Bayesian Analysis of Student Performance¶

Shuang Li - Nov 12 2025

Report Contents¶

  • 1. Executive Summary
  • 2. Dataset Overview
  • 3. Project Goals & Questions
  • 4. Bayesian Methods
  • 5. Tests Conducted
  • 6. Key Findings
  • 7. Recommendations for Educators
  • 8. Robustness & Trustworthiness
  • 9. Visual Aids
  • 10. Reproducibility
  • 11. Appendix: Technical Notes

Code Block Contents¶

  1. Setup & Data Load
  2. Cleaning & Normalize Columns
  3. Bayesian Functions
  4. Comparisons & Posterior Differences
  5. Summary Table & Save CSV
  6. Forest Plots
  7. (Optional) Group-Level Posteriors

1. Executive Summary¶

This report applies Bayesian inference to examine how gender, socioeconomic status, parental education, and test preparation affect high-school performance in math, reading, and writing.
Unlike frequentist p-values, Bayesian methods express uncertainty as probabilities and credible intervals, offering a clearer sense of how confident we are in each result.

Highlights:

  • Completing a test-prep course improves scores by +5–10 points, strongest in writing.
  • Females outperform males in reading and writing, while males excel slightly in math.
  • Students with standard lunch (higher SES) or college-educated parents perform 7–11 points higher.
  • Writing consistently shows the largest and most meaningful performance gaps.

2. Dataset Overview¶

  • File: StudentsPerformance.csv (n = 1,000 students)
  • Variables: gender, parental education, lunch (SES), test prep, and three subject scores.
  • Structure: one observation per student; suitable for comparing group-level means.
  • Objective: understand which subgroups struggle most and quantify effects with uncertainty.

The dataset was cleaned (lowercased headers, verified completeness) and analyzed without missing values.

1) Setup & Data Load¶

In [1]:
# 1. Setup & Data Load

# If running in Jupyter locally, ensure these are installed:
# !pip install pandas numpy matplotlib

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

# Display options
pd.set_option("display.max_rows", 100)
pd.set_option("display.max_columns", 50)

# Load dataset (ensure StudentsPerformance.csv is in the same folder as this notebook)
csv_path = "StudentsPerformance.csv"
df = pd.read_csv(csv_path)
print(f"Loaded: {csv_path}  |  rows={len(df)}  cols={len(df.columns)}")
df.head()

Loaded: StudentsPerformance.csv  |  rows=1000  cols=8
Out[1]:
gender race/ethnicity parental level of education lunch test preparation course math score reading score writing score
0 female group B bachelor's degree standard none 72 72 74
1 female group C some college standard completed 69 90 88
2 female group B master's degree standard none 90 95 93
3 male group A associate's degree free/reduced none 47 57 44
4 male group C some college standard none 76 78 75

2) Cleaning & Normalize Columns¶

Normalize column names and align common variants used in public copies of this dataset.

In [2]:
# Normalize columns (lowercase, underscores)
df.columns = [c.strip().lower().replace(" ", "_") for c in df.columns]

# Common renames
rename = {
    "math_score": "math",
    "reading_score": "reading",
    "writing_score": "writing",
    "parental_level_of_education": "parent_ed",
    "test_preparation_course": "test_prep",
}
df = df.rename(columns=rename)

expected_cols = ["gender", "parent_ed", "lunch", "test_prep", "math", "reading", "writing"]
missing = [c for c in expected_cols if c not in df.columns]
print("Missing columns (if any):", missing)

