This paper reports an interlaboratory comparison that evaluated a protocol for measuring and analysing the particle size distribution of discrete, metallic, spheroidal nanoparticles using transmission electron microscopy (TEM). The study was focused on automated image capture and automated particle analysis. NIST RM8012 gold nanoparticles (30 nm nominal diameter) were measured for areaequivalent diameter distributions by eight laboratories. Statistical analysis was used to (1) assess the data quality without using size distribution reference models, (2) determine reference model parameters for different size distribution reference models and nonlinear regression fitting methods and (3) assess the measurement uncertainty of a size distribution parameter by using its coefficient of variation. The interlaboratory areaequivalent diameter mean, 27.6 nm ± 2.4 nm (computed based on a normal distribution), was quite similar to the areaequivalent diameter, 27.6 nm, assigned to NIST RM8012. The lognormal reference model was the preferred choice for these particle size distributions as, for all laboratories, its parameters had lower relative standard errors (RSEs) than the other size distribution reference models tested (normal, Weibull and Rosin–Rammler–Bennett). The RSEs for the fitted standard deviations were two orders of magnitude higher than those for the fitted means, suggesting that most of the parameter estimate errors were associated with estimating the breadth of the distributions. The coefficients of variation for the interlaboratory statistics also confirmed the lognormal reference model as the preferred choice. From quasilinear plots, the typical range for good fits between the model and cumulative numberbased distributions was 1.9 fitted standard deviations less than the mean to 2.3 fitted standard deviations above the mean. Automated image capture, automated particle analysis and statistical evaluation of the data and fitting coefficients provide a framework for assessing nanoparticle size distributions using TEM for image acquisition.
Nanotechnology research is accelerating innovation. For example, the number of nanoparticle patents has an exponential growth rate of >30% in recent years. Nanoobjects are materials with one, two or three external dimensions on the nanoscale, nominally ranging from 1 nm to 100 nm [
There are a wide variety of analytical methods for particle size measurements, including electron microscopy, dynamic light scattering [
Because we are interested in more than a single point representation of the sample size, we compared appropriate reference distributions, such as the normal, lognormal and Weibull distributions, with particle size distribution data. TEM particle size data were converted directly to cumulative numberbased distributions. This information is useful for both nanoparticle applications, for which the surface properties may be distinctly different below a specific length scale, and regulatory requirements, for which the fraction of particles below a length scale of 100 nm would be related to whether the sample is on the nanoscale. Size distribution reference models generally have two parameters, representing the size and the shape of the distribution. For the normal distribution, these would be the sample mean and the sample standard deviation. The number of particles needed for highaccuracy estimates of the average diameter is known to depend on the spread of the particle size distribution [
An important step in the process is visualizing the fitted model prediction relative to the actual data. This step helped us answer the following question: where does the model deviate from the data, i.e. over what diameter range do we know the distribution well? This has relevance for the application and regulatory communities. Because TEM can be a costly method, automated image capture, particle analysis and statistical assessment were preferred.
A metrology checklist [
This case study is intended to provide a scientific foundation for an International Organization for Standardization (ISO;
The case study protocol for 30 nm gold nanoparticles was based on a National Institute for Occupational Safety and Health (NIOSH) internal interlaboratory comparison [
ISO standards exist for the measurement of particle size distributions of powders (ISO TC/24), including representation of particle size analyses [
NIST RM8012 has 30 nm nominal diameter gold nanoparticles stabilized by citric acid in a water dispersion. The NIST Report of Investigation [
Two sample preparation objectives are as follows: disperse the nanoparticles over the 3 mm TEM sample grid so that particles do not touch (so the edges of the particles will be clearly visible in the imaging system) and uniformly distribute the nanoparticles across the sampling medium [
The typical time required to receive the sample, acquire over 500 data points, analyse the data with ImageJ and assemble the framewise data into a master table for analysis exceeded 20 h. Each lab received one grid; no grids were shared between laboratories.
ISO 133221:2004 [
set the accelerating voltage according to the material to be measured (120 kV);
select the sample working distance specified by the electron microscope manufacturer for highresolution imaging;
mount the sample flat on the specimen holder with the stage tilt set to zero;
switch off the dynamic focus and tilt correction;
align the instrument according to the manufacturer’s procedures;
select operating conditions to minimize drift.
While no SEM instruments were used in this study for particle size determination, it is notable that a recent good practice guide [
Since TEMs have wide ranges of magnification and many operating modes, the actual magnification at any given instrument settings may differ from the indicated magnification by up to 10%. Calibration of the instrument to a known length scale under optical conditions similar to those used for analysis is preferred. Standards should be run near the time of the study to provide verification of correct instrument operation within manufacturer specifications and to validate measurement procedures. Typical examples are given in a good practice guide ([15, chapter 4]).
Each participating lab used the loading procedure specific for their instrument to mount the TEM grid in the system. The loading procedure was to minimize the eucentric height adjustment required. The images were to be of sufficient quality such that individual particles can be resolved and their dimensions measured. Each lab analysed one wafer, measured at least 500 particles, and reported the results to the team. The specific instructions were the following.
