An Exploratory Examination of the Use of Co-Occurrence Network Analysis to Assess the Anxiety and Beliefs in College Students When Performing Mathematical Computation

Christopher A. MagalisTowson University cmagalis@towson.edu

Rick Parente Towson University fparente@towson.edu

Jaclyn Kenney Towson Universityjkenne20@students.towson.edu

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______________________________________________________________________________

This study explored differences in students self-reported math and statistics anxiety. We used a Co-Occurrence Network to generate hypotheses regarding the experience of solving an algebra or a statistics problem. Students were given a computational task that required solving either a quadratic algebraic equation, or a standard deviation computation. After solving the problem, the students were asked to generate words that described their computational experience. These data were subsequently analyzed using a co-occurrence network that clustered the words into related groupings. Results indicated that words such as anxious/confused co-occurred when describing the statistics problem. However, words such as relieved, calm co-occurred when describing the algebra problem. The generated words for the statistics problem were consistent with what Landbeck (1983) described as “Math Helplessness”. Convergent assessment using parametric analyses of Math Anxiety (MA), Math Helplessness (MH), and Statistics Anxiety (SA) were conducted to test these hypotheses that derived from the network findings. These analyses indicated that MA directly predicted SA whereas MH had an indirect effect on statistics anxiety via the math anxiety variable.

Keywords: mathematics anxiety, statistical anxiety, math helplessness, co-occurrence network analysis, data mining, text mining, regression, college students

PLEASE NOTE: THIS STUDY WAS APPROVED BY THE TOWSON UNIVERSITY INSTITUTIONAL REVIEW BOARD: APPROVAL NUMBER : #1703016533

INTRODUCTION

1.1 BASED ANXIETY IN COLLEGE STUDENTS RELATED TO MATH

Mathematics often creates tension and anxiety in college students (Hlalele 2012; Kim, Park and Cozart 2014). For example, statistics is a particularly anxiety-provoking math-related subject (Lester, 2016; Phillips & Phillips 2016; Rock, Coventry, Morgan and Loi, 2016). Statistics anxiety (SA) is “a feeling of anxiety when taking a statistics course or doing statistical analyses; that is gathering, processing, and interpreting data” (Cruise, Cash, & Bolton, 1985, p. 92). Though some researchers have argued that anxiety related to statistics, and to the general fear of any mathematical material (e.g. algebra, geometry, etc.) are distinct fears (e.g. Baloğlu 2004 ) SA and Mathematics anxiety (MA) are correlated (Paechter, Macher , Martskvishvili , Wimmer and Papousek, 2017). Students who report high levels of SA tend to also report high levels of MA (e.g. Baloğlu 2004, Maysick 1984, Lalayant 2012; Lalonde and Gardner 1993). Research has been lacking regarding what links these types of fears.

1.2 Statistics and Mathematics Anxiety as a Computational Fear The fear of having to manipulate numbers to generate solutions to problems is shared by students with SA and MA (Birenbaum and Eylath 1994). (Onwuegbuzie and Seaman,1995 ; Paechter et al. 2017; Richardson and Suinn 1972; Tobias,1978). Although both MA and SA are associated with this common fear, it is unclear if other factors affect this shared fear. For example, computational anxieties associated with SA and MA may derive from a variety of pre-existing

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beliefs about numerical analyses which have developed over many years of experience with math, long before the student’s encounter with statistics (e.g. Landbeck, 1983). Alternatively, a college student who has not had any problems with math may become frightened by fellow students’ descriptions of their experiences with statistics computations. Difficulty computing solutions to math and statistics problems may underlie students’ beliefs that any attempt will result in failure (Yates 2009). This belief defines the psychological disorder known as “learned helplessness” (Dweck and Repucci 1973; Landbeck, 1983).

1.3 Mathematics Helplessness Underlying Computational AnxietyLearned helplessness accumulates as a person ascends from elementary arithmetic through higher level math (Landbeck, 1983; Onwuegbuzie, DaRos, and Ryan 1997). It may also result from an experience with a specific type of computational topic like statistics. Helplessness was formally conceptualized by Seligman (Seligman, 1990). His theory of learned helplessness suggests that learning is a process whereby certain behavioral responses become associated with an expectation of a successful outcome. If a response becomes independent of its outcome, the person comes to believe that behavior is uncorrelated with consequence (Spendlove, Gavetek, and MacMurray, 1981) which produces a feeling of helplessness.

