A Reference number ISO 5725 5 1998(E) INTERNATIONAL STANDARD ISO 5725 5 First edition 1998 07 15 Accuracy (trueness and precision) of measurement methods and results — Part 5 Alternative methods for t[.]
Applications of the split-level design
ISO 5725-2's uniform level design involves testing two or more identical samples of a material in each laboratory and at each experiment level, but this setup risks operator influence between measurements, potentially distorting precision estimates Specifically, such influence can lead to underestimated repeatability standard deviation (s_r) and overestimated between-laboratory standard deviation (s_L) To address this, the split-level design provides each laboratory with samples of two similar but not identical materials at each experiment level, without specifying the exact differences This approach helps accurately determine the repeatability and reproducibility standard deviations of measurement methods while minimizing bias caused by sample influence.
Split-level experiment data can be used to create comparative graphs plotting one material's results against another's, as demonstrated in figure 1 These graphs are valuable tools for identifying laboratories with the greatest biases, enabling targeted investigation into potential causes Understanding and addressing major biases helps improve laboratory accuracy and data reliability.
Repeatability and reproducibility standard deviations in measurement methods typically vary with the material's level, such as increasing proportion of an element in chemical analysis For split-level experiments, it is essential that similar materials used at each level produce comparable measurement results and standard deviations Ensuring that materials are nearly identical at each level helps maintain consistent measurement performance, and deliberately introducing significant differences offers no added benefit.
In chemical analysis, the matrix containing the analyte can impact measurement accuracy, necessitating the use of two similar materials at each experimental level Preparing sufficiently similar materials, such as by spiking a sample with a small amount of the analyte, can help achieve this When analyzing natural or manufactured products, finding two highly similar samples can be challenging; using two batches of the same product is a practical solution The primary goal in selecting materials for split-level experiments is to provide samples that are expected to be different but comparable, ensuring valid assessment of experimental performance.
Layout of the split-level design
4.2.1 The layout of the split-level design is shown in table 1.
The p participating laboratories each test two samples at q levels.
The two samples within a level are denoted a and b, where a represents a sample of one material, and b represents a sample of the other, similar, material.
4.2.2 The data from a split-level experiment are represented by: y ijk where subscript i represents the laboratory (i = 1, 2, , p); subscript j represents the level (j = 1, 2, , q); subscript k represents the sample (k = a or b).
Organization of a split-level experiment
4.3.1 Follow the guidance given in clause 6 of ISO 5725-1:1994 when planning a split-level experiment.
Subclause 6.3 of ISO 5725-1:1994 provides essential formulae, involving the parameter A, to determine the optimal number of laboratories required for a reliable split-level experiment These formulas are crucial for designing accurate and statistically sound inter-laboratory studies, ensuring consistent and comparable measurement results across multiple labs Implementing these standardized calculations helps researchers improve experiment accuracy and reliability, adhering to international quality and metrology standards.
NOTE — These formulae have been derived by the method described in NOTE 24 of ISO 5725-1:1994.
To assess the uncertainties of the estimates of the repeatability and reproducibility standard deviations, calculate the following quantities.
Taking the number of replicates as two in equations (9) and (10) of ISO 5725-1:1994 reveals that these equations are essentially the same as equations (1) and (2), with the only minor difference being the use of p - 1 instead of p in some instances This slight variation means that tables and figures (such as Table 1 and Figures B.1 and B.2) in ISO 5725-1:1994 can be effectively used to evaluate the uncertainty associated with the estimates of repeatability and reproducibility standard deviations in split-level experiments.
To evaluate the uncertainty in the bias estimate of the measurement method in a split-level experiment, calculate the quantity A as specified in equation (13) of ISO 5725-1:1994 with n = 2 or refer to table 2 of ISO 5725-1:1994 This calculated value should then be utilized according to the guidelines provided in ISO 5725-1 to ensure accurate assessment of measurement bias uncertainty.
To assess the uncertainty in estimating laboratory bias in a split-level experiment, calculate the quantity A_w as specified by equation (16) in ISO 5725-1:1994 using n = 2 Since a split-level experiment effectively involves only two replicates, increasing the number of replicates cannot reduce the uncertainty of the laboratory bias estimate If reducing this uncertainty is necessary, employing a uniform-level design is recommended.
