits to delineate transitions in the consistency of finegrained soils [
Since Terzaghi introduced Atterberg’s limits into modern soil mechanics practice and Casagrande standardized the testing devices, the liquid and plastic limit tests have been extensively performed in geotechnical engineering and soil science fields worldwide [
In 1932, Casagrande developed a device to standardize the liquid limit (LL) test and in 1949 further refined the design to overcome inherent shortcomings [
The fall cone test device was also evaluated for determination of the plastic limit (PL) of soils by [
An example of using fall cone test results to determine LL and PI.
PaigeGreen and Ventura concluded that the bar linear shrinkage (BLS) test result is a good indicator of the plasticity of soils [
Trough designs and drying methods for various BLS testing standards.
Parameter  South Africa TMH1 A4  British standard BS1377  Australia P6A/1  Texas 107E  California CTM228 

Crosssection shape  Square  Semicircular  Semicircular  Square  Tapered 
Crosssection dimension (mm)  10 × 10  25 diameter  25 diameter  19 × 19  Top width 19.05, bottom width 17.48 
Length (mm)  150  140  135 or 250  127  127 
Drying method  Oven dry (110°C)  Air dry + oven dry (65 and 110°C)^{a}  Air dry (24 h) + oven dry (110°C)  Air dry^{b} + oven dry (110°C)  Air dry^{b} + oven dry (110°C) 
^{a}Place the mold where the soil/water can airdry slowly in a position free from drafts until the soil has shrunk away from the walls of the mold. Then complete the drying, first at a temperature not exceeding 65°C until shrinkage has largely ceased and then at 105°C to 110°C to complete the drying. ^{b}Airdry the soil bar at room temperature until color changes slightly.
In this study, a total of five samples were prepared by incorporating different percentages of pure bentonite powder into the minus No. 40 fraction of existing crushed limestone fines. The incorporated bentonite content by dry mass of the minus No. 40 material was increased from 0% to 12% in 3% increments. The sieve analysis, hydrometer analysis, and Atterberg limits test results of the initial full granularsurface material gradation are shown in Figure
Particle size distribution of the granularsurface material.
The chemical composition and mineralogy of the bentonite were determined using Xray fluorescence (XRF) Xray diffraction (XRD) analyses, respectively. The XRD results showed that the bentonite was sodium montmorillonite (Na_{0.3}(Al,Mg)_{2}Si_{4}O_{10}(OH)_{2}·4H_{2}O) with calcite (CaCO_{3}) and quartz (SiO_{2}). The chemical composition determined by the XRF results is shown in Table
Chemical composition of the bentonite used in the lab study.
Chemical component  Percent 

SiO_{2}  58.77 
Al_{2}O_{3}  20.66 
Fe_{2}O_{3}  3.81 
SO_{3}  0.86 
CaO  2.42 
MgO  3.61 
Na_{2}O  2.45 
K_{2}O  0.62 
P_{2}O_{5}  0.08 
TiO_{2}  0.18 
SrO  0.03 
BaO  0.02 


Total  93.50 
LOI  6.15 
Bulk moisture  7.60 
The liquid and plastic limits of the bentonite determined using the methods of ASTM D4318 [
The conventional LL and PL tests were performed in accordance with ASTM D431810 [
Test devices used in this study: (a) Casagrande LL test device, (b) PL rolling device, (c) fall cone test device, (d) BLS mold, and (e) BLS test specimen in the oven.
The fall cone LL test was conducted in accordance with the British Standard 13772 [
To determine the PI using the fall cone device, the testing and calculation methods recommended by Wroth and Wood were followed [
The BLS test specimens were also prepared during the fall cone LL tests, because the initial water content of the BLS specimens should be close to the LL that results in a cone penetration of 20 mm. As recommended by Sampson et al., the aluminum BLS molds customfabricated for this study are open on two sides, with a length of 150 mm and a 10 mm by 10 mm square cross section (Figures
In this study, a total of five samples were prepared at the same time to minimize possible variations caused by sample preparation. For each of the samples, three welltrained operators performed three replicate tests each. The three operators were all trained on all the different tests at the same time in order to minimize errors associated with the interoperator variability.
The correlation between the liquid limits determined using the Casagrande cup (LL_{cup}) and the fall cone (LL_{cone}) was determined using a total of 45 tests for each device (five bentonite contents times three operators times three replicates per operator). For each bentonite content, the average LL values from the replicate tests are shown in Figure
Correlation between LL values determined using Casagrande cup and fall cone device (3 operators × 3 replicates for each bentonite content).
A strong linear correlation can be observed between the two testing methods. The bestfit line is very close to the 1 : 1 line, but on average, the fall cone test yields higher LL_{cone} values for LL_{cup} values below 33, and lower LL_{cone} values above LL_{cup} equals 33. Both tests yield progressively larger variations with the increasing LL values that result from increasing the bentonite content. However, the variations in the fall cone test results are much smaller than those of the Casagrande cup, as clearly demonstrated by the smaller range of the vertical error bars compared to the horizontal ones. The standard deviation of the test results determined using the two test methods are summarized in Table
Summary of the standard deviation of the liquid limit test results determined using the two tests.
Bentonite content (%)  Standard deviation (%)  

