Visualizing Numerical Variables: Box Plots

While histograms give us a detailed view of the frequency distribution, box plots (also known as box-and-whisker plots) offer a concise summary of a numerical variable's distribution, focusing on its central tendency, spread, and potential outliers. They are particularly effective for quickly comparing distributions across different categories, although we'll focus on the single variable case here, building on the concepts of central tendency and dispersion discussed earlier.

A box plot visualizes the following key statistics:

1. Median (Q2): The line inside the box represents the median or the 50th percentile ($Q_2$) of the data. Half the data points are below this value, and half are above.
2. Interquartile Range (IQR): The box itself spans from the first quartile (Q1, 25th percentile) to the third quartile (Q3, 75th percentile). The length of the box thus represents the IQR ($IQR = Q3 - Q1$), which measures the spread of the middle 50% of the data. A wider box indicates greater variability in the central portion of the data.
3. Whiskers: Lines (whiskers) extend from the box to indicate the range of the data, excluding outliers. Commonly, the whiskers extend to the minimum and maximum values within $1.5 \times IQR$ from the edges of the box (i.e., $Q1 - 1.5 \times IQR$ and $Q3 + 1.5 \times IQR$). Points outside this range are considered potential outliers.
4. Outliers: Individual points plotted beyond the whiskers are flagged as potential outliers. This visual identification is a direct application of the IQR rule, which we will discuss further as a statistical method for outlier detection.

Box plots are excellent for getting a quick sense of:

• Central Location: Where the median lies.
• Spread: How dispersed the data is (via IQR and whisker length).
• Skewness: Asymmetry in the distribution. If the median is closer to Q1, the distribution is likely right-skewed. If it's closer to Q3, it's likely left-skewed. Similarly, unequal whisker lengths can suggest skewness in the tails.
• Outliers: Data points that are unusually far from the bulk of the data.

Creating Box Plots with Seaborn

Seaborn provides a straightforward function, sns.boxplot(), to create informative box plots. It integrates well with Pandas DataFrames.

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

# Sample DataFrame (assuming you have one loaded, e.g., 'df')
# Let's create some sample data for demonstration
np.random.seed(42)
data = {
    'Age': np.random.normal(loc=40, scale=10, size=150).astype(int),
    'Salary': np.random.lognormal(mean=np.log(50000), sigma=0.4, size=150)
}
# Introduce a few outliers
data['Salary'][10] = 180000
data['Salary'][50] = 195000
data['Age'][20] = 85
df = pd.DataFrame(data)
df['Age'] = df['Age'].clip(18, 85) # Ensure ages are reasonable


# Create a box plot for the 'Salary' column
plt.figure(figsize=(6, 4)) # Control figure size
sns.boxplot(y=df['Salary'], color='#74c0fc') # Using a blue color from the palette
plt.title('Distribution of Salary')
plt.ylabel('Salary')
plt.grid(axis='y', linestyle='--', alpha=0.7) # Add horizontal grid lines
plt.show()

# Create a box plot for the 'Age' column
plt.figure(figsize=(6, 4))
sns.boxplot(y=df['Age'], color='#69db7c') # Using a green color
plt.title('Distribution of Age')
plt.ylabel('Age')
plt.grid(axis='y', linestyle='--', alpha=0.7)
plt.show()

Executing this code will generate box plots for the 'Salary' and 'Age' columns.

Interpreting the Box Plot

Observe the generated plot for 'Salary':

• The line inside the box shows the median salary.
• The box shows the range where the middle 50% of salaries fall (Q1 to Q3).
• The whiskers extend to cover the bulk of the data range, typically within $1.5 \times IQR$ of the box edges.
• Points plotted individually above the top whisker represent potential high-salary outliers. In our synthetic data, the values we manually set to 180,000 and 195,000 should appear as outliers.

Similarly, analyze the 'Age' plot to understand its distribution, median, spread, and any potential outliers.

Here is an example representation using Plotly for the 'Salary' data:

Box plot summarizing the Salary distribution, highlighting the median, IQR (box), typical range (whiskers), and potential outliers (individual points).

Box plots provide a compact yet powerful way to grasp the essential features of a numerical variable's distribution and are an indispensable tool in the univariate analysis toolkit. They visually flag potential outliers based on the IQR rule, complementing the statistical methods discussed next.