Box Plots (Box-and-Whisker Plots)

A box plot, invented by John Tukey, summarises a numeric distribution using the five-number summary and explicitly marks outliers. It is the most efficient chart for comparing distributions across groups.

Anatomy of a Box Plot

                    outlier
                       ○

              ┌────────┴────────┐  <- upper whisker (max within fence)
              │
        ┌─────┴─────┐  Q3 (75th percentile)
        │           │
        ├───────────┤  Median (Q2, 50th percentile)
        │           │
        └─────┬─────┘  Q1 (25th percentile)
              │
              └────────┬────────┘  <- lower whisker (min within fence)


   Box height  = IQR = Q3 − Q1  (the middle 50% of the data)
   Whiskers    extend to the most extreme point within 1.5 × IQR of the box
   Points beyond the whiskers are plotted individually as OUTLIERS

The Five-Number Summary

Outlier Fences

   IQR = Q3 − Q1

   Lower fence = Q1 − 1.5 × IQR
   Upper fence = Q3 + 1.5 × IQR

   Extreme (far) outliers use 3.0 × IQR instead of 1.5

Worked Example — By Hand

Data: 12, 15, 18, 22, 24, 25, 28, 30, 35, 42, 78 (n = 11, already sorted)

Median (Q2) = 6th value            = 25

Lower half: 12, 15, 18, 22, 24     ->  Q1 = 18
Upper half: 28, 30, 35, 42, 78     ->  Q3 = 35

IQR = 35 − 18 = 17

Lower fence = 18 − 1.5(17) = 18 − 25.5 = −7.5
Upper fence = 35 + 1.5(17) = 35 + 25.5 = 60.5

Outliers: 78  (it exceeds 60.5)

Whiskers: lower = 12 (smallest value ≥ −7.5)
          upper = 42 (largest value ≤ 60.5)

Five-number summary: 12, 18, 25, 35, 42  with 78 plotted as an outlier

Reading Shape from a Box Plot

Box Plot vs Histogram

Key limitation: a box plot cannot show multi-modality. Two datasets — one unimodal, one strongly bimodal — can produce identical box plots. Pair box plots with histograms or use a violin plot (a box plot with a mirrored KDE) when shape matters.

Python

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

data = pd.Series([12, 15, 18, 22, 24, 25, 28, 30, 35, 42, 78])

Q1, Q2, Q3 = data.quantile([0.25, 0.5, 0.75])
IQR = Q3 - Q1
lower, upper = Q1 - 1.5 * IQR, Q3 + 1.5 * IQR

print(f"Q1 = {Q1}, Median = {Q2}, Q3 = {Q3}, IQR = {IQR}")
print(f"Fences: [{lower}, {upper}]")
print("Outliers:", data[(data < lower) | (data > upper)].tolist())
# Q1 = 20.0, Median = 25.0, Q3 = 32.5, IQR = 12.5
# Fences: [1.25, 51.25]
# Outliers: [78]

print("Five-number summary:\n", data.describe()[["min", "25%", "50%", "75%", "max"]])

Note on quartile methods: pandas uses linear interpolation by default, which gives Q1 = 20.0 here, while the "median of the lower half" method taught in most textbooks gives Q1 = 18. Both are accepted; state which method you used.

# Single box plot
fig, ax = plt.subplots(figsize=(7, 5))
bp = ax.boxplot(data, vert=True, patch_artist=True, widths=0.4,
                boxprops=dict(facecolor="#168B99", alpha=0.7),
                medianprops=dict(color="#ef4444", linewidth=2),
                flierprops=dict(marker="o", markerfacecolor="#ef4444", markersize=8))
ax.set_title("Box Plot with One Outlier", fontweight="bold")
ax.set_ylabel("Value")
plt.show()

# COMPARING GROUPS — where box plots genuinely shine
tips = sns.load_dataset("tips")

fig, axes = plt.subplots(1, 3, figsize=(16, 4.5))

sns.boxplot(data=tips, x="day", y="total_bill", ax=axes[0], palette="Set2")
axes[0].set_title("Total Bill by Day")

sns.boxplot(data=tips, x="day", y="total_bill", hue="sex", ax=axes[1], palette="Set1")
axes[1].set_title("Total Bill by Day and Sex")

# Violin plot — box plot + distribution shape
sns.violinplot(data=tips, x="day", y="total_bill", ax=axes[2], palette="Set3", inner="box")
axes[2].set_title("Violin Plot — Shape Revealed")

plt.tight_layout()
plt.show()

# Group-wise five-number summary in numbers
summary = tips.groupby("day")["total_bill"].describe()[["min", "25%", "50%", "75%", "max"]].round(2)
summary["IQR"] = (summary["75%"] - summary["25%"]).round(2)
print(summary)
#       min    25%    50%    75%    max    IQR
# day
# Thur 7.51  12.44  16.20  20.16  43.11   7.72
# Fri  5.75  12.09  15.38  21.75  40.17   9.66
# Sat  3.07  13.91  18.24  24.74  50.81  10.83
# Sun  7.25  14.99  19.63  25.60  48.17  10.61
# INSIGHT: weekend bills are both higher (median) and more variable (IQR)

# Automated outlier report across every numeric column
def outlier_report(df):
    rows = []
    for col in df.select_dtypes(include=np.number).columns:
        Q1, Q3 = df[col].quantile([0.25, 0.75])
        IQR = Q3 - Q1
        lo, hi = Q1 - 1.5 * IQR, Q3 + 1.5 * IQR
        out = df[(df[col] < lo) | (df[col] > hi)][col]
        rows.append({"column": col, "Q1": round(Q1, 2), "Q3": round(Q3, 2),
                     "IQR": round(IQR, 2), "lower_fence": round(lo, 2),
                     "upper_fence": round(hi, 2), "n_outliers": len(out),
                     "pct": round(len(out) / len(df) * 100, 2)})
    return pd.DataFrame(rows)

print(outlier_report(tips))

When to Use a Box Plot

Histograms and box plots describe one variable. The next chart handles two.