Cronbach's alpha is affected by exam duration. Methods: Cronbach's alpha and ordinal alpha were compared in . Coefficient Alpha: a reliability coefficient for the 21st Century? doi: 10.1007/s11336-008-9098-4, Green, S. B., and Yang, Y. The assumption of tau-equivalence (i.e., the same true score for all test items, or equal factor loadings of all items in a factorial model) is a requirement for to be equivalent to the reliability coefficient (Cronbach, 1951). Nurs. To solve this issue, there must be at least two to three indexes to ensure the reliability of the exam. It can also be described simply as a measure of how closely related a set of items are as a collective. Cronbachs alpha was created to measure the internal consistency of the exams [24]. Asia Pac. For more information, please visit our Permissions help page. Res. The lowest score was 18.1 and the highest was 43.1 (out of 50%) for the 4th-year students, with a mean of 33.6, a median of 33.75, an SD of 4.35, and a relative SD of 12.9. ABN 56 616 169 021, (I want a demo or to chat about a new project. 25, 6976. The general rule of thumb is that a Cronbach's alpha of .70 and above is good, .80 and above is better, and .90 and above is best. The validity, which refers to how well a test measures what it is purported to measure, was measured by Pearsons correlation. Psychometric properties of the 8-item english arthritis self-efficacy scale in a diverse sample. Cronbach's Alpha 4E - Practice Exercises.doc. 1 Cronbach's alpha is a measure of inter-item reliability. All authors read and approved the final manuscript. Multivariate Behav. Psychometrika 69, 613625. Al-Homidan, S. (2008). In general the trend is maintained for both 6 and 12 items. No use, distribution or reproduction is permitted which does not comply with these terms. The OSCE had 18 clinical stations (with no repeated stations) and covered history, physical examination, communication skills, and data interpretation. The coefficient is the most widely used procedure for estimating reliability in applied research. doi: 10.1007/s40299-013-0075-z, Wilcox, S., Schoffman, D. E., Dowda, M., and Sharpe, P. A. Standartlatrlm Maddelere (Sorulara) Dayal Cronbach's . 66, 930944. Quantile lower bounds to population reliability based on locally optimal splits. The study aimed to use the Multi-Theory Model (MTM) for health behavior change to explain the intention of initiating and sustaining the behavior of COVID-19 vaccination among the Hispanic and Latinx populations that expressed and did not express hesitancy towards the vaccine in . Meas. In this paper, using Monte Carlo simulation, the performance of these reliability coefficients under a one-dimensional model is evaluated in terms of skewness and no tau-equivalence. Cronbach L. Coefficient alpha and the internal structureof tests. If people were treated more equally in this country we would have many fewer problems. If you do have lots of items, Cronbachs Alpha tends to be the most frequently used estimate of internal consistency. doi: 10.1002/jae.1278, Raykov, T. (1997). In order to evaluate the accuracy of the various estimators in recovering reliability, we calculated the Root Mean Square of Error (RMSE) and the bias. Res. Psychol. Copyright 2016 Trizano-Hermosilla and Alvarado. (2013). A high alpha value is often used (along with substantive arguments and possibly . Pearsons correlation was 0.63, which demonstrates that the OSCE is a valid exam. Coefficient alpha and beyond: issues and alternatives for educational research. R: A Language and Environment for Statistical Computing. As stated by Sijtsma (2009), its popularity is such that Cronbach (1951) has been cited as a reference more frequently than the article on the discovery of the DNA double helix. Construction of the methodological framework (IT, JA). On the reliabilityof a dental OSCE, using SEM:effect of different days. figured out a way to get the mathematical equivalent a lot more quickly. doi: 10.1177/01466216010251005, Reise, S. P. (2012). Descriptive statistics for modern test score distributions: skewness, kurtosis, discreteness, and ceiling effects. In this study four factors were manipulated: tau-equivalence or congeneric model, sample size (250, 500, and 1000), the number of test items (6 and 12) and the number of asymmetrical items (from 0 asymmetrical items to all the items being asymmetrical) in order to evaluate robustness to the presence of asymmetrical data in the four reliability coefficients analyzed. Despite this, the impact of skewness on reliability estimation has been little studied. We know that if we measure the same thing twice that the correlation between the two observations will depend in part by how much time elapses between the two measurement occasions. *Correspondence: Italo