Acquire images that have histograms centred and wide enough to cover at least 80% of the possible grey levels.
Select a magnification/image resolution combination that will provide a minimum of two pixels/nm, i.e. >2 pixel nm^{−1} or <0.5 nm/pixel.
Ensure that a scale bar is visible in each digital image/frame.
Do not exclude irregularly shaped particles or particles with sharp corners.
Do not report data for any touching particles (note: overlapping particles can rest on one another, reducing their projected 2D area in a topdown view).
Do not report data for any particles that appear cut by the frame (note: for the case in which there is a difference in particle diameters greater than an order of magnitude, it may be necessary to establish a frame, divide it via a grid pattern, and measure large particles at a lower magnification. The protocol did not address this issue).
Count and report at least 500 particles in frames that are well spaced across the sample (note: the number of particles measured directly affects the measurement uncertainty of the sample mean and standard deviation. In general, the user will select precisions for the sample mean and standard deviation, and then estimate how many particles might be needed. There is guidance on how the sample mean is affected by the sample size [
For all selected particles in each frame, report the particle number, the frame number and all measurand data.
Save images as lossless (like tagged image file format, tiff or dmc) image file type. Do not save images as lossy image file type, like jpg.
The areaequivalent average diameters of all reported particles were used to generate numberbased, cumulative particle size distributions.
Since a large number of nanoparticles are needed for a highquality particle size distribution, the work will be facilitated when image analysis software is used. Both commercial and open source software are available. For a typical sample, an appropriate reference model for the data may not be known, the data may not be monomodal, and the sample may be contaminated with nanoparticles of different sizes and shapes. Multiple models might need to be compared with the data and multiple measurands might be needed to help screen for the desired nanoparticles. Therefore, we have used a more general analysis approach that estimates the sample mean and standard deviation from a nonlinear fit of the reference model to sample population data. The minimum number of particles for analysis was set at 500 for each lab, based on the experience from prior studies.
This protocol assumed that all images were taken in digital format. ImageJ, open source software with a suite of analysis routines (
Create working copies of all images/frames (preserve the original unmodified images).
Open ImageJ and open the frame file.
Set the measurement scale using the scale bar or another measurement of pixel size, returning to the original scale prior to continuing.
Crop the image to remove scale bars and other image artefacts that might affect contrast or particle analysis.
Check and correct brightness and contrast to ensure that all images have histograms centred and wide enough to cover at least 80% of the possible grey levels.
The thresholding operation may result in frame files with single pixel artefacts or poor image quality, e.g. rough particles or uneven background due to nonuniform electron beam illumination. In the case of the former, apply the despeckle and erode/dilate processes to remove these artefacts and save the changes. In the case of poor image quality, the operator could clean up the edges of particles or correct for uneven background by applying special filters. Assess the image transformation and save changes.
Touching particles should not be addressed by using automated separation algorithms (Watershed, in the case of ImageJ). Rather, all particle analyses should be recorded, and touching particles should be removed manually from the spreadsheet of the results.
Select the measurands (such as area, shape descriptors, Feret’s diameter). Note: several size and shape descriptors will help identify imaging and measurement issues as well as assist with the characterization of the sample.
Analyse the particles (ImageJ specific settings should include the following: show outlines, display results, include holes and exclude on edges).
Save each image file that shows particle outlines and their number sequence (filename.tif) and the spreadsheet (Results.xls), which reports all measurand values, the particle number and the frame number associated with each particle.
There are three major applications for statistical analysis of particle size data: assessment of data robustness, fitting reference models to the size distributions and assessment of measurement uncertainty.
Analysis of variance (ANOVA) was used to assess the intralaboratory repeatability (variation with one operator and one instrument; section 2.20 of [
For intralaboratory repeatability, the objective, metric and software were as follows.
Null hypothesis: for each lab, all frames have the same mean.
Alternative hypothesis: for each lab, not all frames have the same mean.
For interlaboratory reproducibility, the objective, metric and software were as follows.
Null hypothesis: all labs have the same mean.
Alternative hypothesis: not all labs have the same mean.
Three reference models are commonly fitted to cumulative particle size distribution data: lognormal, Rosin–Rammler–Bennett and Weibull. These three, plus the normal distribution, were compared with the cumulative frequency data generated in this case study. In all cases, two parameter models were used. Differential probability distributions were not used as information is lost when the data are binned, often obscuring the details near the ends of the distributions.
Three different visualization methods [
The software provided the
For each parameter and its associated statistics, it is possible to construct a ‘grand’ statistic for the interlaboratory study. For example, the fitted lognormal means that each lab will have a grand mean and a grand standard deviation. The ratio of the grand standard deviation to the grand mean is the coefficient of variation for that parameter or statistic (for example,