Follow the guidance outlined in clauses 5 and 6 of ISO 5725-2:1994 to organize a split-level experiment effectively In this context, the number of replicates (n) corresponds to the number of split-levels in the design, typically two Adhering to these standards ensures accurate measurement of precision and reproducibility in your analytical methods This approach aligns with international testing protocols, enhancing the reliability of your experimental results.
The a samples should be allocated to the participants at random, and the b samples should also be allocated to the participants at random and in a separate randomization operation.
In split-level experiments, it is essential for statisticians to clearly identify which results correspond to material A and which to material B at each experimental stage Proper labeling of samples ensures accurate data analysis and interpretation Additionally, it is crucial to maintain participant blindness by not disclosing this information to the study subjects, thereby preserving the integrity of the experimental design.
Table 1 — Recommended form for the collation of data for the split-level design
Statistical model
The basic model in ISO 5725-1:1994, clause 5, describes measurement accuracy by assuming each result is composed of three components: the general average (m_j) at a specific level, a bias component (B_ij), and a random error (e_ijk) This approach helps in estimating the trueness and precision of measurement methods by analyzing these components.
B_ij represents the laboratory component of bias under repeatability conditions in laboratory i (i = 1, , p) at level j (j = 1, , q) The term e_ijk denotes the random error of test result k (k = 1, , n), obtained in laboratory i at level j under repeatability conditions Understanding these components is essential for analyzing measurement variability and ensuring the accuracy and reliability of laboratory test results.
4.4.2 For a split-level experiment, this model becomes: y ijk = m jk + B ij + e ijk (4)
This variation from equation (3) in section 4.4.1 introduces a key feature: the subscript k in m_jk indicates that, according to equation (4), the overall average now depends on the material type—either a or b—within level j This change allows for a more nuanced analysis by accounting for material-specific differences within the hierarchical structure Understanding this distinction is essential for accurately modeling variations in the data and ensuring precise results in material-related studies.
In the absence of a subscript k in B_ij, it is assumed that the laboratory bias does not vary based on materials a or b within a given level This assumption highlights the importance of using similar materials to ensure consistent and reliable results Understanding this relationship is crucial for accurate experimental comparisons across different laboratories and materials.
4.4.3 Define the cell averages as: y ij = (y ija + y ijb ) / 2 (5) and the cell differences as:
4.4.4 The general average for a level j of a split-level experiment may be defined as: m j = (m ja + m jb ) / 2 (7)
Statistical analysis of the data from a split-level experiment
4.5.1 Assemble the data into a table as shown in table 1 Each combination of a laboratory and a level gives a
“cell” in this table, containing two items of data, y ija and y ijb
To analyze cell differences, calculate Dij by subtracting b from a (Dij = a - b) and record the results in a table similar to Table 2 Ensure that each difference is calculated consistently in the same order, maintaining the sign of the difference to accurately reflect the variation between cells This method preserves data integrity and facilitates precise comparison in the analysis process.
Calculate the cell averages y ij and enter them into a table as shown in table 3.
If a cell in Table 1 lacks two test results—due to sample spoilage or data exclusion after outlier tests—the corresponding cells in Tables 2 and 3 should remain empty This ensures data integrity and clarity in the reporting process.
4.5.3 For each level j of the experiment, calculate the average D j and standard deviation s Dj of the differences in column j of table 2:
Here, S represents summation over the laboratories i = 1, 2, , p.
If there are empty cells in table 2, p is now the number of cells in column j of table 2 containing data and the summation is performed over non-empty cells.
4.5.4 For each level j of the experiment, calculate the average y j and standard deviation s yj of the averages in column j of table 3, using: y j =∑ y ij p (10)
Here, S represents summation over the laboratories i = 1, 2, , p.
If there are empty cells in table 3, p is now the number of cells in column j of table 3 containing data and the summation is performed over non-empty cells.