Trizano-Hermosilla, italo.trizano@ufrontera.cl, http://ftp.daum.net/CRAN/web/packages/GPArotation/GPArotation.pdf, https://www.webmedcentral.com/wmcpdf/Article_WMC001649.pdf, http://personality-project.org/r/psych/help/glb.algebraic.html, http://personality-project.org/r/html/guttman.html, http://www.crame.ualberta.ca/docs/April 2012/AERA paper_2012.pdf, Creative Commons Attribution License (CC BY). Methods 18, 207230. Cited by lists all citing articles based on Crossref citations.Articles with the Crossref icon will open in a new tab. No single reliability index can be considered a perfect assessment tool to solve this issue. doi: 10.1007/BF02289858, Teo, T., and Fan, X. Cite this article. Surv. R syntax to estimate reliability coefficients from Pearson's correlation matrices. Article Am J Surg. Meas. The number of medical students accepted into medical programs is increasing, which has made the traditional long/short case style of examination difficult to conduct. Cronbachs alpha is not a measure of dimensionality, nor a test of unidimensionality. This website is using a security service to protect itself from online attacks. All 207 students took the clinical and written exams. The study was approved by the Institutional Review Board of the University of Dammam (Approval number: IRB-2014-01-317). The std option standardizes items in the scale to have a mean of 0 and a variance of 1 (again, whether or not you use this option might depend on whether or not youve already standardized the variables Q1-Q6), the detail option will list individual inter-item correlations and covariances, and gen(SCALE) will use these six items to generate a scale and save it into a new variable called SCALE (or whatever else you specify in between the parentheses). As an alternative, you could look at the correlation of ratings of the same single observer repeated on two different occasions. Tau-equivalent model with = 0.558 for the six items > library(psych) > library(Rcsdp) > Cr <-matrix(c(1.00, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 1.00, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 1.00, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 1.00, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 1.00, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 0.3114, 1.00), ncol = 6), > omega(Cr,1)$alpha # standardized Cronbach's [1] 0.731, > omega(Cr,1)$omega.tot # coefficient total [1] 0.731, > glb.fa(Cr)$glb # GLB factorial procedure [1] 0.731, > glb.algebraic(Cr)$glb # GLB algebraic procedure [1] 0.731, # Example 2. CM DART, University Veterinary Centre, Department of Veterinary Clinical Sciences, The University of Sydney, Werombi Road, Camden, New South Wales 2570. different types of reliability, on the advantages and disadvantages of different reliability indices, and on the methods for obtaining them (e.g., Bentler, 2009; Cortina, 1993; Revelle, & Zinbarg, 2009; Schmitt, 1996; Sijtsma, 2009). Springer Nature. The above syntax will produce only some very basic summary output; in addition to the \( \alpha \) coefficient, SPSS will also provide the number of valid observations used in the analysis and the number of scale items you specified. Table 1. RMSE and Bias with tau-equivalence and congeneric condition for 6 items, three sample sizes and the number of skewed items. National University of Distance Education (UNED), Spain. Is the most common test of neuropsychological function and is well used in research. An examination of theory and applications. doi: 10.1177/0013164406288165, Green, S. B., and Yang, Y. Although the standards for what makes a good \( \alpha \) coefficient are entirely arbitrary and depend on your theoretical knowledge of the scale in question, many methodologists recommend a minimum \( \alpha \) coefficient between 0.65 and 0.8 (or higher in many cases); \( \alpha \) coefficients that are less than 0.5 are usually unacceptable, especially for scales purporting to be unidimensional (but see Section III for more on dimensionality). As a result, this may have produced a misleading value that is not as reliable, and this is the main disadvantage of Cronbachs alpha (Table1) [3, 5, 13]. The exams were conducted for 34.3h/day over 7days for all three groups. The parallel forms approach is very similar to the split-half reliability described below. After each exam, the coordinator of the course met with faculty and students to assess and correct any problems with the OSCE to ensure better reliability in the future and they were confidents with OSCE. The findings could help internal medicine departments in our institute and in other medical colleges to improve the OSCE station reliability by considering multiple tools to assess the reliability of the stations and not focus solely on one index, especially given the disadvantages of each measurement tool. CAS In young Mexican university students, the instrument obtained Cronbach's Alpha of 0.86 for the barriers scale and 0.84 for the resources scale. BMC Research Notes 3:34. doi: 10.3389/fpsyg.2012.00034, Sijtsma, K. (2009). Article Cronbachs Alpha is mathematically equivalent to the average of all possible split-half estimates, although thats not how we compute it. In the long test of 12 items the reliability was set at 0.845 taking the same values as in the short test for both tau-equivalence and the congeneric model (in this case there were two items for each value of lambda). It was shown that the reliance on Cronbach's alpha as a sole index of reliability is no longer sufficiently warranted. 