Standards organizations require statements on measurement uncertainty. There are differences between CEN (Comité Européen de Normalisation), ISO and ASTM (American Society for Testing and Materials) approaches [
For the areaequivalent diameter, elements of the pooled measurement uncertainty, (
For the normal distributions, we have the information needed to compute each of these components for the sample mean. However, the lognormal mean of RM8012 has not been reported or certified, so only the interlaboratory reproducibility and the image resolution error can be computed.
Although each laboratory used a different TEM instrument, the sample grids were suitable for each one. Sample preparation, a known element of variability for TEM analysis, was not varied in this study. RM8012 is known to have discrete, nonaggregated gold nanoparticles in its suspension medium. This is confirmed by its Report of Investigation, in which the average particle size by dynamic light scattering is essentially the same as the size by TEM. In general, the gold nanoparticles were well separated on the functionalized silicon TEM grid surfaces, but a number of touching particles were observed on all grids. There are at least two mechanisms by which this could occur: agglomeration, which is a general phenomenon for colloidal particles in solution and increases with particle concentrations, and random deposition of one nanoparticle near another. Touching nanoparticles were assumed to be agglomerated. For Laboratory H, a total of 672 nanoparticles were imaged. Of these, there were 530 discrete nanoparticles. Fiftyone dimer, 11 trimer and one septamer nanoobjects were judged to be touching and were not counted, i.e. only 79% of the imaged nanoparticles could be used directly for the automated particle analysis. It is likely that the identification of touching particles could be automated by using shape factor or aspect ratio measurands, but this was not addressed in this interlaboratory comparison.
One lab reported a calibration error, which occurred when two different magnifications were used for particle imaging (one of these magnifications had not been properly calibrated). This type of error was easily detected by the ANOVA analysis of frametoframe particle diameter means. In general, the labs could achieve good contrast between nanoparticles and the background, and the sample preparation method [
One laboratory reported thresholding problems that resulted in a number of small particle artefacts being reported. This was detected by reviewing the cumulative particle size distribution of all the data; about 5% of the ‘particles’ were less than 6 nm, which was known not to be characteristic of this reference material. The issue was corrected by redoing the thresholding and ensuring that the despeckle and erode/dilate steps were used. The effects of the erode/dilate step can be checked directly during the particle analysis process. The artefacts also could be removed manually, but this would reduce the benefits of automated analyses.
Touching particles were not analysed in this interlaboratory comparison, although ImageJ has a tool to do so (Watershed). The Watershed tool separates touching particles by inserting a linear boundary at the ‘necks’ between particles. This approach tends to reduce the total area attributed to each particle and to lower the average areaequivalent diameter reported for the sample.
The protocol used for the US TAG interlaboratory comparison on gold nanoparticles provided no guidance for postprocessing review of the raw data from the automated image capture and particle analysis process. Interlaboratory comparisons of the data showed that there were differences in the ranges of particle sizes reported as well as the cumulative particle size distributions. These differences appeared to be due to differences in how operators treated the data and/or set thresholding parameters.
Statistical methods used to assess data precision and accuracy assume homogeneous variance and normally distributed residuals.
The oneway ANOVA test provides a rapid way to assess whether the data in all frames within a lab are best represented by the same mean. If the null hypothesis is not rejected, we have more confidence in the data. If the null hypothesis is rejected, the particle data (images plus measurand results) can be reviewed to determine whether any artefacts or unusual particles exist in any frame with a mean not equal to the lab grand mean. Therefore, intralaboratory statistical assessment can help identify the following: (1) repeatability, (2) particles that may be outside the expected range for the distribution, (3) frames with dissimilar mean particle sizes, (4) calibration errors at different magnification levels, (5) thresholding to eliminate small ‘ghost’ particles, (6) thresholding to eliminate large particles, (7) touching particles measured as one particle, and others.
Any particles added to or removed from the data set would need to be justified on technical grounds. We prefer not to use traditional outlier tests to identify particles that appear to be outside the distribution—we are trying to determine its true ‘breadth’. In addition, showing the data range and mean for each frame can trigger reviews even when the frame mean may be similar (an unusually large and an unusually small particle might offset each other, for example). The variation in the sample means provides an indication of the intralaboratory repeatability.
Different measurands will have different
RM8012 has been certified for its mean value only (
The grand standard deviation,
where Δ_{m} = 
The interlaboratory assessment used a model with frames nested within labs to identify differences between pairwise laboratories. Similarly to the intralaboratory assessment, the null hypothesis is that every lab has the same mean. Results from the ANOVA test can vary depending on the measurand. For the areaequivalent diameter measurand, the frametoframe (nested) ANOVA analyses gave similar results when using either the areaequivalent diameter (corresponding to the normal distribution) or the log transformed measurand (corresponding to the lognormal distribution). For most lab pairs, the null hypothesis was rejected; only 4 of 28 pairs fail to reject the null hypothesis. For this comparison, frames with means less than the 10th percentile or greater than the 90th percentile were excluded, i.e. data that might be questionable were not considered. This suggests that the lab means are, in general, different from each other, possibly related to the use of different instruments and different operators.