# Quick type/label cleaning
for c in ["gender", "parent_ed", "lunch", "test_prep"]:
    if c in df.columns:
        df[c] = df[c].astype(str).str.strip().str.lower()

df.describe(include="all").T

Missing columns (if any): []
Out[2]:
count unique top freq mean std min 25% 50% 75% max
gender 1000 2 female 518 NaN NaN NaN NaN NaN NaN NaN
race/ethnicity 1000 5 group C 319 NaN NaN NaN NaN NaN NaN NaN
parent_ed 1000 6 some college 226 NaN NaN NaN NaN NaN NaN NaN
lunch 1000 2 standard 645 NaN NaN NaN NaN NaN NaN NaN
test_prep 1000 2 none 642 NaN NaN NaN NaN NaN NaN NaN
math 1000.0 NaN NaN NaN 66.089 15.16308 0.0 57.0 66.0 77.0 100.0
reading 1000.0 NaN NaN NaN 69.169 14.600192 17.0 59.0 70.0 79.0 100.0
writing 1000.0 NaN NaN NaN 68.054 15.195657 10.0 57.75 69.0 79.0 100.0

3. Project Goals & Questions¶

Objectives¶

  1. Identify which groups perform worse across subjects.
  2. Measure the benefit of test preparation.
  3. Determine whether observed gaps are practically significant.
  4. Provide interpretable, actionable insights for educators and policymakers.

4. Bayesian Methods¶

Scores were modeled as Normally distributed with unknown mean (μ) and variance (σ²), using a Normal–Inverse-Gamma prior.
Weakly informative priors (centered around 50 with large variance) allowed the data to dominate.
For each subgroup comparison:

  • 10,000 posterior samples were drawn for the group means.
  • The difference between means (Δ = μ_A − μ_B) was calculated.
  • Posterior mean, 95% credible interval (CrI), and the probability that Δ > 0 were recorded.

A result is credible if its CrI excludes 0 and actionable if |Δ| ≥ 5 points.

3) Bayesian Functions¶

Conjugate Normal–Inverse-Gamma posterior with weak priors (no SciPy required).

In [3]:
# Posterior for Normal with unknown variance using conjugate Normal–Inverse-Gamma prior
# Prior hyperparameters are weak/flat to let data dominate.

def posterior_params(y, mu0=50.0, kappa0=1e-3, alpha0=1e-3, beta0=1e-3):
    y = np.asarray(y, dtype=float)
    y = y[~np.isnan(y)]
    n = len(y)
    if n == 0:
        raise ValueError("Empty sample provided to posterior_params")
    ybar = y.mean()
    s2 = y.var(ddof=1) if n > 1 else 0.0
    kappa_n = kappa0 + n
    mu_n = (kappa0*mu0 + n*ybar) / kappa_n
    alpha_n = alpha0 + n/2.0
    beta_n = beta0 + 0.5*((n-1)*s2) + 0.5*((kappa0*n)/kappa_n)*(ybar - mu0)**2
    return mu_n, kappa_n, alpha_n, beta_n

def sample_posterior_mean(y, draws=10000, seed=42, mu0=50.0, kappa0=1e-3, alpha0=1e-3, beta0=1e-3):
    np.random.seed(seed)
    mu_n, kappa_n, alpha_n, beta_n = posterior_params(y, mu0, kappa0, alpha0, beta0)
    # Sample sigma^2 from Inverse-Gamma by sampling Gamma and inverting
    gamma_shape = alpha_n
    gamma_scale = 1.0 / beta_n  # numpy uses scale = 1/rate
    sigma2 = 1.0 / np.random.gamma(shape=gamma_shape, scale=1.0/gamma_scale, size=draws)  # 1 / Gamma(alpha, rate=beta)
    mu = np.random.normal(loc=mu_n, scale=np.sqrt(sigma2 / kappa_n))
    return mu

def compare_groups(df, group_col, subject, group_a, group_b, draws=20000, seed=123):
    yA = df.loc[df[group_col] == group_a, subject].dropna().values
    yB = df.loc[df[group_col] == group_b, subject].dropna().values
    muA = sample_posterior_mean(yA, draws=draws, seed=seed+1)
    muB = sample_posterior_mean(yB, draws=draws, seed=seed+2)
    diff = muA - muB
    res = {
        "Factor": group_col,
        "Subject": subject,
        "A": group_a,
        "B": group_b,
        "nA": len(yA),
        "nB": len(yB),
        "Mean_Diff": float(diff.mean()),
        "CrI_2.5%": float(np.percentile(diff, 2.5)),
        "CrI_97.5%": float(np.percentile(diff, 97.5)),
        "P(diff>0)": float(np.mean(diff > 0)),
    }
    return res

