ANALYSIS AND 
INTERPRETATION OF DATA

Data analysis is the process of extracting information from data. Data analysis is a method in which data is collected and organized so that no one can derive helpful information from it. In other words, the main purpose of data analysis to look at what is trying to tell us. Analysis of data means studying the organized material in order to discover inherent facts. The data was studied from as many angles as possible to explore new facts. Statistical techniques have contributed greatly in gathering, organizing, analyzing and interpreting numerical data. (Koul, 2014)
Data analysis is the process of systematically applying statistical and logical techniques to describe and illustrate, condense and recap and evaluate data. An essential component of ensuring data integrity is the accurate and appropriate analysis of research findings. Data analysis is the process of applying statistical techniques to organize, represent, describe, evaluate and interpret data. It aims at evaluating the data using analytical and logical reasoning to examine each component of the data provided.
The main purpose of the study is to find out the interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics among secondary school students. This chapter describes the details of the statistical analysis of the data collected. The collected data were analyzed statistically on the basis of the objectives of the study and the results are presented and discussed in this chapter.
Objectives of the Study
The objectives of the study are:
•	To find out whether there exist any significant difference in the Mathematical Beliefs of secondary school students for the subgroups based on 
o	Gender
o	Socio-Economic Status
•	To find out whether there exist any significant difference in Achievement Motivation of secondary school students for the subgroups based on 
o	Gender
o	Socio-Economic Status
•	To find out whether there exist any significant difference in Achievement in Mathematics of secondary school students for the subgroups based on 
o	Gender
o	Socio-Economic Status
•	To study the main effect and interaction effect of  Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics of secondary school students for the  total sample 
Hypotheses of the Study
The study was carried out to test the following hypotheses:
•	There exist significant gender differences in the mean scores of Mathematical Beliefs of secondary school students.
•	There exist significant difference in the mean scores of Mathematical Beliefs of secondary school students with respect to their Socio-Economic Status.
•	There exist significant gender difference in the mean scores of Achievement Motivation of secondary school students.
•	There exist significant difference in the mean scores of Achievement Motivation of secondary school students with respect to their Socio-Economic Status.
•	There exist significant gender difference in the mean scores of Achievement in Mathematics of secondary school students.
•	There exist significant difference in the mean scores of Achievement in Mathematics of secondary school students with respect to their socio-economic status.
•	There exist significant main effect of the independent variables namely: Mathematical Beliefs and Achievement Motivation on the dependent variable, Achievement in Mathematics of secondary school students for the total sample.
•	There exist significant interaction effect of the independent variables namely: Mathematical Beliefs and Achievement Motivation on the dependent variable,  Achievement in Mathematics of secondary school students for the total sample
Preliminary Analysis
As the first step of analysis the descriptive statistics such as mean, median, mode, standard deviation, skewness and kurtosis were worked out for the variables, Mathematical Beliefs, Achievement Motivation and Achievement in Mathematics among secondary school students for total sample to know the basic properties of the variables.
The results of descriptive statistics for the distribution of scores for Mathematical Beliefs of secondary school students for total sample are calculated and presented in Table 6.
Table 6
Descriptive Statistics of the Variable Mathematical Beliefs of Secondary School Students for Total Sample       
Variables	Category	Number	Mean	Median	Mode	SD	Skewness	Kurtosis	
Mathematical Beliefs	Total	600	26.62	27	29	3.78	-0.77	0.63	
Table 6 shows that the obtained value of mean, median and mode of the independent variable, Mathematical Beliefs of secondary school students are 26.62, 27 and 29 for the total sample. It indicates that the value of mean, median and mode coincide approximately for the total sample. The indices of skewness (sk = -0.77) show that the distribution of the scores of Mathematical Beliefs of the secondary school students is slightly negatively skewed for the total sample. The indices of kurtosis for Mathematical Beliefs reveals that the distribution of scores of Mathematical Beliefs (K=0.63) is slightly leptokurtic in nature for the total sample of the secondary students. Thus, the distribution of the scores of Mathematical Beliefs of secondary school students shows that the distribution is almost normal for the total sample. 
The graphical representation of the distribution of scores of Mathematical Beliefs for the total sample is given in Figure 3.