2023 BioMed Central Ltd unless otherwise stated. 2010;32:80211. We use cookies to improve your website experience. doi: 10.1007/s11336-008-9099-3, Green, S. B., and Yang, Y. Did you know that with a free Taylor & Francis Online account you can gain access to the following benefits? Future of psychometrics: ask what psychometrics can do for psychology. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. The R2 coefficient increased in the second group and then decreased in the third, which may have been because the examiner made the checklist score correspond to the global score in the second group. In both examples the true reliability is 0.731. We misinterpret. When we look at the effect of progressively incorporating asymmetrical items into the data set, we observe that the coefficient is highly sensitive to asymmetrical items; these results are similar to those found by Sheng and Sheng (2012) and Green and Yang (2009b). 2008;12:1317. PubMed Central On the use, the misuse, and the very limited usefulness of Cronbach's alpha. 7:769. doi: 10.3389/fpsyg.2016.00769. 2014;26:37986. The major difference is that parallel forms are constructed so that the two forms can be used independent of each other and considered equivalent measures. Hacettepe University. As it is the first round of testing a new product or software solution goes through, alpha testing is concerned with finding any possible issues, bugs or mistakes, before progressing to user testing or market launch. You learned in the Theory of Reliability that its not possible to calculate reliability exactly. Cronbach's alpha and more. By using this website, you agree to our In general, the test-retest and inter-rater reliability estimates will be lower in value than the parallel forms and internal consistency ones because they involve measuring at different times or with different raters. The amount of time allowed between measures is critical. Received: 22 September 2015; Accepted: 09 May 2016; Published: 26 May 2016. To evaluate whether a single reliability index is enough to assess the OSCE and to ensure fairness among all participants. 30, 121144. We look forward to having very strong validity in the next few years. In conditions of tau-equivalence, the and coefficients converge, however in the absence of tau-equivalence (congeneric), always presents better estimates and smaller RMSE and % bias than . Consequently t corrects the underestimation bias of when the assumption of tau-equivalence is violated (Dunn et al., 2014) and different studies show that it is one of the best alternatives for estimating reliability (Zinbarg et al., 2005, 2006; Revelle and Zinbarg, 2009), although to date its functioning in conditions of skewness is unknown. The /STATISTICS line provides several additional options as well: DESCRIPTIVE produces statistics for each item (in contrast to the overall statistics captured through /SUMMARY described above), SCALE produces statistics related to the scale resulting from combining all of the individual items, CORR produces the full inter-item correlation matrix, and COV produces the full inter-item covariance matrix. Other authors, such as Revelle and Zinbarg (2009) and Green and Yang (2009a), recommend the use of , however this coefficient only produced good results in the condition of normality, or with low proportion of skewness items. Imagine that we compute one split-half reliability and then randomly divide the items into another set of split halves and recompute, and keep doing this until we have computed all possible split half estimates of reliability. Working with data which comply with this assumption is generally not viable in practice (Teo and Fan, 2013); the congeneric model (i.e., different factor loadings) is the more realistic. Educ Psychol Measur. Organ. The action you just performed triggered the security solution. the analysis of the nonequivalent group design), the fact that different estimates can differ considerably makes the analysis even more complex. Plasma noradrenaline and renin concentrations are reduced. (2009b). 