CGPA analysis (7 Jan)

Load & see the data

library(readxl)
df <- read_excel("C:\\Users\\Dell\\OneDrive\\Desktop\\STATISTICS\\2nd Year\\R programming\\varsity\\Project_Distribution.xlsx")
# View(df)
summary(df)

    ExamRoll           Sex                 CGPA           Group      
 Min.   :2316601   Length:80          Min.   :2.053   Min.   :1.000  
 1st Qu.:2316622   Class :character   1st Qu.:3.242   1st Qu.:1.000  
 Median :2316643   Mode  :character   Median :3.484   Median :2.000  
 Mean   :2316642                      Mean   :3.415   Mean   :2.275  
 3rd Qu.:2316662                      3rd Qu.:3.769   3rd Qu.:3.000  
 Max.   :2316682                      Max.   :3.988   Max.   :4.000  
 AssignedTeacher   
 Length:80         
 Class :character  
 Mode  :character  
                   
                   
                   

sum(is.na(df))

[1] 0

convert the classes

df$ExamRoll<-as.character(df$ExamRoll)
df$Sex <- as.factor(df$Sex)
df$Group <- as.factor(df$Group)
df$AssignedTeacher <- as.factor(df$AssignedTeacher)
class(df$AssignedTeacher)

[1] "factor"

See Data again

summary(df)

   ExamRoll         Sex         CGPA       Group  AssignedTeacher
 Length:80          F:30   Min.   :2.053   1:23   A      : 4     
 Class :character   M:50   1st Qu.:3.242   2:23   B      : 4     
 Mode  :character          Median :3.484   3:23   C      : 4     
                           Mean   :3.415   4:11   D      : 4     
                           3rd Qu.:3.769          E      : 4     
                           Max.   :3.988          FT     : 4     
                                                  (Other):56     

Descriptive Statistics For CGPA (i) Mean (ii) Median (iii) Standard deviation

library(psych)
describe(df$CGPA)

describe(df$CGPA)$mean

[1] 3.415225

Frequency Distributions (i) Sex (ii) Group (iii) AssignedTeacher

table(df$Sex)

 F  M 
30 50 

table(df$Group)

 1  2  3  4 
23 23 23 11 

table(df$AssignedTeacher)

 A  B  C  D  E FT  G  H  I  J  K  L MT  N  O  P  Q  R  S  T  U  V  W 
 4  4  4  4  4  4  4  4  4  4  4  3  3  3  3  3  3  3  3  3  3  3  3 

barplot(table(df$Sex))

barplot(table(df$Group))

barplot(table(df$AssignedTeacher))

Group Comparisons (i) CGPA by Sex (mean, SD, count) (ii) CGPA by Group (mean, SD, count)

library('dplyr')
# CGPA by Sex

result_sx <- df %>%
  group_by(Sex) %>%
  summarise(
    Count = n(),
    Mean_CGPA = round(mean(CGPA, na.rm = TRUE), 2),
    SD_CGPA = round(sd(CGPA, na.rm = TRUE), 2),
    .groups = 'drop'
  )
result_sx

# CGPA by Group
result_grp <- df %>%
  group_by(Group) %>%
  summarise(
    Count = n(),
    Mean_CGPA = round(mean(CGPA, na.rm = TRUE), 2),
    SD_CGPA = round(sd(CGPA, na.rm = TRUE), 2),
    .groups = 'drop'
  )
result_grp

NA

Data Visualization

Distribution Plots (i) Histogram of CGPA (ii) Density plot of CGPA

library(ggplot2)

# Histogram of CGPA
ggplot(df, aes(x=CGPA)) +
  geom_histogram(bins=30, fill='Steelblue', color = 'white') +
  labs(title = "Histogram of CGPA", x="CGPA", y="Count") +
  theme_minimal()

# Density plot of CGPA
ggplot(df, aes(x=CGPA)) +
  geom_density(fill='Steelblue') +
  labs(title = "Density plot of CGPA", x="CGPA", y="Density") +
  theme_minimal()

Categorical Data Plots: (i) Bar plot of Sex (ii) Bar plot of Group (iii) Bar plot of AssignedTeacher