Mathematical Beliefs

                                       Mathematical Beliefs
Figure 3. Graphical representation of the distribution of scores of Mathematical Beliefs for the total sample

From Figure 3 it is evident that the distribution of scores of Mathematical Beliefs of secondary school students is approximately normal.
The results of descriptive statistics for the distribution of scores for Achievement Motivation of secondary school students for total sample are calculated and presented in Table 7.

Table 7
Descriptive Statistics of the Variable Achievement Motivation of Secondary School Students for Total Sample
Variables	Category	Number	Mean	Median	Mode	SD	Skewness	Kurtosis	
Achievement Motivation	Total	600	80.87	80	82	10.85	0.06	0.50	
Table 7 shows that the obtained value of the mean, median, and mode of the independent variable, Achievement Motivation of secondary school students are 80.87, 80 and 82 respectively for the total sample. It indicates that the value of mean, median and mode coincide approximately for the total sample. The indices of skewness (sk=0.06) show that the distribution of the scores of Achievement Motivation of the secondary school students is slightly positively skewed for the total sample. The indices of kurtosis for Achievement Motivation reveals that the distribution of scores of Achievement Motivation (K=0.50) is slightly leptokurtic in nature for the total sample of the secondary students. Thus, the distribution of the scores of Achievement Motivation of secondary school students shows that the distribution is almost normal for the total sample.
	The graphical representation of the distribution of scores of Achievement Motivation for the total sample is given in Figure 4.

Achievement Motivation

                                      Achievement Motivation
Figure 4. Graphical representation of the distribution of scores of Achievement Motivation for the total sample

From figure 4 it is evident that the distribution of scores of Achievement Motivation of secondary school students is approximately normal. 
The results of descriptive statistics for the distribution of scores for Achievement in Mathematics of secondary school students for total sample are calculated and presented in Table 8.

Table 8
Descriptive Statistics of the Variable Achievement in Mathematics of Secondary School Students for Total Sample
Variables	Category	Number	Mean	Median	Mode	SD	Skewness	Kurtosis	
Achievement in Mathematics	Total	600	34.04	31.00	30.00	14.26	0.82	0.55	
Table 8 shows that the obtained value of the mean, median, and mode of the dependent variable, Achievement in Mathematics of secondary school students are 34.04, 31 and 30 respectively for the total sample. It indicates that the value of mean, median and mode coincide approximately for the total sample. The indices of skewness (sk = 0.82) show that the distribution of the scores of Achievement in Mathematics of the secondary school students is slightly positively skewed for the total sample. The indices of kurtosis for Achievement in Mathematics reveals that the distribution of scores of Achievement in Mathematics (K=0.55) is slightly leptokurtic in nature for the total sample of the secondary students. Thus, the distribution of the scores of Achievement in Mathematics of secondary school students shows that the distribution is almost normal for the total sample.
The graphical representation of the distribution of scores of Achievement in Mathematics for the total sample is given in Figure 5.

Achievement in Mathematics 
Figure 5. Graphical representation of the distribution of scores of Achievement in Mathematics for the total sample.

From Figure 5 it is evident that the distribution of scores of Achievement in Mathematics of secondary school students is approximately normal.
Major Analysis
Mean Difference Analysis 
Mean difference analysis was done in order to find out whether there exist any group difference in scores of independent and dependent variables with respect to gender and socio economic status. The intention was to find out whether exist any significant difference in the mean scores of Mathematical Beliefs, Achievement Motivation, and Achievement in Mathematics for girls and boys, and for High – Average –Low Socio –Economic Status group of higher secondary school students.
	Gender difference 
The test of significance of the difference between mean scores of the independent variables, namely; Mathematical Beliefs, Achievement Motivation and Achievement in Mathematics for girls and boys, and for High – Average –Low Socio –Economic Status group of secondary school students were carried out.
	Comparison of the mean scores of Mathematical Beliefs of boys and girls of secondary schools.
The data and results of the test of significance difference between the mean scores of Mathematical Beliefs for boys and girls of secondary schools are presented in Table 9.
Table 9
Data and Results of the Test of Significance Difference between Mean Scores of Mathematical Beliefs for Boys and Girls of Secondary Schools
Gender	N	Mean	SD	t-value	Level of significance	
Boys
Girls	300
300	26.37
26.87	3.79
3.76	1.63	NS	
	