4.5.5 Use tables 2 and 3 and the statistics calculated in 4.5.3 and 4.5.4 to examine the data for consistency and outliers, as described in 4.6 If data are rejected, recalculate the statistics.
4.5.6 Calculate the repeatability standard deviation s rj and the reproducibility standard deviation s Rj from: s rj =s Dj 2 (12) s Rj 2 =s yj 2 +s rj 2 2 (13)
4.5.7 Investigate whether s rj and s Rj depend on the average y j , and, if so, determine the functional relationships, using the methods described in subclause 7.5 of ISO 5725-2:1994.
Table 2 — Recommended form for tabulation of cell differences for the split-level design
Table 3 — Recommended form for tabulation of cell averages for the split-level design
Scrutiny of the data for consistency and outliers
4.6.1 Examine the data for consistency using the h statistics, described in subclause 7.3.1 of ISO 5725-2:1994.
To check the consistency of the cell differences, calculate the h statistics as:
To check the consistency of the cell averages, calculate the h statistics as:
To analyze inconsistent laboratories, plot both sets of statistics grouped by laboratory and ordered by levels, as illustrated in figures 2 and 3 These graphs help identify laboratories with lower repeatability, indicated by numerous large h statistics in the cell differences graph, or those exhibiting bias, shown by h statistics clustering in one direction in the cell averages graph Proper interpretation of these plots, discussed in ISO 5725-2:1994, subsection 7.3.1, enables pinpointing laboratories that need to investigate and report their findings to the experiment organizer for quality assurance.
4.6.2 Examine the data for stragglers and outliers using Grubbs’ tests, described in subclause 7.3.4 of ISO 5725-2:1994.
To test for stragglers and outliers in the cell differences, apply Grubbs’ tests to the values in each column of table 2 in turn.
To test for stragglers and outliers in the cell averages, apply Grubbs’ tests to the values in each column of table 3 in turn.
ISO 5725-2:1994, subclause 7.3.2, provides guidelines for interpreting test results to identify outliers that significantly skew repeatability and reproducibility statistics Data identified as outliers are typically excluded from calculations to ensure accurate assessment, while stragglers are usually included unless justified otherwise When a value in tables 2 or 3 is excluded from the analysis, the corresponding value in the other table should also be excluded to maintain consistency.
Reporting the results of a split-level experiment
4.7.1 Advice is given in subclause 7.7 of ISO 5725-2:1994 on:
— reporting the results of the statistical analysis to the panel;
— decisions to be made by the panel; and
— the preparation of a full report.
4.7.2 Recommendations on the form of a published statement of the repeatability and reproducibility standard deviations of a standard measurement method are given in subclause 7.1 of ISO 5725-1:1994.
Example 1: A split-level experiment — Determination of protein
Table 4 presents data from an experiment [5] that determined the protein content in feeds through combustion analysis The study involved nine laboratories and included 14 different protein levels For each level, two feeds with similar protein mass fractions were analyzed to ensure accurate and comparable results across the various testing sites.
4.8.2 Tables 5 and 6 show the cell averages and differences, calculated as described in clause 4.5.1, for just Level 14 (j = 14) of the experiment.
Using equations (8) and (9) in 4.5.3, the differences in table 5 give:
D 14 = 8,34 % s D14 = 0,436 1 % and applying equations (10) and (11) in 4.5.4 to the averages in table 6 gives: y 14 = 85,46 % s Y14 = 0,453 4 % so the repeatability and reproducibility standard deviations are, using equations (12) and (13) in 4.5.6: s r14 = 0,31 % s R14 = 0,50 %
Table 7 gives the results of the calculations for the other levels.
Figure 1 illustrates the comparison between samples a and b from Table 4 for Level 14 using a Youden plot Laboratory 5's results are represented by a point in the bottom left corner of the graph, indicating specific measurement characteristics This visualization helps in assessing the consistency and accuracy of laboratory results across different samples.