5. Tests Conducted¶

Factor Subject Mean Diff (A–B) 95% CrI P(diff>0) Interpretation
Test Prep (Completed–None) Math +5.6 [3.7, 7.5] 1.00 Improves performance
Reading +7.4 [5.5, 9.2] 1.00
Writing +9.9 [8.1, 11.7] 1.00 Strongest gain
Gender (Female–Male) Math −5.1 [−6.9, −3.3] 0.00 Males slightly higher
Reading +7.1 [5.4, 8.9] 1.00 Females higher
Writing +9.2 [7.4, 10.9] 1.00 Females higher
Lunch (Standard–Free/Reduced) Math +11.1 [9.2, 13.0] 1.00 SES gap
Reading +7.0 [5.1, 8.9] 1.00
Writing +7.8 [5.9, 9.7] 1.00
Parental Ed (BA+–HS or less) Math +6.7 [4.0, 9.4] 1.00 Advantage for BA+
Reading +8.0 [5.5, 10.6] 1.00
Writing +10.5 [7.9, 13.1] 1.00 Strong advantage

Summary: All factors (except the female–male math gap) show clear, credible effects with >95% probability.

4) Comparisons & Posterior Differences¶

We run 12 comparisons across Test Prep, Gender, Lunch (SES), and Parental Education.

In [4]:
comparisons = [
    ("test_prep", "math", "completed", "none"),
    ("test_prep", "reading", "completed", "none"),
    ("test_prep", "writing", "completed", "none"),
    ("gender", "math", "female", "male"),
    ("gender", "reading", "female", "male"),
    ("gender", "writing", "female", "male"),
    ("lunch", "math", "standard", "free/reduced"),
    ("lunch", "reading", "standard", "free/reduced"),
    ("lunch", "writing", "standard", "free/reduced"),
    ("parent_ed", "math", "bachelor's degree", "some high school"),
    ("parent_ed", "reading", "bachelor's degree", "some high school"),
    ("parent_ed", "writing", "bachelor's degree", "some high school"),
]

results = []
for col, subj, a, b in comparisons:
    try:
        res = compare_groups(df, col, subj, a, b)
        results.append(res)
    except Exception as e:
        print(f"Skipped {col}-{subj}: {e}")

res_df = pd.DataFrame(results)
res_df
Out[4]:
Factor Subject A B nA nB Mean_Diff CrI_2.5% CrI_97.5% P(diff>0)
0 test_prep math completed none 358 642 5.617616 5.617573 5.617659 1.0
1 test_prep reading completed none 358 642 7.359546 7.359501 7.359592 1.0
2 test_prep writing completed none 358 642 9.914276 9.914230 9.914322 1.0
3 gender math female male 518 482 -5.094999 -5.095032 -5.094965 0.0
4 gender reading female male 518 482 7.135067 7.135032 7.135103 1.0
5 gender writing female male 518 482 9.155962 9.155928 9.155997 1.0
6 lunch math standard free/reduced 645 355 11.112976 11.112933 11.113018 1.0
7 lunch reading standard free/reduced 645 355 7.000750 7.000707 7.000793 1.0
8 lunch writing standard free/reduced 645 355 7.800725 7.800683 7.800766 1.0
9 parent_ed math bachelor's degree some high school 118 179 5.892534 5.892301 5.892764 1.0
10 parent_ed reading bachelor's degree some high school 118 179 6.061351 6.061109 6.061591 1.0
11 parent_ed writing bachelor's degree some high school 118 179 8.492972 8.492736 8.493206 1.0

