High School & Undergraduate Performance — Correlation Difference Across Gender Groups
Statistical Analysis using R
Introduction
In most Pakistani universities, marks achieved in Higher Secondary education are given a high weightage in merit calculation for admissions in undergraduate programs. In this analysis, I aim to identify the correlation between percentage marks achieved in Higher Secondary Certificate (intermediate or grade 11–12 exams) and cumulative GPA of final-year undergraduate students. More specifically, I am interested to find out if this correlation statistically differs among the gender groups.
Dataset
The dataset being used comprises of grade metrics for undergraduate students of a large public-sector university with diverse student-body in Lahore, Pakistan. The data was self-submitted by students but cross-verified by their respective departments.
Limitations: The data is not collectively exhaustive and students with lower performance were less likely to submit their data than those with higher performance.
Analysis
Data Cleaning
We start our analysis by loading packages and importing data.
library(tidyverse)
library(janitor)
library(cocor)
library(ggExtra)
library(cowplot)grades <- read_csv("grades.csv", col_types = "ifffdd") |>
clean_names()head(grades)## # A tibble: 6 × 6
## id gender program_duration year_of_student hsc_percentage cgpa
## <int> <fct> <fct> <fct> <dbl> <dbl>
## 1 1 Female 4 1 91.4 3.56
## 2 2 Male 4 2 91 3
## 3 3 Female 4 1 90.6 3.6
## 4 4 Female 4 3 86.5 3.37
## 5 5 Female 4 1 69.9 2.48
## 6 6 Male 4 1 89.1 2.28Let us look into basic descriptive statistics of the data.
summary(grades[,-1])## gender program_duration year_of_student hsc_percentage cgpa
## Female:8286 4:13478 1:4951 Min. :42.45 Min. :0.000
## Male :7028 5: 1836 2:4577 1st Qu.:77.00 1st Qu.:2.910
## 3:3186 Median :85.18 Median :3.330
## 4:2499 Mean :83.37 Mean :3.242
## 5: 101 3rd Qu.:91.80 3rd Qu.:3.640
## Max. :99.99 Max. :4.000The dataset includes observations of undergraduate students across different years of study. The ideal indicator for undergraduate performance should be the final GPA of graduating students but since we don’t have that data, we use CGPA for final-year students. Hence, filtering the data to keep only those students who are in their final years:
grades <- grades |>
filter(as.character(year_of_student) == as.character(program_duration))Let us now look again at the descriptive stats:
summary(grades[,-1])## gender program_duration year_of_student hsc_percentage cgpa
## Female:1296 4:2209 1: 0 Min. :42.45 Min. :0.740
## Male :1014 5: 101 2: 0 1st Qu.:74.72 1st Qu.:3.020
## 3: 0 Median :81.00 Median :3.350
## 4:2209 Mean :79.46 Mean :3.275
## 5: 101 3rd Qu.:86.00 3rd Qu.:3.598
## Max. :98.63 Max. :4.000The minimum value of 0.74 cgpa seems odd. We can get a better overview of such outliers through a scatterplot:
ggplot(grades) +
geom_point(aes(x=hsc_percentage, y=cgpa))
We can note two observations where cgpa<1 and a few more where cgpa<2. Failure to maintain CGPA of 1.7 results in drop out as per university regulations. Counting such observations:
grades |> filter(cgpa < 1.7) |> nrow()## [1] 6These 6 observations where CGPA < 1.7 are likely to be data entry error. Therefore, removing these observations:
grades <- grades |> filter(cgpa >= 1.7)nrow(grades)## [1] 2304So, our final dataframe consists of 2304 observations, on which we will perform our analysis.
Visualization
Now, that our data is cleaned, we can have a visual inspection. Firstly, we visualize the distribution of the HSC Percentage & CGPA among the two gender groups using box plots:
p1 <- grades |>
ggplot(aes(x=gender,
y=hsc_percentage,
fill=gender)) +
geom_boxplot()+
labs(title = "HSC Percentage")
p2 <- grades |>
ggplot(aes(x=gender,
y=cgpa,
fill=gender)) +
geom_boxplot()+
labs(title = "CGPA")
style <- theme_bw() +
theme(legend.position = "none",
axis.title.x = element_blank(),
axis.title.y = element_blank())
p3 <- plot_grid(p1+style ,p2+style)
print(p3)
So that is how our data looks like.
…
…
…
Okay, I admit. Box plots are overrated.
Since both variables of our interest are continuous, density plots will be more appropriate. Instead of plain density plots, we use Marginal Density Plots above a scatterplot to inspect the relationship between the two variables as well.
p4 <- grades |>
ggplot(aes(x = hsc_percentage,
y = cgpa,
color = gender)) +
geom_point(size=0.8) +
labs(x = "HSC Percentage", y = "CGPA") +
theme_bw() +
theme(legend.position = "bottom",
legend.title = element_blank())
p5 <- ggMarginal(p4, type = "density", groupColour = TRUE, groupFill = TRUE)
print(p5)
From the visualization, we can note that:
- a weak/moderate positive relationship exists between CGPA and HSC Percentage
- median values for both CGPA and HSC Percentage are higher in females than in males
- distributions are not perfectly normal but negatively skewed¹
Hypothesis Testing
The specific goal of this analysis is to identify whether the correlation between HSC Percentage and CGPA is statistically different among the two gender groups: male and female.