The visual comparison of the data to a reference model is valuable in selecting which models are appropriate.
The standard error of a statistic is expected to decrease as the number of data points increases (SE_{x̄} ~ 1/√
The coefficient of variation represents the standard deviation of a statistic, and can be used to compare reference model choices across the interlaboratory study (using the ‘grand mean’ approach). For normal and lognormal distributions, the mean and standard deviation can be computed both from the standard definitions and the nonlinear regression approach (fitted parameters).
In this study, the preferred measurand is the areaequivalent diameter, primarily because the sample itself has been well studied with respect to
A comparison of coefficient of variation values for the areaequivalent and maximum Feret diameters showed no statistically significant difference between these measurands. The
Three fitting methods are described in ISO 92763 [
Lab F had the lowest coefficients of variation for its parameters. The boxplot for Lab F (
Based on these examples, the cumulative distribution plot appears to be a reasonable method to develop a general fit to the data. The residual deviation plot would be most sensitive to differences in the middle of the distribution. The quantile plot is very efficient for identifying deviations at the edges of the distribution. In general, the choice of method will depend on the application for the data and model.
When we evaluate the areaequivalent diameter using the normal model, we can generate the following measurement uncertainty components for the sample mean: the interlab reproducibility, the trueness and the image resolution. However, since the reference material was not certified for a lognormal reference model, it is not possible to determine the trueness of the preferred reference model parameters of this study. Rather, we computed expanded measurement uncertainties for the reference model parameters, mean and standard deviation of the interlaboratory comparison. Using coefficients of variation allowed the comparison of lognormal and normal parameters on a relative basis. The equation used was [
where
Automated analysis of particle size measurands is an important objective for interpreting particle size distributions by TEM. Properly implemented, automated image analysis should reduce the time needed for evaluation and provide protocols with documented precision. Statistical analysis of the test results can be used to (1) assess the data quality with no use of reference models, (2) determine reference model parameters for different models and fitting methods, and (3) assess essential parts of measurement uncertainty of parameters by using their coefficients of variation. These quality measures can be used for a variety of applications, such as process quality control, regulation [
Comparing the particle size distributions of nontouching and touching particles demonstrated that the deconvoluting routine reduced both the mean and the standard deviation for the processed data set. Selfreview of particle size distribution data can be improved by the use of statistical analysis tools that quickly identify particle images that should be reviewed for consistency. The ANOVA test can be used to evaluate intralaboratory and interlaboratory data quality independent of a model choice for the distribution. If these methods are used with a reference material, then the trueness of the protocol to the value assigned to the reference material measurand can be determined.
In this interlaboratory comparison, only two of the eight labs did not reject the null hypothesis of similar frametoframe means for three different measurands (areaequivalent diameter, maximum Feret diameter and shape factor). Yet, the interlaboratory areaequivalent diameter mean (27.6 nm; coefficient of variation = 2.6%; expanded measurement uncertainty = 5.5%) was quite similar to that of RM8012. With respect to visualization tools, the cumulative distribution plot was used to verify general agreement between the data and model, the residual deviation plot was helpful in showing deviations near the sample mean, and the quantile plot was used to show differences near the ends of the distribution. Quasilinear plots of the eight data sets showed that the average range for good fits between the model and the cumulative numberbased distributions was −1.9
The RSEs of the fitted parameters provided a good starting point for evaluating intralab data quality. The RSEs aided in the selection of preferred reference models, the comparison of different measurands and the selection of the fitting methods. The RSEs did not appear to correlate with the number of frames analysed or the pixels/nm of the frame scale, which was tightly controlled. RSEs for lognormal model parameters, the mean and standard deviation, generally decreased as the number of particles measured increased. However, the standard deviation RSEs were about two orders of magnitude larger than those of the mean. Therefore, most of the error of the reference models appears to be associated with the breadths of the distributions.
Interlaboratory results were analysed by constructing grand averages of the parameter values from all labs. The coefficients of variation (as percentages) could be used to evaluate quality of the parameter estimates across the ILC. In general, the grand mean is better known than the grand standard deviation. The coefficient of variation for a parameter could be used to estimate its relative expanded measurement uncertainty as part of a measurement uncertainty budget.
Statements in this paper reflect the opinions of the authors and do not necessarily reflect the opinions of the National Institute of Standards and Technology (NIST), the National Institute for Occupational Safety and Health (NIOSH) or the US Food and Drug Administration (FDA). Certain commercial equipment, instruments or materials are identified in this paper. Such identification does not imply recommendation or endorsement by NIST, NIOSH or the FDA, nor does it imply that the products identified are necessarily the best available for the purpose.
Metrology checklist for particle size distribution by TEM.
Question  Response 