# Barplot of Sex
ggplot(df, aes(x=Sex, fill = Sex)) +
  geom_bar() +
  labs(title = 'Bar plot of Sex', x= "Sex", y="Count")

# Barplot of Group
ggplot(df, aes(x=Group, fill = Group)) +
  geom_bar() +
  labs(title = 'Bar plot of Group', x= "Group", y="Count")

# Barplot of AssignedTeacher
ggplot(df, aes(x=AssignedTeacher, fill = AssignedTeacher)) +
  geom_bar() +
  labs(title = 'Bar plot of AssignedTeacher', x= "Teacher", y="Count")

Comparative Plots

1. Boxplot of CGPA by Sex
2. Boxplot of CGPA by Group

# Boxplot of CGPA by Sex
ggplot(df, aes(y= CGPA, x= Sex, fill= Sex)) +
  geom_boxplot(outliers= TRUE, outlier.color = "Red") +
  stat_summary(fun = mean,
               fill = "yellow",
               geom = "point",
               shape = 23,
               size = 3)

# Boxplot of CGPA by Group
ggplot(df, aes(y= CGPA, x= Group, fill= Group)) +
  geom_boxplot(outliers= TRUE, outlier.color = "Red") +
  stat_summary(fun = mean,
               geom = "point",
               shape = 23,
               size = 3,
               fill = "yellow")

Inferential Statistics

Mean Comparisons

# t-test: CGPA by Sex
t.test(df$CGPA ~ df$Sex)

    Welch Two Sample t-test

data:  df$CGPA by df$Sex
t = 1.7854, df = 70.874, p-value = 0.07848
alternative hypothesis: true difference in means between group F and group M is not equal to 0
95 percent confidence interval:
 -0.01995516  0.36152849
sample estimates:
mean in group F mean in group M 
       3.521967        3.351180 

# ANOVA: CGPA by Group
anova(lm(CGPA ~ Group, data = df))

Analysis of Variance Table

Response: CGPA
          Df  Sum Sq Mean Sq F value    Pr(>F)    
Group      3 13.6697  4.5566  203.13 < 2.2e-16 ***
Residuals 76  1.7048  0.0224                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# ANOVA: CGPA by AssignedTeacher
anova(lm(CGPA ~ AssignedTeacher, data = df))

Analysis of Variance Table

Response: CGPA
                Df Sum Sq Mean Sq F value Pr(>F)
AssignedTeacher 22  1.780 0.08091  0.3392 0.9966
Residuals       57 13.595 0.23850               

Association Tests

# Chi-square test: Sex vs Group
chisq.test(table(df$Sex, df$Group))

    Pearson's Chi-squared test

data:  table(df$Sex, df$Group)
X-squared = 4.0095, df = 3, p-value = 0.2604

# Chi-square test: Sex vs AssignedTeacher
chisq.test(table(df$Sex, df$AssignedTeacher))

    Pearson's Chi-squared test

data:  table(df$Sex, df$AssignedTeacher)
X-squared = 25.244, df = 22, p-value = 0.2855

Regression Analysis Simple Linear Regression Model 1: CGPA ~ Sex Model 2: CGPA ~ Group Multiple Regression Full model: CGPA ~ Sex + Group + AssignedTeacher

model1 <- lm(df$CGPA ~ df$Sex)
model2 <- lm(df$CGPA ~ df$Group)
full_model <- lm(df$CGPA ~ df$Sex + df$Group + df$AssignedTeacher)
library(performance)
check_model(model1)

check_model(model2)

check_model(full_model)

summary(model1)

Call:
lm(formula = df$CGPA ~ df$Sex)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.29818 -0.22513  0.09743  0.28478  0.60182 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   3.5220     0.0796  44.244   <2e-16 ***
df$SexM      -0.1708     0.1007  -1.696   0.0938 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.436 on 78 degrees of freedom
Multiple R-squared:  0.03557,   Adjusted R-squared:  0.02321 
F-statistic: 2.877 on 1 and 78 DF,  p-value: 0.09384

summary(model2)