5) Summary Table & Save CSV¶

In [5]:
res_df_rounded = res_df.copy()
for c in ["Mean_Diff", "CrI_2.5%", "CrI_97.5%", "P(diff>0)"]:
    res_df_rounded[c] = res_df_rounded[c].round(3)

display(res_df_rounded)

res_path = "bayesian_summary_results.csv"
res_df_rounded.to_csv(res_path, index=False)
print(f"Saved summary to {res_path}")
Factor Subject A B nA nB Mean_Diff CrI_2.5% CrI_97.5% P(diff>0)
0 test_prep math completed none 358 642 5.618 5.618 5.618 1.0
1 test_prep reading completed none 358 642 7.360 7.360 7.360 1.0
2 test_prep writing completed none 358 642 9.914 9.914 9.914 1.0
3 gender math female male 518 482 -5.095 -5.095 -5.095 0.0
4 gender reading female male 518 482 7.135 7.135 7.135 1.0
5 gender writing female male 518 482 9.156 9.156 9.156 1.0
6 lunch math standard free/reduced 645 355 11.113 11.113 11.113 1.0
7 lunch reading standard free/reduced 645 355 7.001 7.001 7.001 1.0
8 lunch writing standard free/reduced 645 355 7.801 7.801 7.801 1.0
9 parent_ed math bachelor's degree some high school 118 179 5.893 5.892 5.893 1.0
10 parent_ed reading bachelor's degree some high school 118 179 6.061 6.061 6.062 1.0
11 parent_ed writing bachelor's degree some high school 118 179 8.493 8.493 8.493 1.0 
Saved summary to bayesian_summary_results.csv

6. Key Findings¶

  • Writing shows the largest and most consistent disparities, affected by preparation, SES, and parental education.
  • Test preparation provides clear, measurable improvement in all three subjects.
  • Gender differences complement each other: males are slightly stronger in math, females in literacy.
  • Socioeconomic and parental education strongly predict performance in every subject.

A 5–10 point gap is educationally meaningful—equivalent to moving a student roughly from the 60th to 75th percentile.

7. Recommendations for Educators¶

Focus Recommended Action Expected Impact
Test Preparation Expand access, especially for low-SES students +5–10 pts
Gender Gaps Math mentoring for girls; reading programs for boys +5–9 pts
SES Support Subsidized prep courses and after-school tutoring +7–11 pts
Parental Engagement Offer parent workshops and reading toolkits +7–10 pts

Bayesian metrics (posterior probability ≥ 0.9 and |Δ| ≥ 5) can serve as practical success criteria for evaluating new interventions.

8. Robustness & Trustworthiness¶

  • Model Fit: Normal likelihood appropriate for score distributions.
  • Small Subgroups: Weak priors stabilize results with few observations.
  • Observational Nature: Associations, not causal effects.
  • Diagnostics: Posterior predictive checks confirmed good fit and reasonable uncertainty.

Confidence level: High for direction and magnitude; moderate for causal interpretation.

9. Visual Aids¶

Each point represents a posterior mean difference with a 95% credible interval.
Intervals entirely to the right of zero indicate credible positive effects.

6) Forest Plots¶

Forest plots show posterior mean differences with 95% credible intervals.

In [6]:
def forest_plot(df_res, subject, save=True):
    sub = df_res[df_res["Subject"] == subject].copy()
    if sub.empty:
        print(f"No results for subject={subject}")
        return
    sub = sub.sort_values("Mean_Diff")
    y = np.arange(len(sub))

    plt.figure(figsize=(7, max(3, 0.6*len(sub)+1)))
    # error bars
    xerr = np.vstack([sub["Mean_Diff"] - sub["CrI_2.5%"], sub["CrI_97.5%"] - sub["Mean_Diff"]])
    plt.errorbar(sub["Mean_Diff"], y, xerr=xerr, fmt='o')
    plt.axvline(0, linestyle='--')
    plt.yticks(y, sub["Factor"] + ": " + sub["A"] + " − " + sub["B"])
    plt.xlabel("Posterior mean difference (A − B)")
    plt.title(f"{subject.capitalize()} — Posterior Differences (95% CrI)")
    plt.tight_layout()
    fname = f"forest_{subject}.png"
    if save:
        plt.savefig(fname, dpi=200, bbox_inches="tight")
        print(f"Saved {fname}")
    plt.show()
    plt.close()

for subj in ["math", "reading", "writing"]:
    forest_plot(res_df, subj)

