A Review and Discussion of the Mean – Median Protocol Acceptability of Data and Reporting Test Result as Arithmetic Mean

International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 6, 2025 DOI: 10.5281/zenodo.17855945 Original Article ©2025 RS Publication, [email protected] 207 A Review and Discussion of the Mean – Median Protocol Acceptability of Data and Reporting Test Result as Arithmetic Mean 1. Dr Abhilash Babu G - Research Officer Regional Analytical Laboratory , Kuthuparamba PO , Kannur District , Kerala orcid id : 0009-0000-9283-0973 2. Anil Kumar T - Government Analyst, Regional Analytical Laboratory , Malaparamba PO , Kozhikode District , Kerala ARTICLE INFO ABSTRACT ©2025 RS Publication Paper ID: IJASTR6935A931BE594 Published: 2025-12-10 DOI: https://dx.doi.org/10.5281 /zenodo.17901712 Page No: 207-213 The testing laboratories , often , expressed their final quoted test result as arithmetic mean of replicate measurements . Whether all the data obtained during the replicate measurements are correct and can be used for calculating arithmetic mean ?. , yet under noticed . This article describes and exemplifies the Mean – Median method for determining the acceptability of data , applicable for reporting of test result as arithmetic mean or median , the measurement uncertainty , verification and validation of test methods and for defining acceptability criteria for replicate measurements in quality control activities . KEYWORDS Arithmetic Mean , Precision , Median , Replicate , Range , Critical Range factor , outlier , verification , validation , uncertainty INTRODUCTION The evaluation of data obtained during replicate measurements is of great importance to a number of laboratory activities . The arithmetic mean of such data is used for reporting test results , calculating measurement uncertainty , defining criteria for replicate measurements and as specification in validation and verification of test methods . The acceptability , accuracy and validity of test result is of great importance as it is an embodiment of the competence of a laboratory . The laboratories have a general thinking that if the result of test parameter is expressed as arithmetic mean of two or more replicates , error can be minimised and the result becomes more accurate . Is it correct ?. Whether arithmetic mean is the only solution ?. No lab personnel is bothered about it and yet the question remains under discussed . In our opinion , merely expressing the test result as arithmetic mean is totally wrong . The collected data has to be evaluated before reporting the value . The result can be expressed as either arithmetic mean or median . When and where the arithmetic mean or median is used for expressing test result can be determined by a number of international methods . International Journal of Advanced Scientific and Technical Research Available online on http://www.rspublication.com/ijst/index.html ISSN 2249-9954 Cite This Paper: Dr Abhilash Babu G and Anil Kumar T (2025). "A Review and Discussion of the Mean – Median Protocol Acceptability of Data and Reporting Test Result as Arithmetic Mean". INTERNATIONAL JOURNAL OF ADVANCED SCIENTIFIC AND TECHNICAL RESEARCH (IJASTR) , vol. 15, no. 6, 2025, pp. 207-213. DOI: https://dx.doi.org/10.5281/zenodo.17901712 International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 6, 2025 DOI: 10.5281/zenodo.17855945 Original Article ©2025 RS Publication, [email protected] 208 Through this article , we made an earnest attempt to discuss the procedure based on Mean – Median method for evaluating the data of replicate measurements and expressing the result as either arithmetic mean or median . The protocol is well illustrated and hypothesis are used at certain places . DEFINITION Some important definitions are included . 