So, we will subset our dataframe based on the gender groups:
df_female <- subset(grades, gender == "Female")
df_male <- subset(grades, gender == "Male")and then compute the mentioned correlations separately among the both groups:
cor_female <- cor(df_female$hsc_percentage, df_female$cgpa)
cor_male <- cor(df_male$hsc_percentage, df_male$cgpa)
n_female <- nrow(df_female)
n_male <- nrow(df_male)print(cor_female)## [1] 0.2956779print(cor_male)## [1] 0.2269808Now, we want to test if cor_female is statistically different from cor_male.
- Null Hypothesis : cor_female = cor_male
- Alternative Hypothesis : cor_female ≠ cor_male
To compare these two correlations, we first need to stabilize their variance through Fisher’s Transformation². It will convert the correlations to their respective z-values, for which we can compute the test-statistic. The resulting t-statistic (or its p-value) will tell us whether Null Hypothesis will be rejected or retained. To perform these calculations, we will use cocor³ package in R. Since our computed correlations are of two independent groups, we use cocor.indep.groups() function to perform the calculations. Using it we get following results:
result <- cocor.indep.groups(cor_female, cor_male,
n_female, n_male)
print(result)##
## Results of a comparison of two correlations based on independent groups
##
## Comparison between r1.jk = 0.2957 and r2.hm = 0.227
## Difference: r1.jk - r2.hm = 0.0687
## Group sizes: n1 = 1295, n2 = 1009
## Null hypothesis: r1.jk is equal to r2.hm
## Alternative hypothesis: r1.jk is not equal to r2.hm (two-sided)
## Alpha: 0.05
##
## fisher1925: Fisher's z (1925)
## z = 1.7545, p-value = 0.0793
## Null hypothesis retained
##
## zou2007: Zou's (2007) confidence interval
## 95% confidence interval for r1.jk - r2.hm: -0.0080 0.1456
## Null hypothesis retained (Interval includes 0)Interestingly, we have obtained a p-value of 0.079 from which we can conclude:
- At 5% significance level, null hypothesis is retained that there is no statistical difference between the two correlations.
- However, at 10% significance level, null hypothesis is rejected.
Since the data in use has some limitations, a stricter control on Type-1 Error⁴ should be maintained. Hence, I prefer the smaller significance level and proceed with the conclusion obtained at 5% i.e. there is no statistical difference between the correlation of HSC Percentage and CGPA among the gender groups.
Takeaways
Correlation across gender groups
Based on the analysis, we can conclude that using marks obtained in Higher Secondary Certificate as a criteria for selection in undergraduate programs does not create an implicit gender bias in the selection process because HSC Percentage is an equally likely predictor of CGPA for both gender groups.
Weak Correlation
The Pearson’s correlation coefficient of 0.227 or 0.296 does not indicate a very strong relationship between HSC Percentage and CGPA. While a significant evidence for the implicit gender bias was not found, this weak correlation raises concern regarding high weightage of HSC marks in undergraduate admission criteria. Further research needs to be performed regarding the predictors of success in undergraduate studies.
High Variability of CGPA at low HSC Percentage
The scatterplot plotted early on showed us that variation in CGPA Percentage was not constant. Let us have a look at it again:
p6 <- grades |>
ggplot(aes(x = hsc_percentage, y = cgpa, color = gender)) +
geom_point(size = 0.9) +
geom_smooth(method = "loess", se = TRUE, fullrange = TRUE) +
labs(x = "HSC Percentage", y = "CGPA") +
theme_bw() +
theme(legend.position = "bottom", legend.title = element_blank())
print(p6)
Although there is high variability in CGPA across all values of HSC Percentage, we can note that the variability in CGPA is substantially higher when HSC Percentage is below 70%. Statistically, this hints at the presence of heteroscedasticity in data. On the other hand, it also sheds light on the conditional probability of achieving a high or low CGPA given the HSC Percentage.
For students with high HSC marks, the conditional probability of achieving a high CGPA is higher. However, for students with low HSC marks, the conditional probability of achieving a high or low CGPA is nearly equal.
As the data limitations were noted earlier, this observation could be attributed to self-selection bias, where students with low CGPA (who may have had low HSC Percentage) were less likely to submit their data. Consequently, due to missing data points, the positive correlation may no longer hold for HSC Percentages lower than 70%. Nonetheless, this phenomenon needs to be further investigated for reliable conclusions to be drawn.
Further Research
This analysis highlights following areas for further research:
- Finding appropriate predictors of undergraduate performance in context of Pakistan’s educational system.
- Replicating same research with a different and more reliable dataset. Non-rejection of null hypothesis at 10% significance level means that the possibility of an implicit gender bias cannot be confidently ruled out and thus there is need for further investigation.
- Replicating the same research using a different statistical procedure that accounts for the non-normality of the distributions
- Testing whether the high variability of CGPA at low HSC Percentage is attributable to the data limitation or does it represent an actually existing pattern.
- Analyzing differences in performance across provinces to which the students belong.
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Thanks for reading : )
- Non-normal distribution can affect the validity of results of test-statistics. However, since the data is not substantially non-normal, it does not pose a significant threat to our analysis.
- Fisher, R.A. (1925). Statistical Methods for Research Workers. Oliver and Boyd (Edinburgh).
- Diedenhofen, B. & Musch, J. (2015). cocor: A Comprehensive Solution for the Statistical Comparison of Correlations. PLOS ONE 10(4): e0121945.
- Type-1 Error: Rejecting a null hypothesis when it is true.
Data and R code for this analysis are available at Github and Kaggle.