47, 667696. Is coefficient alpha robust to non-normal data? Discussion of the results in light of current theoretical background (JA, IT). Cronbach's coefficient alpha: well known but poorly understood. Google Scholar. In fact the exact opposite is the case, as was shown by Sijtsma (2009), and its application in such conditions may lead to reliability being heavily overestimated (Raykov, 2001). New York: McGraw-Hill; 1994. Psychometrika 74, 145154. Many reliability index measures have been used for the OSCE, including Cronbachs alpha, Spearmans rank correlation, and R2 coefficient determinants. For each observation, the rater could check one of three categories. While Cronbach's Alpha coefficient recorded a value greater than 0.70 and compared: 0.899 on the E-learning/advantages axis, and 0.837 on the E- . We have gone too far in pushing equal rights in this country. You will want to assess the scales face validity by using your theoretical and substantive knowledge and asking whether or not there are good reasons to think that a particular measure is or is not an accurate gauge of the intended underlying concept. Psychol. R Development Core Team (2013). Yes! Package psych. Available online at: http://org/r/psych-manual.pdf, Revelle, W., and Zinbarg, R. (2009). Cronbach (1951) showed that in the absence of tau-equivalence, the coefficient (or Guttman's lambda 3, which is equivalent to ) was a good lower bound approximation. Analysis of quality and feasibility of an objective structured clinical examination (OSCE) in preclinical dental education. 74, 7481. Alternatively, you might want to use the option reverse(ITEMS) to reverse the signs of any items/variables you list in between the parentheses. Available online at: http://personality-project.org/r/html/guttman.html, Revelle, W. (2015b). statement and 15, 2335. 64, 128136. This paper discusses the limitations of Cronbach's alpha as a sole index of reliability, showing how Cronbach's alpha is analytically handicapped to capture important measurement errors and scale dimensionality, and how it is not invariant under variations of scale length, interitem correlation, and sample characteristics. Cronbach's alpha. Spearmans rank correlation and the R2 coefficient determinant values did not differ, which indicated good internal consistency. For the GLB and GLBa coefficients, as the sample size increases the RMSE and the bias tend to diminish; however they maintain a positive bias for the condition of normality even with large sample sizes of 1000 (Shapiro and ten Berge, 2000; ten Berge and Soan, 2004; Sijtsma, 2009). Consequently, before calculating it is necessary to check that the data fit unidimensional models. (2011). This was a pilot study conducted in the Internal Medicine department of Dammam University in 2014. Additional documentation for the psy package can be found here. Dong T, Swygert KA, Durning SJ, Saguil A, Gilliland WR, Cruess D, et al. Cronbach's alpha quantifies the level of agreement on a standardized 0 to 1 scale. Meas. The unicorn, the normal curve, and other improbable creatures. doi: 10.5093/ejpalc2014a4. Med Educ. What are the advantages and disadvantages of the nonequivalent control group pretest-posttest design? variables, using Cronbach's alpha reliability coefficient. Is Cronbachs alpha sufficient for assessing the reliability of the OSCE for an internal medicine course?. Register a free Taylor & Francis Online account today to boost your research and gain these benefits: Cronbach's Alpha: Review of Limitations and Associated Recommendations, /doi/epdf/10.1080/14330237.2010.10820371?needAccess=true. 2023 by the Rector and Visitors of the University of Virginia. In interpreting a scales \( \alpha \) coefficient, remember that a high \( \alpha \) is both a function of the covariances among items and the number of items in the analysis, so a high \( \alpha \) coefficient isnt in and of itself the mark of a good or reliable set of items; you can often increase the \( \alpha \) coefficient simply by increasing the number of items in the analysis. As the duration increases, reliability will increase [ 3, 5, 6 ]. Pallant (2001) states Alpha Cronbachs value above 0.6 is considered high reliability and acceptable index (Nunnally and Bernstein, 1994). Coefficient alpha and the internal structure of tests. 2 and were calculated based on a total possible score of 100. This would result in false inflation of the R2 because the global rating would score the students confidence, organization and professional application of clinical skills, which might not be included in the checklist sheets [14]. Advantages: Can compare scores before and after a treatment in a group that receives the treatment and in a group that does not. The GLB coefficient presents better estimates when the test skewness value of the test is around 0.30; GLBa is very similar, presenting better estimates than with an test skewness value around 0.20 or 0.30. Such research can lead to a more reliable and valid OSCE in the future. The advantage of this perspective over the notion of a high average correlation among the items of a test - the perspective underlying Cronbach's alpha - is that the average item correlation is affected by skewness (in the distribution of item correlations) just as any other average is. This increase occurred over a short period as a first experience for the department of internal medicine. In the example, we find an average inter-item correlation of .90 with the individual correlations ranging from .84 to .95. 