	Table 9 indicates that the mean score of Mathematical Beliefs of boys is 26.37 and the mean score of Mathematical Beliefs of girls is 26.87. The standard deviation obtained for boys is 3.79 and female students is 3.76. The t-value obtained is 1.63, which is less than the tabled value at 0.05 level (1.96). Since the t-value obtained is less than the tabled value, it can be concluded that there exists no significant difference in the mean scores of Mathematical Beliefs of boys and girls of   secondary schools.
	Discussion	
	The mean scores of Mathematical Beliefs of boys and girls secondary schools were analyzed. It was found that there is no significant difference in the mean scores of Mathematical beliefs of boys and girls of secondary schools. So it can be concluded that boys and girls of secondary schools have the same level of Mathematical Beliefs.
	Comparison of the mean scores of Achievement Motivation of boys and girls of secondary schools.
The data and results of the test of significance difference between the mean scores of Achievement Motivation in for boys and girls of secondary schools are presented in Table 10.
Table 10
Data and Results of the Test of Significance Difference between Mean Scores of Achievement Motivation for Boys and Girls of Secondary Schools
Gender	N	Mean	SD	t-value	Level of Significance	
Boys
Girls	300
300	81.67
80.07	11.42
10.20	1.82	NS	
	
Table 10 indicates that the mean score of Achievement Motivation of boys is 81.67 and the mean score of Achievement Motivation of girls is 80.07. The standard deviation obtained for boys is 11.42 and girls is 10.20. The t-value obtained is 1.82, which is less than the tabled value at 0.05 level (1.96). Since the t-value obtained is less than the tabled value, it can be concluded that there exists no significant difference in the mean scores of Achievement Motivation of boys and girls of secondary schools.
	Discussion.
	The mean scores of Achievement Motivation of boys and girls of secondary schools were analyzed. It was found that there is no significant difference in the mean scores of Achievement Motivation of boys and girls of secondary schools. So it can be concluded that boys and girls of secondary schools have the same level of Achievement Motivation in Mathematics.
	Comparison of the mean scores of Achievement in Mathematics of boys and girls of secondary schools students.
The data and results of the test of significance difference between the mean scores of Achievement in Mathematics for boys and girls of secondary schools are presented in Table 11.
Table 11
Data and Result of the Test of Significance Difference between Mean Scores of Achievement in Mathematics for Boys and Girls of Secondary Schools
Gender	N	Mean	SD	t-value	Level of significance	
Boys
Girls	300
300	33.08
35.00	13.54
14.91	1.66	NS	
					
	Table 11 indicates that the mean scores of Achievement in Mathematics of boys is 33.08 and the mean scores of Achievement in Mathematics of girls is 35.00. The standard deviation obtained for boys is 13.54 and girls is 14.91. The t-value obtained is 1.66, which is less than the tabled value at 0.05 level (1.96). Since the t-value obtained is less than the tabled value, it can be concluded that there exists no significant difference in the mean scores of Achievement in Mathematics of boys and girls of secondary schools.
	Discussion
	The mean scores of Achievement in Mathematics of boys and girls of secondary schools were analyzed. It was found that there is no significant difference in the mean scores of Achievement in Mathematics of boys and girls of secondary schools. So it can be concluded that boys and girls of secondary schools have the same level of Achievement in Mathematics.
	Difference in the mean scores of Mathematical Beliefs on the basis of Socio-Economic Status
	Data and results of the test of one way ANOVA of Mathematical Beliefs for the subgroups based on Socio-Economic Status of secondary school students are presented in the Table 12.
Table 12        
Summary of Analysis of Variance of Mathematical Beliefs for the Subgroups based on Socio-Economic Status               
	Sum of squares	df	Mean square	F	Sig	
Between groups	3.03	2	1.51	0.11	NS	
Within groups	8552.57	597	14.33			
Total	8555.60	599				