Is the system/body/substance that will be subjected to the  The objective is to measure the particle size distribution 
Is the definition of the system/body/substance  The definition is not unnecessarily restrictive. The 
Is the measurand clearly described?  The numberbased cumulative distribution of 
Has it been clearly indicated whether the measurand is  Yes. The measurement is performed in a vacuum, which 
Is the measurement unit defined? Are the tools  Length. Areaequivalent diameter is one of several 
Has the method already been validated in one or  No. While many journal articles report particle size 
What are the quality control tools available to  The test method requires that the TEM has been 
Have results of measurements using the proposed method  Yes. The protocol is based on several methods reported 
Is the required instrumentation widely available?  Yes. TEM is widely available, but is costly to operate. 
Does the document propose a measurement  Type A and B measurement uncertainties of the fitted 
Standard definition: the
Fitted model: estimates for the fitted mean begin with the standard definition and then are iteratively updated to minimize the sum of differences between the reference model and the data.
Standard definition: the
Fitted model: estimates for the fitted mean begin with the standard definition and then are iteratively updated to minimize the sum of differences between the reference model and the data.
Standard definition: the
Fitted model
Fitted model: computed using Wald confidence intervals
The RSE is the standard error divided by its statistic and expressed as a percentage.
Standard definition: example—RSE of the mean.
If the null hypothesis were true and if the experiment were repeated many times, a
Note: in hypothesis testing, a statement claiming that the null parameter is the true parameter is called the
Δ_{m} = 
Quotient of the bias divided by the expected value.
The variance, Var(
The
A
Diameter of a circle that has an area equivalent to the area reported for the particle
Distance between parallel tangents; corresponds to ‘length’;
Distance between parallel tangents; corresponds to ‘breadth’;
Ratio of the maximum and minimum Feret diameters for a particle (inverse of aspect ratio)
For the areaequivalent diameter, elements of the pooled measurement uncertainty (
The Report of Investigation for RM8012 [
where
SEM (lefthand side) and TEM (righthand side) images of RM8012 [
Comparison of data for nontouching particles (solid black squares) and the Watershed algorithm for separation of touching particles (open red circles). Lab E, (
ANOVA boxplots comparing frame means and data ranges, Lab G. Vertical line = overall mean; solid diamond = frame mean; grey box = 25th–75th percentile for each frame (interquartile range (IQR)); black error bars =
Shape factor of gold nanoparticles. Lab H: data (open circles) fitted to a Rosin–Rammler–Bennett model (solid red line). For fitting purposes, the shape factor (SF) was transformed to SF_{t} = 1 − SF.
RSE of the sample means and standard deviations for all labs. Lognormal model parameters for the areaequivalent diameter distribution; open squares = standard deviation, open circles = mean.
Comparison of fitting methods (lognormal distribution). Lab B: data = open squares. (
Instrument factors.
Organization  A  B  C  D  E  F  G  H 