Laboratory 1's data point in the top right corner indicates a consistent positive bias over samples a and b, while Laboratory 5 exhibits a corresponding negative bias, highlighting systematic discrepancies between the labs Such patterns are common in split-level experimental designs, as shown in figure 1, which illustrates biases across different laboratories Notably, Laboratory 4's results are unique, with its data point far from the line of equality, suggesting potential anomalies The remaining laboratories cluster centrally on the plot, emphasizing the need to investigate the underlying causes of biases at Laboratories 1, 4, and 5 for more accurate data interpretation and quality control.
NOTE — For further information on the interpretation of “Youden plots”, see references [7] and [8].
4.8.4 The values of the h statistics, calculated as described in 4.6.1, are shown in tables 5 and 6, for only Level 14. The values for all levels are plotted in figures 2 and 3.
Figure 3 demonstrates that Laboratory 5 consistently exhibits negative h statistics across all levels, indicating a persistent negative bias in their data Conversely, Laboratories 8 and 9 predominantly display positive h statistics, reflecting a steady positive bias that is less pronounced than Laboratory 5's negative bias Additionally, Laboratories 1, 2, and 6 show h statistics that fluctuate with different levels, suggesting interactions between laboratory effects and data levels which may help identify the sources of bias In contrast, Figure 2 does not display any significant or noteworthy patterns related to bias.
4.8.5 Values of the Grubbs’ statistics are given in table 8 These tests again indicate that the data from Laboratory 5 are suspect.
At this stage of the analysis, the statistical expert should investigate Laboratory 5 for potential causes of suspect data before proceeding further If the root cause remains unidentified, all data from Laboratory 5 may need to be excluded when calculating repeatability and reproducibility standard deviations The analysis can then focus on exploring possible functional relationships between these standard deviations and the overall mean This process aligns with the guidelines outlined in ISO 5725-2 and does not introduce new considerations beyond that standard.
Table 4 — Example 1: Determination of mass fraction of protein in feed, expressed as a percentage
Table 5 — Example 1: Cell differences for Level 14
Table 6 — Example 1: Cell averages for Level 14
Table 7 — Example 1: Values of averages, average differences, and standard deviations calculated from the data for all 14 levels in table 4
Average difference Standard deviations j p y j % D j % s yj % s Dj % s rj % s Rj %
Table 8 — Example 1: Values of Grubbs' statistics
Level One smallest Two smallest Two largest One largest
Grubbs' statistics for cell averages
Level One smallest Two smallest Two largest One largest
NOTE — Numbers in brackets indicate the laboratories that give rise to the stragglers or outliers.
The critical values of the Grubbs' test statistics for 9 laboratories, whether applied to the differences or the cell averages, are as follows.
Grubbs' test for a single outlier 2,215 2,387
Grubbs' test for a pair of outliers 0,149 2 0,085 1
Figure 1 — Example 1: Data obtained at Level 14
Figure 2 — Example 1: Consistency check on cell differences (grouped by laboratory)
Figure 3 — Example 1: Consistency check on cell averages (grouped by laboratory)
Applications of the design for a heterogeneous material
Leather exemplifies a heterogeneous material, as no two hides are identical and its properties vary significantly within a single hide A standard test for leather is the tensile strength test outlined in BS 3144, which involves specimen preparation and testing protocols that specify cutting from specific positions within the hide When conducting precision experiments following ISO 5725-2's uniform level design, sending one hide per test level allows for the assessment of laboratory variability, but the variation between hides can influence reproducibility measurements To accurately estimate the test method's reproducibility, each laboratory should test two hides per level, enabling the separation of variation due to differences between hides from the measurement variability itself.
Sand, a common heterogeneous material used in concrete production, is naturally layered with variations in particle size, making particle size distribution critical in construction applications In concrete technology, this distribution is typically measured through sieve testing according to standards such as BS 812-103 To assess sand properties, a representative bulk sample (around 10 kg) is collected and subdivided into smaller test portions (approximately 200 g) Due to natural variability, multiple bulk samples of the same product will show some differences; therefore, sending one bulk sample per laboratory introduces variability that affects the reproducibility of test results Using two bulk samples per laboratory allows for the calculation of reproducibility standard deviation that accounts for this inherent natural variation, ensuring more accurate and consistent testing outcomes.
