Bivariate Numerical Analysis, PyDA Course

When you examine how two numerical variables relate to each other, you enter the domain of bivariate analysis. This lesson covers scatter plots (the workhorse of bivariate analysis), regression lines that quantify the relationship, joint plots that combine marginal and joint distributions, and grouped plots that introduce a categorical dimension.

Key Concepts

Scatter plots

The scatter plot is the most fundamental bivariate visualization. Each point represents one observation, plotted on two numerical axes:

python

import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt

# students-performance.csv ships with the course — load it from the browser file system.
df = pd.read_csv("students-performance.csv")

fig, ax = plt.subplots(figsize=(8, 6))
ax.scatter(df["math score"], df["reading score"], alpha=0.5, edgecolors="black", linewidth=0.5)
ax.set_title("Math vs Reading Scores")
ax.set_xlabel("Math Score")
ax.set_ylabel("Reading Score")
plt.show()

Alpha transparency (alpha=0.5) is critical, it reveals point density where points overlap.

Regression plots

Seaborn’s regplot adds a regression line that quantifies the linear relationship:

python

fig, axes = plt.subplots(1, 3, figsize=(15, 5))

pairs = [
    ("math score", "reading score"),
    ("math score", "writing score"),
    ("reading score", "writing score"),
]

for i, (x, y) in enumerate(pairs):
    sns.regplot(data=df, x=x, y=y, ax=axes[i], scatter_kws={"alpha": 0.4})
    axes[i].set_title(f"{x.split()[0].title()} vs {y.split()[0].title()}")

plt.tight_layout()
plt.show()

Reading a regression plot:

Slope: positive slope means positive correlation; steeper means stronger
Confidence interval (shaded region): wider means more uncertainty
Residuals: points far from the line are poorly predicted

Joint plots

Joint plots combine the scatter plot with marginal distributions on each axis:

python

sns.jointplot(
    data=df,
    x="math score",
    y="reading score",
    kind="scatter",      # or "reg", "kde", "hist"
    height=7,
    alpha=0.4
)
plt.suptitle("Math vs Reading (Joint Plot)", y=1.02)
plt.show()

The kind parameter changes the joint plot type:

"scatter": raw scatter with marginal histograms
"reg": scatter with regression line and marginal histograms
"kde": 2D kernel density with marginal KDEs
"hist": 2D histogram with marginal histograms

Hexbin plots for density

When datasets are large, scatter plots become overplotted. Hexbin plots solve this:

python

fig, ax = plt.subplots(figsize=(8, 6))
hb = ax.hexbin(df["math score"], df["reading score"], gridsize=20, cmap="YlOrRd")
ax.set_title("Math vs Reading (Hexbin Density)")
ax.set_xlabel("Math Score")
ax.set_ylabel("Reading Score")
plt.colorbar(hb, label="Count")
plt.show()

Numerical-categorical comparisons

When one variable is categorical, compare distributions across groups:

python

fig, axes = plt.subplots(1, 3, figsize=(15, 5))

# Grouped box plots
sns.boxplot(data=df, x="gender", y="math score", ax=axes[0], palette="Set2")
axes[0].set_title("Math Scores by Gender")

# Grouped violin plots
sns.violinplot(data=df, x="lunch", y="reading score", ax=axes[1], palette="Set3")
axes[1].set_title("Reading Scores by Lunch Type")

# Grouped with multiple categories
sns.boxplot(
    data=df,
    x="test preparation course",
    y="writing score",
    hue="gender",
    ax=axes[2],
    palette="Set1"
)
axes[2].set_title("Writing Scores by Test Prep & Gender")

plt.tight_layout()
plt.show()

Swarm plots and strip plots

For smaller datasets, show individual points with jitter:

python

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Strip plot (jittered points)
sns.stripplot(data=df, x="gender", y="math score", ax=axes[0],
              alpha=0.3, jitter=True, palette="Set2")
axes[0].set_title("Math Scores — Strip Plot")

# Swarm plot (non-overlapping points — slower for large datasets)
sns.swarmplot(data=df, x="gender", y="math score", ax=axes[1],
              size=3, palette="Set2")
axes[1].set_title("Math Scores — Swarm Plot")

plt.tight_layout()
plt.show()