Landbeck (1983) extended Seligman’s notion of learned helplessness as an explanation of MA. She proposed that mathematics helplessness (MH) derived from feelings of hopelessness that accompanied encounters with computational tasks. Succinctly, she hypothesized that some student learn that their encounters with math will generally produce failure regardless of their efforts. In her study of both high school and college students, scaled measures of MH accounted for more than 50% of the variance in MA which was consistent with her hypothesis. Further, MH seems to be also associated with SA. For example, Onwuegbuzie, DeRos, and Ryan (1997) found that when students had to use complex formulas in statistics courses, they reported a sense of “being out of control”. Malik (2015) reported that students expressed feelings of “inadequacy” and giving up when confronted with statistical equations involving computations.

1.4 Measurement and Analysis Limitations for Computational AnxietyComputational anxiety related to statistics and math is usually assessed using quantitative rating scales (Onwuegbuzie and Wilson 2003; Zeidner, 1991). For example, the Mathematical Anxiety Ratings Scale (Richardson and Suinn, 1972) is an instrument that has been used to index anxiety that results from solving math problems (e.g. Zettle and Houghton 1998; Zientek,, Yetkiner, and Thompson, 2010). The Statistical Anxiety Rating Scale (STARS) (e.g. Cruise Cash and Bolton 1985) has been one of the most frequently used instruments to measure anxiety associated with solving statistics problems. The STARS has a sub-scale that measures anxiety related to computing numerical solutions to statistics problems.

Although these rating scales have been frequently used in math anxiety studies, they have several limitations. First, they do not measure MH Landbeck (1983; Malik 2015; Yates 2009.) Second, the scales constrain the students’ self-report to a predetermined domain that is defined by the items on the scale. That is, numerical quantitative scaling methods can prohibit qualitative description of the students’ unique experience with the computational process. They are therefore insensitive to the qualitative nuances of real-time experience that can only be obtained with verbal description (Davis 2017; Jain and Dowson 2009; Leppävirya 2011). Thus, a qualitative technique that requires verbal description of computational emotions may provide a

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useful augmentation to conventional rating scale measures when exploring the relationship between MA, SA, SH.

We propose that the best way to investigate the emotional and attitudinal content of the experience of computational anxiety is to have students actively perform math versus statistics problems and then generate their own verbal descriptors of their experience. This exploratory technique was then used to generate hypotheses that were tested with conventional quantitative procedures. Specifically, the research design used here required students to compute either a math or a statistics problem and then generate words that described their unique experience in either domain. The words were then analyzed with a co-occurrence network (Kim, Lee, and Jang 2018) to visualize and quantify the communalities among words that most frequently describe these experiences. This methodology (Sheydaei, Saraee, & Shahgholian, 2015) was used to detect patterns of word choice in the generated text (Shinde and Gill 2012).

1.5 Co-occurrence Network Analysis

Co-occurrence network analysis creates a visual image of associations among various types of data, such as words. This technique allows researchers to extract the semantic content of the participant’s verbal descriptions of their experience (Milka 2005; Stuart and Botella 2009). The generated network is composed of nodes representing certain words that are inter-connected with lines (called edges). Co-occurrence statistics are then used to assess the strength of associations between nodes connected by the edges (Higuchi 2016). Co-occurrence networks group together the nodes that have the strongest associations with one another. Nodes with the shortest and bolded edges form a “community”, (Borgatti, Mehra Brass and Labianca, 2009). A co-occurrence analysis then draws a picture of these communities of related nodes (Zhao Wang Zhang and Zhu, 2015; Newman and Girvan 2004; Shikalgar and Dixit 2014 ).

1.6 The Present StudyThis study began with an exploratory investigation (Oquendo et al. 2012) that used a co-occurrence network to investigate the subjective experience of college students after solving an algebra versus a statistics problem. We asked college students to solve either a math or a statistics problem after which they generated words that described their experience. The primary goal of this portion of the study was to reveal the pattern of words college students generated to describe their unique math and statistics computational experience. The co-occurrence analysis would allow a visual assessment of the network of words the students chose that suggested different anxieties related to solving these two types of problems. Hypotheses regarding ratings of SA, MA, and MH when solving statistics and math problems could then be formulated from network analysis results, and tested with conventional scaled measures of SA, MA, and MH. Our expectation was that the two types of computation would produce different networks of descriptors. We then generated hypotheses based on the observed networks and tested these hypothesis using rating scales and convention linear modeling procedures.