5.1.3 The above examples also highlight another characteristic of heterogeneous materials: because of the variability of the material, the specimen or test portion preparation can be an important source of variation Thus with leather, the process of cutting specimens from a hide can have a large influence on the measured tensile strength, and with sieve tests on sand the process of preparing test portions from bulk samples is usually the major source of variability in the test method If specimens or test portions are prepared for a precision experiment in a way that does not correspond to normal practice (in an attempt to produce identical “samples”) then the values of repeatability and reproducibility standard deviations produced by the experiment will not be representative of the variability experienced in practice There are situations in which it can be desirable to produce identical “samples” by some special process designed to eliminate, as far as possible, the variability of the material (for example, for a proficiency test, or when a precision experiment is used as part of a programme of work during the development of a measurement method) However, when the aim of the precision experiment is to discover the variability that will be experienced in practice (for example, when vendors and purchasers test samples of the same product) then it is necessary for the variability arising as a consequence of the heterogeneity of the material to be included in the measures of the precision of the measurement method.
Care should also be taken to ensure that each test result in an experiment is obtained by carrying out the test procedure independently of other tests This will not be so if some stages of the specimen preparation are shared by several specimens, so that a bias or deviation introduced by the preparation will have a common influence on the test results derived from these specimens.
The heterogeneous material design outlined in this clause provides valuable insights into sample variability that are not accessible through the uniform level design in ISO 5725-2 While this approach involves higher testing costs due to the need for additional samples, the resulting information can significantly enhance decision-making For example, in the leather quality assessment, understanding variability between hides can inform the optimal number of hides to test or whether to focus on testing more hides with fewer specimens each Similarly, in the sand sample analysis, variability data between bulk samples can help evaluate and improve sampling procedures, ensuring more accurate and reliable results. -**Sponsor**Looking to optimize your article and ensure it's SEO-friendly? [Blogify](https://pollinations.ai/redirect-nexad/RVskEcSD) can help! Blogify is an AI-driven platform that can transform your existing content, like that article excerpt, into engaging, SEO-optimized blog posts It can identify the core meaning, rewrite key sentences for coherence, and ensure your content is ready to rank, saving you time and effort.
This section discusses experimental designs involving three hierarchical factors, with variations such as “laboratories” at the top level, followed by “samples within laboratories,” and then “test results within samples.” Alternatively, in some experiments, “test results within laboratories” may be the second level, with “determinations within test results” as the lowest, especially when laboratories conduct multiple tests per sample The formulas outlined in sections 5.5, 5.6, and 5.9 can be applied to data from such experiments, but the calculation of repeatability and reproducibility standard deviations must be adjusted accordingly It is crucial to specify the number of determinations averaged to produce a test result, as this directly impacts the standard deviation values.
Layout of the design for a heterogeneous material
5.2.1 The layout of the design for a heterogeneous material is shown in table 9.
Participating laboratories receive two samples at specified q levels and perform two tests on each sample Each experimental cell thus yields four test results, comprising two results per sample This setup ensures comprehensive data collection for accurate analysis.
This article discusses the generalization of a simple measurement design, accommodating more than two samples per laboratory per level and multiple test results per sample While the calculations for this broader design are more complex, their fundamental principles remain consistent with the simpler design Therefore, the article provides detailed formulas and examples specifically for the basic design, with extended formulas for repeatability and reproducibility standard deviations presented later in section 5.9 and their practical application in section 5.10.
In analyzing heterogeneous materials, the data are characterized by the notation y_ijtk, where each subscript has a specific meaning: 'i' denotes the laboratory (i = 1, 2, , p), 'j' indicates the level (j = 1, 2, , q), 't' represents the sample number (t = 1, 2, , g), and 'k' corresponds to the test result (k = 1, 2, , n) This structured data representation facilitates comprehensive analysis across different laboratories, levels, samples, and test outcomes, ensuring accurate evaluation of the material's heterogeneity.
Usually, g = 2 and n = 2 In the more general design, g or n or both are greater than two.