2008;13:47993. The hospital anxiety and depression scale: a meta confirmatory factor analysis. How do I interpret Cronbach's alpha? Tavakol M, Dennick R. Making sense of Cronbachs alpha. Share Cite Improve this answer Follow answered Mar 3, 2016 at 11:23 The correlation was 0.63, which indicated a strong correlation between the OSCE score and the written exam score (Fig. Cronbach's Alpha: Review of Limitations . Medicine, Dentistry, Nursing & Allied Health. However, it did not increase in the same manner as the Cronbachs alpha for stability. After all, if you use data from your study to establish reliability, and you find that reliability is low, youre kind of stuck. To assess the performance of the reliability coefficients (, , GLB and GLBa) we worked with three sample sizes (250, 500, 1000), two test sizes: short (6 items) and long (12 items), two conditions of tau-equivalence (one with tau-equivalence and one without, i.e., congeneric) and the progressive incorporation of asymmetrical items (from all the items being normal to all the items being asymmetrical). Psychometrika 42, 579591. 2. However, when there is a low or moderate test skewness GLBa should be used. Although it is considered a good index for station stability, it has some disadvantages: The measure is affected by exam time and dimensionality. Search for more papers by this author. If all of the scale items you want to analyze are binary and you compute Cronbachs alpha, youre actually running an analysis called the Kuder-Richardson 20. Cronbach's Alpha 4E - Practice Exercises.doc. Cronbach's alpha typically ranges from 0 to 1. Chesser AM, Laing MR, Miedzybrodzka ZH, Brittenden J, Heys SD. 1979;13:3954. Advantages and disadvantages of using alpha-2 agonists in veterinary practice. In the congeneric condition corrects the underestimation of . Psychol. Educ. ScoreA is computed for cases with full data on the six items. It breaks down into two parts: the sum of the inter-item covariance matrix for item true scores Ct; and the inter-item error covariance matrix Ce (ten Berge and Soan, 2004). Meas. Psychometrika. Most tests generally efficient in terms of administration time. With an increasing number of medical students being accepted into programs worldwide, it has become difficult to assess them in a proper and fair manner using the old traditional style (long and short cases). If you do have lots of items, Cronbach's Alpha tends to be the most frequently used estimate of internal consistency. Five of these scales can be summarized in two broader scales: (a) the delinquent behavior and aggressive behavior scales form the externalizing behavior scale and (b) the withdrawn, somatic complaints and anxious/depressed scales are combined in the internalizing behavior scale. It was thus discovered in our study that Cronbachs alpha is not sufficient for measuring reliability. One solution has been to use factorial procedures such as Minimum Rank Factor Analysis (a procedure known as glb.fa). For example, Micceri (1989) estimated that about 2/3 of ability and over 4/5 of psychometric measures exhibited at least moderate asymmetry (i.e., skewness around 1). Compared to other studies reporting the reliability and validity of the OSCE, this is the only report that has focused on the measurement tools and index defects in an internal medicine course. View the entire collection of UVA Library StatLab articles. This value increased with each subsequent exam, which may have been because the exam durations increased progressively.Footnote 2 In particular, the third group took longer because of changing the patients secondary to their request and because of the large number of students. In the example it is .87. Nevertheless, it may be said that for these two coefficients, with sample size of 250 and normality we obtain relatively accurate estimates (Tang and Cui, 2012; Javali et al., 2011). There is therefore an unresolved debate as to which of these two methods gives the best lower bound; furthermore the question of non-normality has not been exhaustively investigated, as the present work discusses. In the short test the reliability was set at 0.731, which in the presence of tau-equivalence is achieved with six items with factor loadings = 0.558; while the congeneric model is obtained by setting factor loadings at values of 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8 (see Appendix I). They range from .82 to .88 in this sample analysis, with the average of these at .85. Correspondence to If the