# of frames  62  49  20  27  11  135  20  55 
# of nanoparticles  706  624  535  513  608  1112  1480  531 
Instrument  JEOL  JEOL  Jeol  JEOL  Jeol  FEI  FEI Tecnai  JEOL 
Acceleration  200 kV  200 keV  80 kV  100 kV  80 kV  300 kV  200 kV  120 kV 
Magnification  20 000×  21 000× nominal  100 000×  20 000×  40 000×  27 000×  19 000×,  20 000× 
Frame size  1040 nm  1217.6 nm×  1320 nm×  550 nm×  1000 nm×  782 nm×  1875 nm×  475 nm× 
Pixel dimension  0.51  0.30  0.51  0.50  0.53  0.38  0.5  0.50 
Image acquisition  3 s  4 s  5 s  4 s  3.5 s  0.5 s  3 s  3 s 
Mean signaltonoise  ~2.5  ~14.2  2  ~2.4  2  ~2  2  ~2.5 
Image analysis  Image J  Image J  Image J  Image J  Image J  Image J  Image J  Image J 
ANOVA for the natural logarithm of three measurands: (A) areaequivalent diameter, (B) maximum Feret diameter and (C) shape factor. The
(A) Natural log of area equivalent  Areaequivalent  
Laboratory 
 