Casagrande cup  Fall cone  
0  0.24  0.21 
3  0.28  0.38 
6  0.57  0.4 
9  1.67  0.67 
12  2.93  0.92 
The linear correlations determined in the previous and present studies for different types of materials are summarized in Table
Correlations between the Casagrande cup and fall cone liquid limit test results.
Reference  Material  LL range (%)  Number of specimens  Correlations 

Sherwood and Ryley [ 
Various clays  30–76  25  LL_{cone} = 0.95 LL_{cup} + 0.95 
Belviso et al. [ 
Natural soils, Southern Italy  34–134  16  LL_{cone} = 0.97 LL_{cup} + 1.19 
Wasti and Bezirci [ 
Turkey natural soils  27–110  15  LL_{cone} = 1.01 LL_{cup} + 4.92 
Dragoni et al. [ 
Clayey soils, Central Italy  28–74  41  LL_{cone} = 1.02 LL_{cup} + 2.87 
Özer [ 
Natural soils, Turkey  29–104  32  LL_{cone} = 0.90 LL_{cup} + 6.04 
Fojtová et al. [ 
Ostrava Basin clay, Czek Republic  20–50  52  LL_{cone} = 1.00 LL_{cup} + 2.44 
Di Matteo [ 
Database of various soils  24–50  >50  LL_{cone} = 1.00 LL_{cup} + 2.20 
Spagnoli [ 
Kaolinite and illitic clay  20–61  50  LL_{cone} = 0.99 LL_{cup} + 1.05 
Present study  Crushed limestone material plus bentonite  20–45  45  LL_{cone} = 0.85 LL_{cup} + 5.51 
Fall cone tests were also performed on the five bentonitetreated samples to determine the PI and thereby the PL, using the previously described method of Wroth and Wood [
The (a) PI and (b) PL values determined using the fall cone and ASTM standardized methods.
A linear correlation can be observed between the two testing methods, but the PI_{cone} values are approximately 40% greater than those determined by the ASTM test method using the ASTM plastic roller device. For the fall cone device, the plastic limit (PL_{cone}) was calculated by subtracting the PI_{cone} values from LL_{cone} values. The resulting plastic limits are compared with those from the conventional ASTMstandardized rolling device in Figure
In this study, the BLS test was also conducted on the five samples with bentonite contents varying from 0 to 12%. The BLS test results compared with the PI values determined using the ASTM standard tests are shown in Figure
Correlation between BLS values and PI determined by ASTM methods (Casagrande cup for LL and plastic limit roller for PL), for five testing samples.
However, as the bentonite content increases from 0% to 12%, the PI determined by the ASTM methods varies from 0 to 28%, whereas the BLS values vary over a much smaller range of 2% to 8%. This indicates that BLS values are much less sensitive than PI values to changes in plasticity. More importantly, the ranges of maximum and minimum values of PI (vertical error bars) for the different bentonite contents do not overlap, whereas most of the BLS ranges (horizontal error bars) do overlap. This means that a BLS measurement on the high end of the range for a bentonite content of 3%, for example, could have the same value as the BLS measurement on the low end of the range for a bentonite content of 12%. In both cases, plugging in the BLS value into the linear equation for converting BLS to PI in Figure
In this study, a twoway ANOVAbased R&R analysis was used to statistically quantify the repeatability, reproducibility, overall variability, and error sources of the various laboratory plasticity tests. This statistical analysis method is detailed in Vardeman and Jobe [
The corresponding variances (
According to the random effects model, the only difference between different measurements for a specific combination of part and operator is the measurement error (
For a fixed part “
Therefore, the overall variation due to repeatability and reproducibility (
To obtain the parameters used in this model, a twoway ANOVA table such as Table
Typical twoway ANOVA table for the R&R study.
Source  Sum of squares, SS  Degrees of freedom, df  Mean square, MS 

Part ( 
SSA 

MSA = SSA/( 
Operator ( 
SSB 

MSB = SSB/( 
Part × operator ( 
SSAB  ( 
MSAB = SSAB/( 
Error  SSE 

MSE = SSE/ 
Total  SSTot 

— 
The number of parts (
The degrees of freedom of the three quantities can be approximately determined using the Satterthwaite method [
The corresponding confidence limits for each of the quantities can be calculated based on the Chisquared distribution (
The contributions of
The twoway ANOVAbased R&R analysis was conducted on the results of the various laboratory plasticity tests detailed in the preceding sections. The testing matrix used for the R&R analysis is shown in Table
Laboratory testing matrix used in this study.
Test method  No. of soil samples ( 
No. of operators ( 
No. of replicate tests per operator ( 