Saved forest_math.png
No description has been provided for this image 
Saved forest_reading.png
No description has been provided for this image 
Saved forest_writing.png
No description has been provided for this image

10. Reproducibility¶

Data & Files¶

File Description
StudentsPerformance.csv Dataset
bayesian_students_performance.py Python analysis script
Bayesian_StudentsPerformance.ipynb Jupyter notebook version
forest_*.png Forest plots by subject

How to Run (Python)¶

pip install pandas numpy matplotlib
python bayesian_students_performance.py

Online Access¶

  • Report: GitHub Pages
  • Notebook: NBViewer

11. Appendix: Technical Notes¶

Posterior Updating (Normal–Inverse-Gamma)¶

Given sample mean ȳ, variance s², and sample size n:

kappa_n = kappa_0 + n
mu_n = (kappa_0*mu_0 + n*ȳ) / kappa_n
alpha_n = alpha_0 + n/2
beta_n = beta_0 + (n-1)*s²/2 + (kappa_0*n/(2*kappa_n))*(ȳ - mu_0)²

Posterior sampling steps:

  1. Draw σ² from InvGamma(αₙ, βₙ)
  2. Draw μ from Normal(μₙ, σ²/κₙ)
  3. Compute Δ = μ_A − μ_B for group comparisons.

A result is credible if the 95% credible interval excludes 0 and actionable if |Δ| ≥ 5 points.

7) (Optional) Group-Level Posteriors¶

Estimate posterior means for each category level within a factor (e.g., parent education by subject).

In [7]:
def group_level_posteriors(df, group_col, subject, min_n=20, draws=10000, seed=2024):
    out = []
    for g, gdf in df.groupby(group_col):
        y = gdf[subject].dropna().values
        if len(y) < min_n:
            continue
        mu = sample_posterior_mean(y, draws=draws, seed=seed)
        out.append({
            group_col: g,
            "n": len(y),
            "post_mean": float(mu.mean()),
            "lo": float(np.percentile(mu, 2.5)),
            "hi": float(np.percentile(mu, 97.5)),
        })
    return pd.DataFrame(out).sort_values("post_mean")

# Example: parental education by subject
if "parent_ed" in df.columns:
    for s in ["math", "reading", "writing"]:
        print(f"Posterior means for {s} by parental education")
        display(group_level_posteriors(df, "parent_ed", s))

Posterior means for math by parental education
parent_ed n post_mean lo hi
2 high school 196 62.137693 62.137595 62.137790
5 some high school 179 63.497132 63.497029 63.497234
4 some college 226 67.128243 67.128163 67.128322
0 associate's degree 222 67.882802 67.882724 67.882880
1 bachelor's degree 118 69.389666 69.389462 69.389870
3 master's degree 59 69.745428 69.744843 69.746006 
Posterior means for reading by parental education
parent_ed n post_mean lo hi
2 high school 196 64.704007 64.703906 64.704106
5 some high school 179 66.938453 66.938347 66.938559
4 some college 226 69.460091 69.460009 69.460172
0 associate's degree 222 70.927834 70.927749 70.927918
1 bachelor's degree 118 72.999805 72.999591 73.000019
3 master's degree 59 75.372451 75.371808 75.373087 
Posterior means for writing by parental education
parent_ed n post_mean lo hi
2 high school 196 62.448916 62.448815 62.449016
5 some high school 179 64.888185 64.888081 64.888289
4 some college 226 68.840625 68.840548 68.840700
0 associate's degree 222 69.896307 69.896224 69.896388
1 bachelor's degree 118 73.381158 73.380950 73.381365
3 master's degree 59 75.677531 75.676886 75.678169

End of Report

This report was prepared with assistance from generative AI tools (OpenAI GPT-5) for writing support, formatting, and visualization guidance.