assumption of tau-equivalence is violated the true reliability value will be underestimated (Raykov, 1997; Graham, 2006) by an amount which may vary between 0.6 and 11.1% depending on the gravity of the violation (Green and Yang, 2009a). Micceri, T. (1989). In this more realistic condition therefore (Green and Yang, 2009a; Yang and Green, 2011), becomes a negatively biased reliability estimator (Graham, 2006; Sijtsma, 2009; Cho and Kim, 2015) and is always preferable to (Dunn et al., 2014). There are a wide variety of internal consistency measures that can be used. Eur. Provided by the Springer Nature SharedIt content-sharing initiative. The reliability for the OSCE exam was in the acceptable range in all groups, but there were differences in the results that support our hypothesis that no single reliability index can be considered a perfect tool for assessing the OSCE.Footnote 1 There was no difference between the male and female groups in the exam reliability results, which means that gender does not affect the results. (reverse worded). Introductory lectures on the OSCE were held for the faculty to explain the stations, the importance of the rubric for the checklist, and the global ratings. J Pers Asses. The Cronbach's alpha is the most widely used method for estimating internal consistency reliability. The blue print for each exam was established. Therefore, the advantages and disadvantages should be strongly considered within the context of the intended use. Values closer to 1.0 indicate a greater internal consistency of the variables in the scale. Eur J Dent Educ. doi: 10.1037/0033-2909.105.1.156, Moltner, A., and Revelle, W. (2015). Thus, at least two to three indexes should be used to ensure the reliability of the OSCE. Alternative Estimates of Test Reliabiity. Nevertheless, its limitations are well known (Lord and Novick, 1968; Cortina, 1993; Yang and Green, 2011), some of the most important being the assumptions of uncorrelated errors, tau-equivalence and normality. J. Psychosom. Analyses of the correlation of each item with its hypothesized scale revealed the Pearson's correlation coefficients to be 0.49-0.73 for the anxiety subscale and 0.56-0.71 for the depression subscale. These results are discussed below. A Simulation Study for Comparing Three Lower Bounds to Reliability. 26, 329367. Adv Health Sci Educ Theory Pract. Finally, the item option will produce a table displaying the number of non-missing observations for each item, the correlation of each item with the summed index (item-test correlations), the correlation of each item with the summed index with that item excluded (item-rest correlations), the covariance between items and the summed index, and what the \( \alpha \) coefficient for the scale would be were each item to be excluded. Ready to answer your questions: support@conjointly.com. Privacy At the end of the semester, each student took the written exam (control exam), which was analyzed (mean, median, and mode) separately for each year. Data analysis and interpretation of data (IT, JA). PubMed Central Semidefinite programming for the educational testing problem. SDC90 were around 8 for PAIN and PI and 4 for PF. 29, 377392. 2023 Analytics Simplified Pty Ltd, Sydney, Australia. You administer both instruments to the same sample of people. There are four general classes of reliability estimates, each of which estimates reliability in a different way. Skewed items: Standard normal Xij were transformed to generate non-normal distributions using the procedure proposed by Headrick (2002) applying fifth order polynomial transforms: The coefficients implemented by Sheng and Sheng (2012) were used to obtain centered, asymmetrical distributions (asymmetry 1): c0 = 0.446924, c1 = 1.242521, c2 = 0.500764, c3 = 0.184710, c4 = 0.017947, c5 = 0.003159. Nunnally J, Bernstein L. Psychometric theory. Hesitancy toward the COVID-19 vaccine has hindered its rapid uptake among the Hispanic and Latinx populations. Has many subtests that may be selected for use. At Dammam University, the program is shifting to the use of the Objective Structural Clinical Examination (OSCE), which may solve some of these difficulties, including issues with reliability, validity index and exam duration. On the other hand, in some studies it is reasonable to do both to help establish the reliability of the raters or observers. There are two major ways to actually estimate inter-rater reliability. Article doi: 10.1007/BF02295979, Javali, S. B., Gudaganavar, N. V., and Raj, S. M. (2011). It is generally used as a measure of internal consistency or reliability of a psychometric instrument. Cronbach's alpha is thus a function of the number of items in a test, the average covariance between pairs of items, and the variance of the total score. Students were divided into groups as shown in Table1.

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