 
A  3.34  0.0066  0.711  28.6 
B  3.33  0.0030 
 27.9 
C  3.33  0.0036  0.274  27.9 
D  3.27  0.0067  0.168  26.4 
E  3.28  0.0031 
 26.7 
F  3.33  0.0023 
 28.0 
G  3.31  0.0026  <0.0001  27.5 
H  3.31  0.0038  0.106  27.4 
 
(B) Natural log of Feret diameter  Feret  
Laboratory 
 
 
A  3.43  0.0069  0.758  31.2 
B  3.40  0.0035 
 30.1 
C  3.41  0.0043  0.342  30.5 
D  3.36  0.0065  0.413  29.0 
E  3.40  0.0033  <0.0001  30.2 
F  3.43  0.0030  0.138  30.9 
G  3.42  0.0027  <0.0001  30.6 
H  3.39  0.0040 
 29.8 
 
(C) Natural log of aspect ratio  Mean aspect  
Laboratory 

 
 
A  0.100  0.0036  0.904  1.11 
B  0.098  0.0034  0.351  1.11 
C  0.123  0.0065  <0.0001  1.15 
D  0.107  0.0042  0.290  1.12 
E  0.107  0.0041 
 1.12 
F  0.109  0.0035  0.961  1.12 
G  0.108  0.0027 
 1.12 
H  0.089  0.0032 
 1.10 
Coefficients of variation for the interlaboratory comparison. Measurand = areaequivalent diameter; normal distribution is assumed.
Normal  Model parameters  Reported range/nm  



 
RM8012  27.6  
Lab  
A  28.6  3.17  21.9  70.7 
B  28.1  2.17  21.6  37.7 
C  27.9  2.30  22.3  36.7 
D  26.5  2.61  19.6  51.2 
E  26.7  2.03  15.9  33.0 
F  28.0  2.19  21.7  35.2 
G  27.6  2.60  13.9  45.4 
H  27.4  2.47  20.8  40.4 
27.6  2.44  19.7  43.8  
0.708  0.361  3.13  12.4  
2.57%  14.8%  15.9%  28.2% 
Relative bias for the interlaboratory comparison (expressed as a percentage). Measurand = areaequivalent diameter; normal distribution is assumed,
Lab 
Δ


A  1.0 
B  0.5 
C  0.3 
D  1.1 
E  0.9 
F  0.4 
G  0.0 
H  0.2 
Bias  0.55 
Relative bias  2.0% 
RSEs of fitted parameters: (A) lognormal distribution and (B) normal distribution.
Mean  Standard deviation  


 SE_{s(fit)}  RSE_{s(fit)} 
 
 
(A) ln(  Lab  
A  3.34  1.32E–04  3.95E–05  0.0806  2.31E–04  2.87E–03  0.998  
B  3.32  1.69E–04  5.09E–05  0.0720  2.99E–04  4.16E–03  0.996  
C  3.32  9.19E–05  2.76E–05  0.0798  1.66E–04  2.02E–03  0.999  
D  3.27  1.13E–04  3.45E–05  0.0823  2.01E–04  2.44E–03  0.999  
E  3.28  9.55E–05  2.91E–05  0.0736  1.65E–04  2.24E–03  0.999  
F  3.33  5.92E–05  1.78E–05  0.0708  1.05E–04  1.48E–03  0.999  
G  3.31  6.25E–05  1.89E–05  0.0745  1.13E–04  1.52E–03  0.999  
H  3.30  2.41E–04  7.30E–05  0.0811  4.26E–04  5.24E–03  0.995  




 




 
 


 SE_{s(fit)}  RSE_{s(fit)} 
 
 
(B)  Lab  
A  28.3  5.34E–03  1.88E–04  2.27  9.18E–03  4.04E–03  0.996  
B  27.9  5.86E–03  2.10E–04  2.01  1.04E–02  5.18E–03  0.994  
C  27.8  2.70E–03  9.71E–05  2.21  4.72E–03  2.14E–03  0.999  
D  26.3  3.89E–03  1.48E–04  2.15  6.91E–03  3.21E–03  0.998  
E  26.6  3.52E–03  1.32E–04  1.96  6.10E–03  3.11E–03  0.998  
F  27.8  2.45E–03  8.80E–05  1.97  4.35E–03  2.21E–03  0.998  
G  27.5  2.06E–03  7.50E–05  2.04  3.72E–03  1.82E–03  0.999  
H  27.1  8.25E–03  3.04E–04  2.19  1.46E–02  6.65E–03  0.993  




 





Comparison of fitted parameters and standard definitions for all labs (grand means, grand standard deviations and their coefficients of variation): (A) lognormal distribution and (B) normal distribution.

 
 
 
Grand  3.31  0.0768  3.31  0.0836 
Grand  .0253  0.004 58  0.0914  0.008 33 
Grand CoV of parameter  0.764%  5.96%  2.76%  9.96% 
 

 
 
 
Grand  27.4  2.10  27.6  2.46 
Grand  0.707  0.121  0.721  0.392 
Grand CoV of parameter  2.58%  5.76%  2.61%  15.9% 