	From table 12, shows that the F value obtained for Mathematical Beliefs of secondary school students for the subgroups based on Socio- Economic Status is 0.11 for (2,599) df is less than the tabled value of F (3.01). Thus, there is no significant difference in the mean scores of Mathematical Beliefs among various Socio-Economic Status group such as High-Average-Low Socio-Economic Status groups of secondary school students.
	Discussion
	The results of one –way ANOVA for Mathematical Beliefs of secondary school students for the subgroup based on Socio-Economic Status reveals that the F value is less than the tabled value even at 0.05 level of significance. Hence, the difference in the mean scores of Mathematical Beliefs among High Socio-Economic Status group, Average Socio-Economic Status group and Low Socio-Economic Status group of secondary school students is not significant. Thus it can be conducted that the secondary school students who belongs to various Socio-Economic Status groups have same level of Mathematical Beliefs.
	Difference in the mean scores of Achievement Motivation on the basis of Socio-Economic Status
Data and results of one-way ANOVA of Achievement Motivation for the sub group based Socio-Economic Status of secondary school students are presented in the Table 13.
Table 13
Summary of Analysis of Variance of Achievement Motivation for the Subgroups based on Socio-Economic Status
	Sum of squares	df	Mean square	F	Sig	
Between groups	217.60	2	108.80	0.92	NS	
Within groups 	70312.26	597	117.78			
Total	70529.86	599				
                    
	Table 13 shows that the F value obtained for Achievement Motivation of Secondary School Students for the subgroups based on Socio-Economic Status is 0.92 which is less than the tabled value of F (3.01) for df (2,599) at 0.05 level of significance. Thus there is no significant difference in the mean scores of Achievement Motivation among various Socio-Economic Status group such as High –Average –Low Socio-Economic Status groups of secondary school students.
	Discussion
The results of one-way ANOVA of Achievement Motivation of secondary school students for the subgroup based on Socio-Economic Status reveals that the F value is less than the tabled value even at 0.05 level of significance. Hence the difference in the mean scores of Achievement Motivation among High Socio-Economic Status group, Average Socio-Economic Status group and Low Socio-Economic Status group of secondary school students is not significant. Thus it can be concluded that the secondary school students who belong to various Socio-Economic Status groups have same level of Achievement Motivation in Mathematics.
	Difference in the mean scores of Achievement in Mathematics on the basis of socio-economic status
	Data and results of one way ANOVA of Achievement in mathematics for the subgroup based on Socio-Economic Status of secondary school students are presented in the Table 14.
Table 14
Summary of Analysis of Variance of Achievement in Mathematics for the Subgroups based on Socio-Economic Status
	Sum of squares	df	Mean squares	F	Sig	
Between groups	3990.331	2	1995.17	10.11	0.01	
Within groups	117776.71	597	197.28			
Total	121767.04	599				
	Table 14 shows that the F value obtained for Achievement in Mathematics of secondary school students for the subgroups based on Socio-Economic Status is 10.11 which is greater than the tabled value of F (4.64) for df (2,599) at 0.01 level of significance. Hence the difference in Achievement in Mathematics among secondary school students for High-Average –Low Socio-Economic Status group differ significantly. 
Discussion
	The results of one-way ANOVA for Achievement in Mathematics of secondary school students for the subgroup based on Socio-Economic Status reveals that the F value is greater than the tabled value at 0.01 level of significance. Hence the difference in the mean scores of Achievement in Mathematics among High Socio-Economic Status group, Average Socio-Economic Status group and Low Socio-Economic Status group of secondary school students differ significantly and the secondary school students belongs to various groups of Socio-Economic Status (High-Average- Low) differ in their scores on Achievement in Mathematics. 
	This prompts for the Post Hoc test. In order to know which groups differ in their scores on Achievement in Mathematics, Scheffé test of Post Hoc comparison of mean scores of Achievement in Mathematics of secondary school students for subgroups based on Socio-Economic Status are presented in Table 15.