1. Precision : The closeness of agreement between independent test results obtained under stipulated conditions . Repeatability and reproducibility are particular sets of extreme conditions 2. Independent test results : Means results obtained in a manner not influenced by any previous result on the same or similar test object. 3. Repeatability : Precision under repeatability condition 4. Repeatability Condition : Conditions where independent test results are obtained with same method on identical test items in the same laboratory by the same operator using the same equipment within short intervals of time . 5. Criteria : A set of requirements used as a reference against which objective evidence is compared . 6. Arithmetic Mean : The sum of a collection of numbers divided by the count of the numbers in the collection . It is simply called mean or Average and is the Central value of data set . 7. Median : The middle value in a data set that has been arranged in ascending or descending order . 8. Standard Deviation : The amount of variation or dispersion in a data set , showing how much the individual data points typically deviate form the mean . 9. Outlier : A member of a set of values which is inconsistent with the other members of that set . ABBREVIATION 1) ISO – International organisation for Standardisation 2) IEC – International Electrotechnical Commission 3) SD – Standard Deviation 4) CR – Critical Range 5) r – Repeatability Limit DESCRIPTION OF METHODOLOGY The International Standard for the competency of testing , ISO/IEC 17025 requires that laboratory has to generate valid test results . The validity of test results can be ensured by internal as well as external quality control activities . The laboratory personnel usually perform replicate analysis , called repeatability , as an internal quality control activity , but never evaluate the data before the final result is being reported . This will , some times , cause reporting of incorrect test results . The acceptability of arithmetic mean can be checked by a number of international methods . Here , we discuss , the Mean – Median protocol with real data evaluation . The protocol uses a new terminology , called Critical Range Factor , f(n) , which is the 95% quantile of the distribution of (X max – X min )/σ where X max and X min are the extreme values in a sample of size n from a normal distribution with standard deviation σ . The critical values are given in Table – 1 . International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 6, 2025 DOI: 10.5281/zenodo.17855945 Original Article ©2025 RS Publication, [email protected] 209 The Critical Range Factor is , then , used to calculate Critical Range ( CR 0.95 (n)) , according to the equation CR 0.95 (n) = f(n) . σ Remember , critical range and range are two different terminology , where the latter is the difference between two extreme results , X max and X min . If only two replicate data are obtained , the comparison is made with Repeatability limit (r) , according to the equation r = 2.8 σ Table – 1 Where f(n) – critical range factor and n – number of replicates , where n ≥ 5 n f(n) n f(n) n f(n) n f(n) n f(n) n f(n) 2 2.8 10 4.5 18 4.9 26 5.2 34 5.4 50 5.6 3 3.3 11 4.6 19 5.0 27 5.2 35 5.4 60 5.8 4 3.6 12 4.6 20 5.0 28 5.3 36 5.4 70 5.9 5 3.9 13 4.7 21 5.0 29 5.3 37 5.4 80 5.9 6 4.0 14 4.7 22 5.1 30 5.3 38 5.5 90 6.0 7 4.2 15 4.8 23 5.1 31 5.3 39 5.5 100 6.1 8 4.3 16 4.8 24 5.1 32 5.3 40 5.5 ------- ------- 9 4.3 17 4.9 25 5.2 33 5.4 45 5.6 ------- ------- DISCUSSION Two schemes , based on Mean – Median protocol , with scenario are discussed with examples : Hypothetical values are used at some places of standard deviation . Scenario – 1 : Two replicates The following scheme can be used to evaluate two replicate tests . 1. Calculate the arithmetic mean , standard deviation , absolute difference and repeatability limit . 