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2 Method2.1 Participants Forty-five undergraduate college students (36 females and 9 males, mean age = 21.40 years, SD = 1.74) participated in the study. The sample consisted of 60.9% Caucasian, 19.6% African American, and 4.3% Hispanic, and 2.2% Asian, 2.2 % Filipino, and 8.7% mixed (e.g. African-Asian decent). The students were enrolled in statistics and research methodology courses Towson University. The courses covered fundamental topics related to the use of descriptive and inferential statistical analyses in psychological research. These topics included: measures of central tendency and dispersion, outlier screening, sampling and probability theory, hypothesis testing, regression analysis, and analysis of variance models applied to experiments. Approximately 52.2% of students in these courses were social-behavioral science majors (n = 24), with the other 45.70% were Health, Biology, and Occupational Therapy and Speech Pathology majors (n = 21).

The participants varied with respect to past mathematics courses taken in high school. While all participants had taken required Algebra coursework, there were differences with respect to the highest-level math course taken in High school which ranged from Algebra II (24%), Calculus (20%), Pre-Calculus (20%), Statistics (16%), Trigonometry (11%), Geometry (11%), College Algebra (2%), and Finite Mathematics (2%).

2.2 Materials Students initially filled out a demographic survey that assessed various characteristics including: age, gender, ethnicity, and mathematics course history. Next, they were assigned to either the math problem or statistics problem computation and computed the problem. Finally, they filled out several questionnaires that measured MA, SH, and MH. The order of presentation of the scales was counterbalanced using Latin Square procedures. The math problem was a quadratic algebraic equation that required participants to solve for an unknown X value. The problem was: X2 - 1 = (35 + 13). The other problem was a statistics application that required participants to

compute a sample standard deviation √ ∑(X−M )2

n−1 (where M was the sample mean (∑X / n) for a

small data set consisting of three values = (X = 2, 4, 6). The first portion of the survey

These types of problems were chosen for three reasons. First, neither problem required little to no use of a calculator. Second, each involved a squaring operation and knowledge of simple arithmetic computation. Third, each student had taken a pre-requisite college algebra course that included practice solving similar algebraic problems. Each student also had recent experience computing similar standard deviations in their course work.

After the problem-solving portion, the participants were directed to write as many words as they could that described their experience solving the problem.

Each student also filled out a rating scale consisting of 12 statements that were rated on their level of agreement using a 5-point Likert scale (1 = strongly disagree, 2 = disagree, 3 = no feeling, 4 = agree, 5 = strongly agree). The statements and scales were based on those used in a thesis study conducted by Landbeck (1983). The statements were divided into three sub-scales

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each consisting of four statements. The sub-scales included 1. An MA sub-scale, 2. An MH sub-scale, and 3. An SA statistics anxiety sub-scale.

The MA sub-scale measured the level of anxiety students reported when exposed to mathematics as an academic subject, as well as when they encountered mathematical material. The items from the MA sub-scale included: “A math test would scare me”, “My mind goes blank and I am unable to think clearly when I think of trying to solve difficult math problems.”, “I get a sinking feeling when I think of trying hard math problems”, and “Mathematics usually makes me feel uncomfortable and nervous”.

The MS sub-scale assessed how helpless participants perceived themselves when confronted with situations involving math. The items from the MS subscale included: “I usually feel helpless when I take a math test”, “I feel helpless when I make change”, “When I balance my checkbook, I feel helpless”, and “If I were an accountant, I would feel helpless”.

Finally, the SA sub-scale measured how anxious participants perceived themselves to be about statistics, as well as how anxious they would be when encountering a statistics test or statistics problems. The items included: “Statistics makes me feel uncomfortable and nervous”, “I am unable to think clearly when I think of attempting to solve hard statistics problems”, “An exam specifically in statistics would scare me”, and “I particularly get a sinking feeling when I think of trying difficult statistics problems”. The 12 statements in the survey were displayed in random order on the survey. Before analysis, each sub-scale was summed to obtain separate mathematics anxiety, mathematics helplessness, and statistics anxiety scores. Larger scores on any scale represented heightened anxiety or helplessness.