In ISO 5725-1 and ISO 5725-2, the symbol p denotes both the number of laboratories and the index in tables of critical values for Cochran’s test When conducting a uniform-level experiment, these two values are identical, simplifying analysis However, for heterogeneous materials, the Cochran’s test index can be a multiple of the number of laboratories, necessitating the use of p¢ to represent the number of laboratories and p to indicate the Cochran’s test index Proper understanding of these distinctions is essential for accurate statistical assessment in calibration and measurement studies.
Organization of an experiment with a heterogeneous material
When planning an experiment with heterogeneous materials, it is essential to follow the guidance outlined in clause 6 of ISO 5725-1:1994 An important consideration is determining the appropriate number of samples to prepare for each laboratory at each level This ensures the reliability and accuracy of the experimental results, accounting for the variability inherent in heterogeneous materials Proper sample sizing enhances the validity of repeatability and reproducibility assessments, contributing to robust and credible measurement data.
Usually, because of considerations of cost, the answer will be two.
ISO 5725-1:1994 provides formulae, tables, and figures in clause 6 and annex B to assist in determining the appropriate number of laboratories, samples, and replicates for measurement studies These tools should be used with the modifications outlined in sections 5.3.2 to 5.3.5 to ensure accurate and reliable experimental design Proper application of these guidelines supports robust statistical analysis and quality assurance in measurement processes.
The uncertainty of the repeatability standard deviation estimate, obtained from experiments on heterogeneous materials, can be assessed by calculating the parameter A_r, as introduced in subclause 6.3 of ISO 5725-1:1994 This calculation provides a quantitative measure of the variability in the repeatability standard deviation, ensuring more accurate and reliable measurement uncertainty assessments.
In ISO 5725-1:1994, the equation for repeatability can be adapted by replacing p with p′ × g, where p′ is derived from the original equation Specifically, the formulas in figure B.1 and the entries in table 1 under repeatability for Ar can be used by substituting p = p′ × g, facilitating consistent application across different sample preparation scenarios When preparing two samples per laboratory at each level, input the value as p = 2p′ into the table or figure to ensure accurate repeatability calculations.
NOTE — The formulae for A r above (and for A R below) have all been derived by the method described in NOTE 24 of ISO 5725-1:1994.
The uncertainty of the reproducibility standard deviation estimate, derived from experiments on heterogeneous materials, can be assessed by calculating the quantity AR, as introduced in subclause 6.3 of ISO 5725-1:1994.
A R 6, D 1 +D 2 +D 3 2g 4 (17) instead of as defined by equation (10) of ISO 5725-1:1994 Here
The values of F and g may be derived from preliminary estimates of the standard deviations s H , s R and s r obtained during the process of standardizing the measurement method.
5.3.4 Follow the guidance given in clauses 5 and 6 of ISO 5725-2:1994 with regard to the details of the organization of an experiment with a heterogeneous material.
Subclause 5.1.2 of ISO 5725-2:1994 specifies the requirements for a group of n tests or measurements, emphasizing that these should be conducted under repeatability conditions In experiments involving heterogeneous materials, these guidelines apply to the entire group of g × n tests within a single laboratory cell, covering all tests performed at one specific level Following these standards ensures the accuracy and consistency of measurement results across different tests and laboratories.
In experiments involving heterogeneous materials, it is essential to prepare a specific number of samples at each level, typically calculated as p¢ multiplied by g (for example, 2p¢ when g equals 2) To ensure unbiased results, these samples should be randomly allocated to participating laboratories, promoting fairness and accuracy in the testing process Proper sample distribution plays a crucial role in experimental integrity and reliable data collection.
Statistical model for an experiment with a heterogeneous material
The basic model utilized in this section of ISO 5725 is detailed in equation (3) of section 4.4.1 When applied to experiments involving heterogeneous materials, this model is extended to account for additional variability, represented by the equation y ijtk = m j + B ij + H ijt + e ijtk. -**Sponsor**Struggling to rewrite your article and ensure it's SEO-friendly? It can be tough! With [Article Generation](https://pollinations.ai/redirect-nexad/uY2G7aqc), you can instantly get 2,000-word, SEO-optimized articles that capture the essence of your content Imagine saving over $2,500 a month compared to hiring a writer, all while ensuring your articles are clear, coherent, and compliant with SEO rules It's like having your own content team, minus the management headaches!