Table 15
Summary of Scheffé Test of Post Hoc comparison for High- Average- Low Socio-Economic Status Group of Secondary School Students on Achievement in Mathematics
Group	N	Subset of alpha =0.05	
		1	2	
Low   Socio Economic Status	333	32.41		
Average  Socio Economic Status	199	34.48		
High  Socio Economic Status	68		40.75	
	
	Table 15 shows that the difference in the mean scores of Achievement in Mathematics for Low Socio Economic Status group (M = 32.41) and High Socio Economic Status group (M = 40.75) of secondary school students is significant. It also shows that the difference in the mean scores of Achievement in Mathematics for Average Socio Economic Status group (M = 34.48) and High Socio Economic Status group (M = 40.75) of secondary school students is significant. Thus it can be concluded that those secondary school students who belongs to High Socio Economic Status group scores high on Achievement in Mathematics than those students who belongs to Average and Low Socio-Economic Status group. It is also evident that those secondary school students who belong to Average Socio-Economic Status group scores high on Achievement in Mathematics than those students who belong to Low Socio-Economic Status group.
Two way ANOVA with 2x3 factorial Design with 2x3 factorial Designs
The main purpose of the study is to find out the interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics of secondary school students. The main effect and interaction effect of two independent variables on the dependent variable, Achievement in Mathematics is studied for this purpose. Two-way ANOVA with 2x3 Factorial design was carried out for the total sample. The 2x3 Factorial design of Two-way ANOVA includes two levels of Mathematical Beliefs and three Levels of Achievement Motivation.  Mathematical Beliefs was classified as High Mathematical Beliefs group and Low Mathematical Beliefs group. Achievement Motivation was categorized as High Achievement Motivation group, Average Achievement Motivation group and Low Achievement Motivation group. 
	The results and discussion are given under the following headings.
•	The main effect of Mathematical Beliefs on Achievement in Mathematics.
•	The main effect of Achievement Motivation on Achievement in Mathematics.
•	The interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics.
	Main effect of Mathematical Beliefs and Achievement Motivation and their interaction effect on achievement in mathematics of secondary school students for total sample. 
 The main effect of the independent variables, viz., Mathematical Beliefs, and Achievement Motivation on the dependent variable, Achievement in Mathematics for total sample was calculated first and their interaction effect on dependent variable was also found out. The results of 2x3 Factorial design ANOVA of the main effect and interaction effects of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics for total sample are presented in Table 16.
Table 16
Summary of 2  3 Factorial Design ANOVA of Achievement in Mathematics by Mathematical Beliefs and Achievement Motivation of Secondary School Students for the Total Sample
	SS	df	MSS	F-value	Level of significance	
Mathematical Beliefs
Achievement motivation 
MB & AM                              	779.25
3177.54
297.16	1
2
2	779.25
1588.77
148.58	4.01
8.18
0.77	0.05
0.01
NS	
Error	115422.8	594	194.32			
Total	8176000.00	600				

	Main effect of Mathematical Beliefs on Achievement in Mathematics of secondary school students for total sample. 
	Table 16 shows that the F value for Mathematical Beliefs on Achievement in Mathematics for total sample is 4.01 which is greater than the tabled value 3.86 for degrees of freedom (1, 594) required for significance at .05 level. It indicates that there exists significant difference in the mean scores of Achievement in Mathematics, F(1,594)=4.01, p≤ .01, for High Mathematical Beliefs group and Low Mathematical Beliefs group of secondary school students for total sample of secondary school students. Therefore, there exists significant influence of Mathematical Beliefs on Achievement in Mathematics of higher secondary school students for total sample.
 	To know whether there exist any significant difference in the mean scores of Achievement in Mathematics among High Mathematical Beliefs group and Low Mathematical Beliefs group, the test of significance difference between means of large independent sample was carried out. The data and results of the test of significance difference between mean scores of Achievement in Mathematics for High Mathematical Beliefs group and Low Mathematical Beliefs group is given in Table 17.
Table 17
Data and Results of the Test of Significance Difference between Mean Scores of Achievement in Mathematics for High Mathematical Beliefs Group and Low Mathematical Beliefs Group
Group	N	Mean	SD	t	Level of Significance	
Low Mathematical Belief Group	311	32.14	14.05	3.41	0.01	
High Mathematical Belief Group	289	36.08	14.22			