2.3 Procedure Students were recruited from undergraduate classes at Towson University. Those who consented to participate read and signed an informed consent statement. Thereafter, the surveys were distributed to groups using block randomization. This was done to ensure that half of them received the version containing the algebra problem, while the remaining participants received the statistics problem.

The students also rated their levels of SA, MA, and MH using the scales contained on the survey. The participants were given as long as they needed to complete the survey. Once completed, the participants returned their surveys to the primary investigator. At the completion of the study, the primary investigator fully debriefed the participants with respect to the purpose of the study, and allowed time for them to ask questions.

Before analyzing these data with co-occurrence analysis, the surveys were inspected to exclude any participants who did not generate any words after performing the problem task, or who generated sentences rather than words. This insured that the analysis would converge on major descriptors of the computational experience (e.g. indicate adjectives), as opposed to irrelevant words, such as indirect articles. Twenty-one participants were excluded; twenty-four were used for the final data analysis. This sample had comparable demographics to the initial sample. There were more women (87.5%) than men (12.5%). The distribution of ethnicity was also comparable: 62.5% Caucasian, 12.5% African American, 4.2% Filipino and Hispanic, as well as

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4.2 % Asian. Math experience was also comparable with the majority of students noting that the highest math courses taken were Algebra and Statistics.

1. RESULTS

3.1 KH Coder ProgramThe participants’ text descriptions were analyzed with a co-occurrence analysis specifically, the KH Coder software package (https://sourceforge.net/projects/khc/) (Higuchi 2016). We selected the Jaccard index as a basis for constructing the edges of the network connecting word nodes. The Coder program converts this index into a line with a certain thickness and length. Although there are other measures that can be used to construct edges (Fletcher and Islam 2018), we chose the Jaccard because the index is optimal for comparing patterns of data (Fletcher and Islam 2018). We expected that the words produced to describe the experience of computing a statistics problem or an algebra problem would reveal different patterns of association. Our goal was to assess which types of words could distinguish the experiential difference between statistics and math computation.

3.2 SPSS Version 23After exploring the word associations, subsequent linear modeling used for hypothesis testing was conducted using The Statistical Package for the Social Sciences (SPSS) version 23. These analyses included independent-samples t tests, and hierarchal regression. Beta weights generated from regression analysis was also used for a path analysis.

3.3 Data PreparationBefore entering and pre-processing the data into KH Coder, words were classed under two specific taxonomic categories: “statisticsproblem” that were words the students used to describe solving the standard deviation, and “algebraproblem” to classify words that described solving the algebra problem.

3.3 Co-occurrence network Figure 1 displays the results of the co-occurrence network analysis. This diagram displays the nodes that have the strongest co-occurrence to algebra problems versus statistics problems. Two “algebraproblem” nodes were revealed in the analysis, while only one node was revealed for the “statisticsproblem”. This may have been due to students producing more words for the algebra problem, than for the statistics problem.

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Figure 1: Co-occurrence communities between-ness network

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The network plot revealed very different co-occurrence networks for the statistics versus the algebra problem. Words such as “nervous” and “confused” “anxious” “uncertain” “unaware” “wrong” and “uneasy” were closely associated with solving the standard deviation problem. Words strongly associated with performing the algebra problem included: “relaxed” “calm”, “relieved “, “satisfied” “easy”, “confident”, “proud”, and “smart”. In general, the statistic problem elicited a sense of heightened anxiety whereas the algebra problem elicited a relative feeling of calmness. The word “confused” was in closest community with “statisticsproblem”. Conversely, the word node “relieved” was in closest community to “algebraproblem”.

Further inspection of the network revealed clear indicators of helplessness. Specifically, the words “wrong”, “confused”, “uncertain”, and “surprised” appeared in community with the “anxious” node. These are precisely the emotions that typically occur when a person loses a sense of predictability in their world which is the root cause of helplessness. The word “wrong” suggests a sense of impending failure before the student even knew the correct answer. These results suggest that although SA may be correlated with MA, MH may also predict SA either directly or indirectly.