2. If the absolute difference does not exceed repeatability limit , then both test results are considered acceptable, and the final quoted result should be quoted as the arithmetic mean of the two test results . 3. If the absolute difference does exceed repeatability limit , the laboratory should obtain two further test results . 4. Arrange the four test results in ascending/descending order 5. If the range (X max – X min ) of the four test results is equal to or less than the critical range for n = 4 , CR0.95(4), the arithmetic mean of the four test results should be reported as the final quoted result . 6. If the range of the four test results is greater than the critical range for n = 4, the median of the four test results should be reported as the final quoted result . The two situations are exemplified in Table – 2 and Table – 3 International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 6, 2025 DOI: 10.5281/zenodo.17855945 Original Article ©2025 RS Publication, [email protected] 210 Table – 2 (Two test results are acceptable) Test Parameter Saponification Value of Coconut Oil Data (2 Replicates) 251.72 , 252.84 Arithmetic Mean 252.28 Standard Deviation σ 0.56 Repeatability Range 2.8 X 0.56 = 1.568 Absolute Difference 252.84 – 251.72 = 1.12 Remarks Since absolute difference does not exceed repeatability range , both data are acceptable The final quoted result = 252.28 Table – 3 (Two test results are insufficient) Test Parameter Total Hardness of Drinking Water (mg/l) Data (2 Replicates) 47.58 , 48.09 with stated standard deviation of 0.16 Mean 47.835 Standard Deviation σ 0.16 (stated) Repeatability Range 2.8 x 0.16 = 0.448 Absolute Difference 48.09 – 47.58 = 0.51 Remarks – 1 Since absolute difference exceeds repeatability range , data are not acceptable . Obtain two more test results , 47.77 , 48.21 Test Results Descending Order 48.21 , 48.09 , 47.77 , 47.58 Mean ( 4 replicates) 47.9125 Standard Deviation σ 0.2504 Critical Range CR 0.95 (4) 3.6 x 0.2504 = 0.9014 Range 48.21 – 47.58 = 0.63 Remarks – 2 Since the range does not exceed critical range , all the four data are acceptable and the arithmetic mean of data is reported as the final quoted result Mean = 47.835 mg/l Scenario – 2 : Replicate Analysis where n ≥ 5 When a sufficiently large number (n ≥ 5) of test results are obtained under repeatability conditions 1. Record n replicate data , such that n ≥ 5 and arrange in ascending/descending order 2. Calculate the arithmetic mean , standard deviation , range and Critical Range 3. If the range does not exceed the critical range , then the arithmetic mean of all the n test results is reported as the final quoted result . 4. If the range exceeds the critical range , further measurements is required . The number of further 5. measurement , say m , will depends on the size of n and the ease of performing the International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 6, 2025 DOI: 10.5281/zenodo.17855945 Original Article ©2025 RS Publication, [email protected] 211 measurements . m has to be chosen as an integer satisfying the condition n/3 ≤ m ≤ n/2 6. If the range of all the n+m results not exceed the critical range , CR0.95(n+m) , then the arithmetic mean of all the n+m test results is reported as the final quoted result . 7. If the range exceeds the critical range , Median of all (n+m) result is quoted as the final test result . The three situations are exemplified in Table – 4 , Table – 5 and Table – 6 Table – 4 (Initial data sufficient) Test Curcumin content (mg/l) in Turmeric Powder Data (12 Replicates) Ascending Order 3.09 , 3.14 , 3.36 , 3.61 , 3.72 , 3.93 , 4.05 , 4.15 , 4.19 , 4.28 , 4.86 , 5.17 Mean 3.9875 Standard Deviation 0.6055 Critical range CR0.95(12) 4.6 x 0.6055 = 2.7853 Range 5.17 – 3.09 = 2.08 Remarks Since the range does not exceed critical range , all data are acceptable and the arithmetic mean of data is reported as the final quoted result Mean = 3.9875 mg/l Table – 5 (Initial