Call:
lm(formula = df$CGPA ~ df$Group)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.48527 -0.07246  0.00333  0.10026  0.44173 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.83774    0.03123 122.887  < 2e-16 ***
df$Group2   -0.26913    0.04417  -6.094 4.23e-08 ***
df$Group3   -0.57900    0.04417 -13.110  < 2e-16 ***
df$Group4   -1.29947    0.05491 -23.668  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1498 on 76 degrees of freedom
Multiple R-squared:  0.8891,    Adjusted R-squared:  0.8847 
F-statistic: 203.1 on 3 and 76 DF,  p-value: < 2.2e-16

summary(full_model)

Call:
lm(formula = df$CGPA ~ df$Sex + df$Group + df$AssignedTeacher)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.37629 -0.07189  0.01049  0.06907  0.29031 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)           3.697292   0.079024  46.787  < 2e-16 ***
df$SexM               0.053618   0.042178   1.271   0.2092    
df$Group2            -0.278455   0.043500  -6.401 4.14e-08 ***
df$Group3            -0.588325   0.043500 -13.525  < 2e-16 ***
df$Group4            -1.321622   0.058344 -22.652  < 2e-16 ***
df$AssignedTeacherB   0.225250   0.102817   2.191   0.0329 *  
df$AssignedTeacherC   0.183846   0.103356   1.779   0.0810 .  
df$AssignedTeacherD   0.066154   0.103356   0.640   0.5249    
df$AssignedTeacherE   0.093191   0.104957   0.888   0.3786    
df$AssignedTeacherFT  0.033096   0.103356   0.320   0.7501    
df$AssignedTeacherG   0.041346   0.103356   0.400   0.6907    
df$AssignedTeacherH   0.130846   0.103356   1.266   0.2111    
df$AssignedTeacherI   0.007691   0.104957   0.073   0.9419    
df$AssignedTeacherJ   0.260404   0.103356   2.519   0.0148 *  
df$AssignedTeacherK   0.264250   0.102817   2.570   0.0130 *  
df$AssignedTeacherL   0.005017   0.114247   0.044   0.9651    
df$AssignedTeacherMT  0.202096   0.111863   1.807   0.0765 .  
df$AssignedTeacherN  -0.007777   0.112184  -0.069   0.9450    
df$AssignedTeacherO   0.175557   0.112184   1.565   0.1236    
df$AssignedTeacherP   0.143302   0.113299   1.265   0.2115    
df$AssignedTeacherQ   0.021557   0.112184   0.192   0.8484    
df$AssignedTeacherR   0.193096   0.111863   1.726   0.0901 .  
df$AssignedTeacherS   0.087351   0.114247   0.765   0.4479    
df$AssignedTeacherT   0.113557   0.112184   1.012   0.3160    
df$AssignedTeacherU   0.023890   0.112184   0.213   0.8322    
df$AssignedTeacherV   0.165351   0.114247   1.447   0.1537    
df$AssignedTeacherW   0.211429   0.111863   1.890   0.0642 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1454 on 53 degrees of freedom
Multiple R-squared:  0.9271,    Adjusted R-squared:  0.8914 
F-statistic: 25.93 on 26 and 53 DF,  p-value: < 2.2e-16

Diagnostic Checks & Assumptions

Outlier Detection

1. Detect outliers in CGPA using the IQR rule
2. Visualize outliers using boxplots

# Detect outliers using IQR rule
Q1 <- quantile(df$CGPA, 0.25)
Q3 <- quantile(df$CGPA, 0.75)
IQR_val <- Q3 - Q1
lower <- Q1 - 1.5 * IQR_val
upper <- Q3 + 1.5 * IQR_val

outliers <- df[df$CGPA < lower | df$CGPA > upper, ]
outliers

boxplot(df$CGPA, col="steelblue", main = "CGPA boxplot")

Regression Diagnostics

plot(full_model)

shapiro.test(residuals(full_model))

    Shapiro-Wilk normality test

data:  residuals(full_model)
W = 0.97797, p-value = 0.1824

library(car)
vif(full_model)

                       GVIF Df GVIF^(1/(2*Df))
df$Sex             1.577656  1        1.256048
df$Group           1.241791  3        1.036752
df$AssignedTeacher 1.723375 22        1.012447