3.4 Hypothesis GenerationAlthough it is tempting to conclude from the network analysis that words produced when computing statistical problem indicate more computational anxiety than words produced when computing the algebra problem, it is possible that those who performed that statistics problem would have higher rated SA, MA, and MH than those who performed the algebra problem. To test this prediction, several independent t tests were conducted. Initial data screening indicated that all parametric assumptions were met, including normality and homogeneity of variance. Power analysis also indicated that power was adequate (above .80). However, findings indicated that though students who computed the statistics problem had slightly higher levels of rated SA (M = 13.67, SEM =1.31) did those who computed the algebra problem (M = 10.00, SEM = 1.17), this difference was not significant, t (22) = 2.01, p > .05. Similarly, while statistics problem students had slightly higher math anxiety ratings (M = 13.22, SEM = 1.40) than the algebra problem students (M = 9.93, SEM = 1.14), the difference was not significant, t (22) = 1.80, p > .05. Likewise, MH did not differ between the statistics group (M = 11.67, SEM = 1.24) compared to algebra group (M = 11.07, SEM =, 41). It is therefore reasonable to conclude that the difference between the network diagrams was not due to overall differences between the two groups in terms of rated MA, MH, or SA. The network diagram for the statistics problem suggests that MH is a prominent factor in the student’s subjective experience. It is unclear, however, how MH interacts with MA and SA. The following analyses investigated this interaction.

3.5 General Linear Modeling of Scaled MeasuresHierarchal regression analysis and path analysis computed on the SA, MA, and MH ratings were used to assess the correlative relationship among these variables. Initial data screening revealed that the assumptions of normality and homoscedasticity were met. Residual plots showed that the linearity assumption was also met. Again, power levels were adequate (above .80). Table 1 presents the results of a stepwise regression computed on these data with SA as the outcome and MH and MA as predictors. The results of this analysis are displayed in Table 1 below:

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Table 1: Hierarchical regression for variables predicating SA.

Variable B SEB β Adjusted R2 ΔR2

Step 1 ..42* .44*

Math Helplessness .99 .24 .67*

Step 2 .70* .28*

Math Helplessness .18 .24 .12

Math Anxiety .77 .17 .76*

*p < .01

The results indicated that MH ratings initially predicted SA ratings, however, MA predicted SA above and beyond helplessness when added to the model.

Figure 2 shows a path modeling of the MA, MH, and SA variables. MA had a direct significant effect of SA. MH had an indirect effect on SA. MH did have a direct on SA.

Figure 1. Path model for SA, MA, and MH variables.

79*.79

.70*

10

Math

Anxiety

Statistics

Anxiety

Math

Helplessness

* p < .05

4 Discussion The first part of this study was an exploratory investigation of the co-occurrence of words that described the subjective experience of performing statistical and algebraic computations. There were two major findings in these data. First, the co-occurrence Analysis revealed different patterns of descriptors for the algebra versus statistics computations. Second, linear modeling indicated that MA directly predicts SA and that MH influences SA indirectly via MA. The co-occurrence network revealed that the statistics computation evoked descriptors such as “nervous” and “confused” “anxious” “uncertain” “unaware” “wrong” and “uneasy”. On the other hand, the participants used words such as “relaxed” “calm”, “relieved “, “satisfied” “easy”, “confident”, “proud”, and “smart”, to describe their experience with the algebra computation. The first set of words implies a feeling of nervousness and confusion which is consistent with an underlying feeling of helplessness (Meyer-Griffith, Reardon, and Hartley 2009) and a feeling that their final answer is destined to be “wrong”. The second set of words describes a sense of confidence and familiarity with the algebraic computational process which may derive from repeated exposure to this type of problem in high school and college. The difference between the statistics and math problem portions of the network diagram suggests that statistics create a sense of discomfort that is more than a math phobia. The fact that the groups did not differ in terms of their average rating scale scores suggests that the difference in the network diagram was not due to inherent correlative differences in MA, MS, of SA between the groups.

The second portion of this study tested Landbeck’s (1983) notion of the relationship of SA to MH. Findings derived from the path model results indicated that SA is directly predicted by MA and is indirectly predicted by MH. This finding is consistent with the results from Landbeck’s (1983) study. MH is related to SA but the relationship is medicated by MA.