From Table 17, it is evident that the obtained t value is 3.14, which is greater than the tabled value at 0.01 level of significance (2.58). Since the t-value obtained is greater than the tabled value, it can be concluded there exists significant difference in the mean scores of Mathematical Beliefs of secondary school students. A close observation of mean scores of two groups of Mathematical Beliefs reveals that the mean score of Achievement in Mathematics of High Epistemological Beliefs group (M=36.08, SD=14.22) compared to that of Low Mathematical Beliefs group (M=32.14, SD=14.05) is significantly greater. This indicates that those secondary school students who are having high Mathematical Beliefs, scores high on Achievement in Mathematics than those students who are having low Mathematical Beliefs among secondary school students. 
	Main effect of Achievement Motivation on Achievement in Mathematics of secondary school students for total sample.
 Table 16 shows that the F value obtained for Achievement Motivation on Achievement in Mathematics for total sample is 8.18 which is greater than the tabled 4.64 for degrees of freedom (2, 594) required for significance at 0.01 level. This indicates that the main effect due to Achievement Motivation on Achievement in Mathematics of secondary schools students for total  sample is significant, F (2, 594) =8.18, p≤ .01. Therefore, there exists significant difference in the mean scores of Achievement in Mathematics of secondary school students for the high Achievement Motivation, average Achievement Motivation and low Achievement Motivation groups for total sample. 
 	The data were further analyzed with the help of Scheffe’s Test of Post Hoc Comparison to know which group’s mean score of Achievement in Mathematics is significantly higher. The results of Scheffe’s Test of Post Hoc comparison of mean scores of Achievement in Mathematics of secondary school students for total sample among three groups of Achievement Motivation are presented in Table 18.
Table 18
Comparison of Mean Scores of Achievement in Mathematics of Secondary School Students for Total Sample among Three Groups of Achievement Motivation
Group	N	1	2	
Low Achievement Motivation group	188	29.76		
Average Achievement Motivation group	215		35.20	
High Achievement Motivation gruop	197		36.86	

Table 18 also indicates that the mean scores of Achievement in Mathematics for Average Achievement Motivation group (M = 35.20) and Low Achievement Motivation group (M = 29.76) are independent. It is also evident that the mean scores of Achievement in Mathematics for the Average Achievement Motivation group is higher than that of Low Achievement Motivation group. The table 18 also shows that the mean scores of Achievement in Mathematics for Average Achievement Motivation group (M = 35.20) and High Achievement Motivation group (M = 36.86) do not differ significantly. It means that there is no difference in Achievement in Mathematics between Average Achievement Motivation group and High Achievement Motivation group. The mean scores of Achievement in Mathematics for Low Achievement Motivation group (M = 29.76) and High Achievement Motivation group (M = 36.86) differ significantly. It is clear that the mean scores of Achievement in Mathematics for the High Achievement Motivation group is higher than that of Low Achievement Motivation group. Thus it can be concluded that High Achievement Motivation group and Average Achievement Motivation group scores high on Achievement in Mathematics than Low Achievement Motivation group among secondary school students.
	Interaction Effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics of secondary school students for total sample. 
From Table 16 it is evident that the F value obtained for interaction between Mathematical Beliefs and Achievement Motivation for total sample is 0.77 which is less than the tabled value 4.64 for degrees of freedom (2, 594) required for significance at 0.05 level. It indicates that the interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics is not significant, F (2,594) =0.77, p>.05, for the total sample even at .05 level. Therefore, the mean scores of Achievement in Mathematics at two levels of Mathematical Beliefs do not vary significantly for High- Average- Low Achievement Motivation groups of secondary school students for total sample. 
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