Identifying relationships from plots

Pattern	What it means	Plot to use
Linear trend	Variables increase together	Scatter + reg plot
Non-linear trend	Relationship changes across range	Scatter with LOWESS
Heteroscedasticity	Spread changes across range	Residual plot
Clusters	Distinct subgroups exist	Scatter with hue
Outliers	Points far from pattern	Scatter with annotations

python

# Highlighting clusters with hue
fig, ax = plt.subplots(figsize=(8, 6))
sns.scatterplot(
    data=df,
    x="math score",
    y="reading score",
    hue="gender",
    style="test preparation course",
    alpha=0.6,
    ax=ax
)
ax.set_title("Math vs Reading: Gender and Test Prep")
plt.show()

Try It

Explore the relationship between math and reading scores, grouped by gender and test preparation.

python

import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt

# students-performance.csv ships with the course — load it from the browser file system.
df = pd.read_csv("students-performance.csv")

# Scatter with regression
fig, axes = plt.subplots(1, 2, figsize=(14, 6))

sns.regplot(data=df, x="math score", y="reading score", ax=axes[0],
            scatter_kws={"alpha": 0.3}, line_kws={"color": "red"})
axes[0].set_title("Math vs Reading (Regression)")

sns.jointplot(data=df, x="math score", y="reading score",
              kind="kde", height=7)
plt.suptitle("Math vs Reading (Density)", y=1.02)
plt.show()

# Grouped comparison
fig, axes = plt.subplots(1, 2, figsize=(12, 5))
sns.boxplot(data=df, x="gender", y="math score", ax=axes[0], palette="Set2")
axes[0].set_title("Math by Gender")
sns.violinplot(data=df, x="lunch", y="math score", ax=axes[1], palette="Set3")
axes[1].set_title("Math by Lunch Type")
plt.tight_layout()
plt.show()

Key Takeaways

Scatter plots are the foundation of bivariate analysis; always use alpha transparency for overlapping points
Regression lines quantify linear relationships; the shaded region shows uncertainty
Joint plots combine scatter plots with marginal distributions for a complete picture
Hexbin plots solve overplotting for large datasets by showing density
Grouped box plots and violin plots compare numerical distributions across categorical groups
Use hue and style to add third and fourth dimensions to scatter plots

Practice Challenge

Create a figure with 4 panels showing: (1) scatter plot of math vs writing scores, (2) scatter with regression line, (3) hexbin density plot, and (4) scatter colored by lunch type. Add appropriate titles and axis labels.

🧩 Challenge, think first, then reveal

python

import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt

# students-performance.csv ships with the course — load it from the browser file system.
df = pd.read_csv("students-performance.csv")

fig, axes = plt.subplots(2, 2, figsize=(14, 12))
fig.suptitle("Math vs Writing Scores — Four Views", fontsize=14, fontweight="bold")

# Panel 1: Basic scatter
axes[0, 0].scatter(df["math score"], df["writing score"], alpha=0.4, edgecolors="black", linewidth=0.5)
axes[0, 0].set_title("Basic Scatter")
axes[0, 0].set_xlabel("Math Score")
axes[0, 0].set_ylabel("Writing Score")

# Panel 2: Regression
sns.regplot(data=df, x="math score", y="writing score", ax=axes[0, 1],
            scatter_kws={"alpha": 0.3}, line_kws={"color": "red"})
axes[0, 1].set_title("With Regression Line")

# Panel 3: Hexbin
hb = axes[1, 0].hexbin(df["math score"], df["writing score"], gridsize=20, cmap="YlOrRd")
axes[1, 0].set_title("Hexbin Density")
axes[1, 0].set_xlabel("Math Score")
axes[1, 0].set_ylabel("Writing Score")
plt.colorbar(hb, ax=axes[1, 0], label="Count")

# Panel 4: Colored by lunch
sns.scatterplot(data=df, x="math score", y="writing score", hue="lunch",
                alpha=0.5, ax=axes[1, 1], palette="Set1")
axes[1, 1].set_title("Colored by Lunch Type")

plt.tight_layout()
plt.show()

Key Concepts

Scatter plots

Regression plots

Joint plots

Hexbin plots for density

Numerical-categorical comparisons

Swarm plots and strip plots

Identifying relationships from plots

Try It

Key Takeaways

Practice Challenge

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