This study has a number of weaknesses that should be acknowledged. First, the sample size was small and there were almost as many exclusions as there were participants. However, though this may have affected power for the parametric analyses, the primary purpose of the t test and regression analyses were to triangulate network results and build hypotheses from such results. Future research will nevertheless replicate these results with a larger sample of students. Second, although the math and statistic problems were similar in terms of the conditions outlined above, (i.e., neither required use of a calculator; both involved a squaring operation; students had prior experience computing each type of problem), the problems were not psychometrically scaled for difficulty. Thus, we will subsequently use the STARS AND MARS to validate network findings, given their high reliability and validity coefiicents.

Perhaps the more important question concerns how these results are useful to both students and teachers of quantitative content. First, they highlight the need to address student fears of quantitative coursework. Students who fear math or statistics typically do not perform well in these courses (Onwuegbuzie & Wilson, 2003) and it is reasonable to suggest that some portion of their lack of success could be obviated by initially discussing the anxiety, helplessness, and confusion they are likely to experience. Second, the data suggest that helplessness is a large part

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of statistics anxiety. This may be lessened if the instructor would first teach the students the computational procedures before explaining the more theoretical content. This approach would give the students the confidence that they can at least compute the various measures under discussion. Finally, it is necessary for the instructor to adopt an attitude of tolerance towards the student’s frustration with the learning experience.

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APPENDIX (DATA)

5.1 Word-Network Data

17

statisticsproblem nervous frazzled overwhelmed confused accomplished fast anxious pressured smartalgebraproblem effortless joy ease pleasurable entertaining satisfying easy straightforward soothing diversion delight algebraproblem smart confident algebraproblem familiar pressure disinterested boringalgebraproblem calm settled confident joyful relaxedalgebraproblem easy nostalgic calm unstressed confident intelligent satisfied relievedalgebraproblem focused statisticproblem distressed anxiety anxious uncertain uneasy unconfident uncertain confused wrongstatisticsproblem nervous confused anxious struggling revitalizing logical confident uncertain proud contentstatisticsproblem scared confused unaware intimidated forgetful disappointedalgebraproblem quick easy basic satisfied relieved relaxed algebraproblem reminiscent unsurestatisticsproblem frustrated annoyed disbelief bored confusedalgebraproblem good competent easilyalgebraproblem concentrated serious pressured stress unsurealgebraproblem uncomfortable uneasy distractedstatisticsproblem confused frustratingalgebraproblem confident good well satisfied happy surprised proudalgebraproblem secure calm reassured prepared ready validated peaceful methodical proud satisfied relieved determined proud curiousstatisticsproblem helpless clueless lost unaware confused dumbfounded anxious surprised shocked unprepared wrong unfitstatisticsproblem uneasyalgebraproblem difficult easyalgebraproblem confident relieved nervousstatisticsproblem nervous angry depressed confused

5.2 SPSS demographics and rating scale data

partic # gender ethnicity age major* problem SA MA MH

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score score score

1 female African Amercian

20 2 stat 9 6 4

2 female Caucasian 21 2 alg 7 8 10

3 female Caucasian 22 1 alg 17 18 15

4 male Caucasian 25 2 alg 6 5 8

5 female Caucasian 21 2 alg 12 9 13

6 female African Amercian

21 2 alg 8 11 14

7 female Caucasian 21 1 alg 17 16 16

8 male Caucasian 28 2 stat 11 8 11

9 female his 21 2 stat 16 17 13

10 female Caucasian 21 2 stat 19 17 17

11 female Caucasian 20 1 alg 6 6 9

12 female Caucasian 19 2 alg 6 7 8

13 female Caucasian 21 2 stat 16 14 13

14 female Filipino 19 1 alg 12 8 10

15 female Caucasian 20 2 alg 7 5 13

16 male Caucasian 20 1 alg 7 11 8

17 female African Amercian

23 2 stat 7 14 12

18 female Caucasian 19 2 alg 8 9 10

19 female Caucasian 20 1 alg 16 16 13

20 female Caucasian 21 1 stat 16 16 11

19

21 female African American

19 2 stat 13 10 9

22 female African American

21 1 alg 16 15 11

23 female African American

22 1 alg 5 5 8

24 female Not Specified

19 2 stat 16 17 15

20