Data insufficient – Quoted Result as Mean) Test Tartrazine Colour(mg/l) in Fruit Drink Data (9 Replicates) 75.98 , 75.46 , 75.06 , 74.79 , 74.35 , 74.13 , 73.91 , 72.97 , 72.39 Ascending Order Mean 74.338 Standard Deviation 0.58 (stated) Critical Range CR0.95(9) 4.4 x 0.58 = 2.552 Range 75.98 – 72.39 = 3.59 Remarks – 1 Since the range of 9 test results exceeds critical range , data are not acceptable . Obtain more Data , such that n/3 ≤ m ≤ n/2 = 9/3 ≤ m ≤ 9/2 = 3 ≤ m ≤ 4.5 Thus 3 or 4 tests can be performed . Collect 3 more data (m=3) 73.91 , 75.06 , 74.88 Data Ascending order 75.98 , 75.46 , 75.06 , 75.06 , 74.88 , 74.79 , 74.35 , 74.13 , 73.91 , 73.91 , 72.97 , 72.39 Mean (9+3 Replicates) 74.4075 Standard Deviation 0.98 Critical Range CR0.95(12) 4.6 x 0.98 = 4.508 Range 75.98 – 72.39 = 3.59 Remarks – 2 Since the range of 12 test results does not exceed critical range , all data are acceptable and the arithmetic mean of data is reported as the final quoted result Mean = 74.4075 mg/l International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 6, 2025 DOI: 10.5281/zenodo.17855945 Original Article ©2025 RS Publication, [email protected] 212 Table – 6 (All Data insufficient - Quoted Result as Median) Test Protein Content (mg/l) in Nutrimix Data ( 6 Replicates) Ascending Order 15.41 , 15.72 , 15.92 , 16.64 , 16.93 , 17.18 Mean 16.2957 Standard Deviation 0.43 (stated) Critical Range CR0.95(6) 4.0 x 0.43 = 1.72 Range 17.18 – 15.41 = 1.77 Remarks – 1 Since the range of 6 test results exceeds critical range , data are not acceptable . Obtain more data such that n/3 ≤ m ≤ n/2 = 6/3 ≤ m ≤ 6/2 = 2 ≤ m ≤ 3 Thus 2 or 3 tests can be performed . Collect 3 more data (m=3) 16.57 , 15.93 , 16.84 Data (9 Replicates) Descending order 17.18 , 16.93 , 16.84 , 16.64 , 16.57 , 15.93 , 15.92 , 15.72 , 15.41 Mean 16.349 Standard Deviation 0.39 (Stated) Critical Range CR0.95(9) 4.4 x 0.39 = 1.716 Range 17.18 – 15.41 = 1.77 Remarks – 2 Range of all the 9 test results exceeds critical range , Arithmetic mean can’t be used . Median of data is reported as final quoted result . Median = 16.57 mg/l CONCLUSION The Mean – Median method for the acceptability of data is well illustrated by examples . The testing laboratories can follow this method to evaluate data where the final result is reported as arithmetic mean . The evaluation is equally beneficial for calculating measurement uncertainty , verification and validation of test methods and defining acceptability criteria for replicate measurements in internal quality control activities . AUTHOR INFORMATION Authors 1. Dr Abhilash Babu G - Research Officer Regional Analytical Laboratory , Kuthuparamba PO , Kannur District , Kerala orcid id : 0009-0000-9283-0973 Email : [email protected] 2. Anil Kumar . T - Government Analyst, Regional Analytical Laboratory , Malaparamba PO , Kozhikode District , Kerala Email : [email protected] Author Contribution Both the authors contributed equally Notes The authors declare that they have no technical , financial and intellectual conflict of interest . International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 6, 2025 DOI: 10.5281/zenodo.17855945 Original Article ©2025 RS Publication, [email protected] 213 REFERENCE 1. ISO 5725 Series 2. IS 15393 Series 3. ISO 3534 Series 4. ISO/IEC 17025 : 2027 5. IS 7920 6. IS 15393 7. AOAC Guidelines for Single Laboratory Validation of Chemical Methods for Dietary Supplements and Botanicals 8. Eurachem Guide : The Fitness for Purpose of Analytical Methods 9. Analytical Quality Control and Method Validation Procedures foe Pesticide Residue Analysis in Food and Feed SANTE/11312/2021 10. AOAC Appendix F: Guidelines for Standard Method Performance Requirements 11. AOAC Method Verification Protocols 12. ASTA UV Method for Curcumin 13. FSSAI Manual of Cereals and Cereal Products 14. FSSAI Manual of Water 15. FSSAI Manual of Oils