Casagrande cup LL test  5^{a}  3  3 
Fall cone LL test  
ASTM PL test  
Bar linear shrinkage test 
^{a}Minus No. 40 sieved granularroad surface material with 0%, 3%, 6%, 9%, and 12% added bentonite.
The results of the analyses are summarized and compared to the R&R reported in ASTM D4318 in Table
Repeatability and reproducibility results reported in ASTM D4318 and determined by twoway ANOVAbased analysis for the various laboratory tests.
Parameters  Liquid limit  Plastic limit  Bar linear shrinkage  

Casagrande cup  Fall cone  ASTM roller^{b}  Fall cone  
 
Singleoperator standard deviation (%) 
0.5  NA  0.3  NA  NA 
Multilaboratory standard deviation (%) 
1.3  0.9  





0.6  0.5  0.4  NA  0.6 

30  30  24  30  
95% confidence interval (%)  0.5–0.8  0.4–0.6  0.3–0.5  0.5–0.9  

1.7  0.5  0.6  0.7  

6  3  5  3  
95% confidence interval (%)  1.1–3.8  0.3–1.7  0.4–1.5  0.4–2.7  

1.8  0.7  0.7  1.0  

8  10  11  10  
95% confidence interval (%)  1.3–3.5  0.5–1.1  0.5–1.2  0.7–1.7  
Fraction of 
11  50  27  43  
Fraction of 
89  50  73  57 
^{a}The R&R analysis was conducted on a USCS: ML soil. ^{b}The ASTM PL test was conducted on four samples, because the 0% bentonite sample was nonplastic.
The twoway ANOVAbased analysis results show that the overall variation (
For the BLS test,
The ASTM standards typically use the d2s limit (i.e.,
The MCR can be used to determine whether a measurement is suitable for verifying the conformance of a measured characteristic to engineering specifications. The MCR can also be considered when setting specification ranges based on measurements. For example, if the lower (
According to Vardeman and Jobe [
In this study, several laboratory soil plasticity tests were selected for demonstration of the use of a twoway ANOVAbased R&R analysis to evaluate the repeatability, reproducibility, overall variation, source of error, and capacity of a given measurement. Such an analysis can provide useful suggestions for improvement of a testing method. The measurement capacity ratio (MCR) was also demonstrated, which considers errors from both the device and the interoperator variability, and it should therefore be considered when selecting QC/QA testing methods. Based on the findings of this study, the authors suggest using the twoway ANOVAbased analysis presented herein to determine the R&R and identify the sources of measurement error, and considering the MCR of a measurement when setting specifications or selecting QA/QC testing methods. Based on the laboratory testing results, some other key findings about correlations between the various testing methods are listed below:
Correlations between the fall cone and Casagrande cup tests determined in the present and previous studies demonstrated that the fall cone test can be used to determine LL of a material with reduced variability between repeated tests. The twoway ANOVAbased repeatability and reproducibility analysis also revealed that the fall cone test can result in smaller overall variation than the Casagrande cup test, which is more prone to betweenoperator errors.
For measuring the PL and PI, the fall cone test and conventional test method using the ASTM plastic roller yielded significant discrepancies for the abraded crushed limestone granular materials with small percentages of bentonite incorporated. The fall cone test showed a dependence of PL on the bentonite content, whereas the conventional method was practically insensitive to the bentonite content. Further studies need to be conducted to evaluate the influence of the different testing mechanisms and whether PL is governed by the dominant minerals of a soil mixture.
The bar linear shrinkage results exhibited a linear correlation with the PI determined by conventional ASTM testing methods. However, as the PI increased significantly from 0 to 28% by incorporating bentonite, the corresponding BLS values were much less sensitive, exhibiting a change of only 6%. Moreover, the ranges of measured BLS values for the different bentonite contents overlapped, prohibiting a reasonably accurate correlation between BLS and PI.
Liquid limit by fall cone tests
Liquid limit by ASTM standardized Casagrande cup tests
Measurement capacity ratio
Plasticity index by ASTM standardized tests
Plasticity index by fall cone tests
Plastic limit calculated based on fall cone test results
Standard deviation
Repeatability standard deviation
Reproducibility (betweenoperators) standard deviation
Combined R&R standard deviation.
The data used to support the findings of this study are available from the first author upon request.
The authors declare that they have no conflicts of interest.
The authors would like to thank the Iowa Department of Transportation for sponsoring this study. The financial supports provided by Natural Science Foundation of Shaanxi Province in China (grant no. 2019JQ498), the Science and Technology Projects of Gansu Transportation Department (grant nos. 201916 and 201917), and Opening Foundation of Research and Development Center of Transport Industry of Technologies, Materials and Equipments of Highway Construction and Maintenance (Gansu Road and Bridge Construction Group) (grant nos. GLKF201804 and GLKF201807), are greatly appreciated.