Univariate Numerical Analysis, PyDA Course

Univariate numerical analysis examines one numerical variable at a time. The goal is to understand its distribution: where values cluster, how spread they are, whether the distribution is symmetric or skewed, and whether outliers exist. This lesson covers the core plot types and summary statistics for numerical data.

Key Concepts

Summary statistics

Before plotting, compute the numbers that describe the distribution:

python

import pandas as pd
import numpy as np

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

scores = df["math score"]

# Central tendency
print(f"Mean:   {scores.mean():.2f}")
print(f"Median: {scores.median():.2f}")

# Spread
print(f"Std:    {scores.std():.2f}")
print(f"IQR:    {scores.quantile(0.75) - scores.quantile(0.25):.2f}")
print(f"Range:  {scores.max() - scores.min()}")

# Shape
print(f"Skewness:  {scores.skew():.2f}")
print(f"Kurtosis:  {scores.kurtosis():.2f}")

Interpretation:

Skewness > 0: right-skewed tail (e.g., most scores low, a few very high)
Skewness < 0: left-skewed tail (e.g., most scores high, a few very low)
Kurtosis > 0: heavy tails (more outliers than normal)
Kurtosis < 0: light tails (fewer outliers than normal)

Histograms

The histogram is the foundation of univariate numerical analysis. It shows the frequency distribution:

python

import matplotlib.pyplot as plt
import seaborn as sns

fig, axes = plt.subplots(1, 2, figsize=(12, 5))

# Basic histogram
axes[0].hist(scores, bins=20, edgecolor="black", alpha=0.7)
axes[0].set_title("Math Score Distribution (histogram)")
axes[0].set_xlabel("Math Score")
axes[0].set_ylabel("Frequency")

# Histogram with KDE overlay
sns.histplot(scores, kde=True, bins=20, ax=axes[1], color="steelblue")
axes[1].set_title("Math Score Distribution (histogram + KDE)")

plt.tight_layout()
plt.show()

Key decisions:

Number of bins: Too few hides detail, too many creates noise. bins=20 is a reasonable default for datasets under 10,000 rows. Use bins="auto" for automatic selection.
Edge color: edgecolor="black" makes bin boundaries visible.

KDE plots (Kernel Density Estimation)

KDE plots smooth the histogram into a continuous curve, making it easier to compare distributions and identify modality:

python

fig, ax = plt.subplots(figsize=(8, 5))

# Single KDE
sns.kdeplot(scores, fill=True, alpha=0.5, ax=ax)
ax.set_title("Math Score KDE")
ax.set_xlabel("Math Score")
plt.show()

# Compare distributions
fig, ax = plt.subplots(figsize=(8, 5))
for subject in ["math score", "reading score", "writing score"]:
    sns.kdeplot(df[subject], fill=True, alpha=0.3, label=subject, ax=ax)
ax.set_title("Score Distributions by Subject")
ax.legend()
plt.show()

Box plots

Box plots show the five-number summary (min, Q1, median, Q3, max) and highlight outliers:

python

fig, ax = plt.subplots(figsize=(8, 5))

sns.boxplot(x=scores, ax=ax, color="lightblue", flierprops=dict(marker="o", markersize=5))
ax.set_title("Math Score Box Plot")
ax.set_xlabel("Math Score")
plt.show()

Reading a box plot:

Box: Interquartile range (IQR), the middle 50% of data
Line inside box: Median
Whiskers: 1.5 × IQR from Q1 and Q3
Points beyond whiskers: Outliers (typically > 1.5 × IQR)

Violin plots

Violin plots combine the box plot with the KDE, showing both summary statistics and the full distribution shape:

python

fig, ax = plt.subplots(figsize=(8, 5))

sns.violinplot(x=scores, ax=ax, inner="quartile", color="lightgreen")
ax.set_title("Math Score Violin Plot")
ax.set_xlabel("Math Score")
plt.show()