The terms m, B and e have the same meaning as in equation (3) in 4.4.1, but equation (19) contains an extra term
Hijt that represents the variation between samples, and a subscript t representing samples within laboratories (the meaning of the other subscripts is given in 5.2.2).
Variations between samples are generally considered to be random and independent of the laboratory, but they may be influenced by the experimental level Consequently, the term H_ijt is assumed to have a zero expectation, reflecting no systematic bias, while its variance captures the degree of variability within the samples This understanding is crucial for accurate statistical analysis and reliable interpretation of experimental results.
In typical scenarios involving two samples per laboratory and two test results per sample (g = n = 2), it is essential to calculate the sample average and the between-test-result range for each laboratory, level, and sample Specifically, for laboratory i, level j, and sample t (where t = 1 or 2), the sample average provides a central measure of the test results, while the range between test results highlights the variability within each sample These metrics are crucial for assessing the consistency and reliability of measurement results in quality control processes.
( ) y ijt = y ijt 1 +y ijt 2 2 (21) w ijt = y ijt 1 −y ijt 2 (22) b) the cell average, and the between-sample range, for laboratory i, and level j
( ) y ij = y ij 1 +y ij 2 2 (23) w ij = y ij 1 −y ij 2 (24) c) the general average, and standard deviation of cell averages, for level j y j y ij p j q
1 (26) where the summation is over the laboratories i = 1, 2, , p¢.
Statistical analysis of the data from an experiment with a heterogeneous material
This section focuses on the standard scenario where two samples are prepared at each testing level in each laboratory, with two test results obtained from each sample These procedures are detailed to ensure accurate and reliable data collection For more complex situations, such as multiple samples or tests per sample, additional guidance is provided in sections 5.9 and 5.10.
Assemble the data into a table as shown in table 9 Each combination of a laboratory and a level gives a “cell” in this table, containing four test results.
Using equations (21) to (26) from section 5.4.2, we calculated the between-test-result ranges and documented them in Table 10, providing critical insights into test variability Additionally, the between-sample ranges were computed and summarized in Table 11, highlighting the consistency across different samples Lastly, cell averages were determined using the same equations and listed in Table 12, offering a clear overview of central tendencies within the dataset These calculations are essential for quality control and statistical analysis in experimental studies.
Record ranges as positive values (i.e ignore the sign).
Table 9 — Recommended form for collation of data from the design for a heterogeneous material
Laboratory Sample Test result number
Table 10 — Recommended form for tabulation of between-test-result ranges for the design for a heterogeneous material
Laboratory Sample Level 1 Level 2 Level j Level q
Table 11 — Recommended form for tabulation of between-sample ranges for the design for a heterogeneous material
Laboratory Level 1 Level 2 Level j Level q
Table 12 — Recommended form for tabulation of cell averages for the design for a heterogeneous material
Laboratory Level 1 Level 2 Level j Level q
When a cell in Table 9 contains fewer than four test results—due to sample spoilage or data exclusion after outlier testing—either apply the standard formulae provided later or omit all data from that cell.
Option a) is to be preferred Option b) wastes data, but allows the simple formulae to be used.
5.5.3 For each level j of the experiment, calculate the following. a) The sum of squared between-test-result ranges in column j of table 10 (summing over p¢ laboratories and over two samples):
(27) b) The sum of squared between-sample ranges in column j of table 11 (summing over p¢ laboratories):
(28) c) The average and standard deviation of the cell averages in column j of table 12, using equations (25) and (26) in 5.4.2.
5.5.4 Use tables 10, 11 and 12, and the statistics calculated in 5.5.3 to examine the data for consistency and outliers, as described in 5.6 If any data are rejected, recalculate the statistics.
5.5.5 Calculate the repeatability standard deviation s rj and the reproducibility standard deviation s Rj from:
( ) ( ) s Rj 2 =s yj 2 + SS rj −SS Hj 4p′ (30)
If this gives s Rj