The inner parameter controls what is drawn inside the violin:

"quartile": shows Q1, median, Q3 lines
"box": shows a miniature box plot
"stick": shows all data points as ticks

Choosing the right plot

Plot	Best for	Shows
Histogram	Frequency distribution, shape	Bins and counts
KDE	Smooth distribution, comparing groups	Continuous density curve
Box plot	Summary statistics, outliers	Five-number summary
Violin plot	Full distribution + summary	KDE + box plot combined

Comparing distributions across groups

python

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

for i, subject in enumerate(["math score", "reading score", "writing score"]):
    sns.boxplot(data=df, x="gender", y=subject, ax=axes[i])
    axes[i].set_title(subject.replace(" score", " Scores").title())

plt.tight_layout()
plt.show()

Try It

Analyze the reading score distribution from the Students Performance dataset.

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")

reading = df["reading score"]

# Summary statistics
print("Summary Statistics:")
print(f"  Mean:   {reading.mean():.2f}")
print(f"  Median: {reading.median():.2f}")
print(f"  Std:    {reading.std():.2f}")
print(f"  Skew:   {reading.skew():.2f}")

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

sns.histplot(reading, kde=True, bins=20, ax=axes[0], color="steelblue")
axes[0].set_title("Histogram + KDE")

sns.kdeplot(reading, fill=True, ax=axes[1], color="coral")
axes[1].set_title("KDE Only")

sns.boxplot(x=reading, ax=axes[2], color="lightgreen")
axes[2].set_title("Box Plot")

plt.tight_layout()
plt.show()

Key Takeaways

Always compute summary statistics before plotting, they tell you what to look for in the visual
Histograms show frequency; KDE plots show density; box plots show summary statistics; violin plots combine both
Skewness and kurtosis quantify distribution shape in numbers
Box plots make outliers obvious; histograms reveal modality (unimodal vs bimodal)
Compare distributions across groups by plotting them side by side with shared axes

Practice Challenge

Create a single figure with four subplots showing the distribution of math score using: (1) a histogram, (2) a KDE plot, (3) a box plot, and (4) a violin plot. Add a vertical line at the mean on each plot. Set the figure title to “Math Score Distribution Analysis”.

🧩 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")

math = df["math score"]
mean_val = math.mean()

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle("Math Score Distribution Analysis", fontsize=14, fontweight="bold")

# Histogram
sns.histplot(math, kde=False, bins=20, ax=axes[0, 0], color="steelblue", edgecolor="black")
axes[0, 0].axvline(mean_val, color="red", linestyle="--", label=f"Mean: {mean_val:.1f}")
axes[0, 0].set_title("Histogram")
axes[0, 0].legend()

# KDE
sns.kdeplot(math, fill=True, ax=axes[0, 1], color="coral")
axes[0, 1].axvline(mean_val, color="red", linestyle="--", label=f"Mean: {mean_val:.1f}")
axes[0, 1].set_title("KDE Plot")
axes[0, 1].legend()

# Box plot
sns.boxplot(x=math, ax=axes[1, 0], color="lightgreen", flierprops=dict(marker="o", markersize=5))
axes[1, 0].axvline(mean_val, color="red", linestyle="--", label=f"Mean: {mean_val:.1f}")
axes[1, 0].set_title("Box Plot")
axes[1, 0].legend()

# Violin
sns.violinplot(x=math, ax=axes[1, 1], inner="quartile", color="lightyellow")
axes[1, 1].axvline(mean_val, color="red", linestyle="--", label=f"Mean: {mean_val:.1f}")
axes[1, 1].set_title("Violin Plot")
axes[1, 1].legend()

plt.tight_layout()
plt.show()

Key Concepts

Summary statistics

Histograms

KDE plots (Kernel Density Estimation)

Box plots

Violin plots

Choosing the right plot

Comparing distributions across groups

Try It

Key Takeaways

Practice Challenge

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