Histograms

A histogram displays the frequency distribution of a single continuous numeric variable by dividing its range into consecutive, non-overlapping intervals (bins) and drawing a bar whose height is the count of values falling in each bin.

Marks:  45, 52, 55, 58, 61, 63, 65, 67, 68, 70, 72, 75, 78, 82, 88

Bin        Frequency   Bar
40-50          1       █
50-60          3       ███
60-70          6       ██████
70-80          3       ███
80-90          2       ██

Histogram vs Bar Chart — The Classic Exam Question

What a Histogram Reveals

1. Central tendency — where the bulk of the data sits
2. Spread — how wide the distribution is
3. Shape/skewness — symmetric, left- or right-skewed
4. Modality — one peak (unimodal), two (bimodal), many
5. Outliers — isolated bars far from the main body
6. Gaps — ranges with no data at all

A bimodal histogram is a signal, not a nuisance — it usually means two distinct populations have been mixed together (e.g. exam marks of a coached batch and an uncoached batch). Split and analyse them separately.

Choosing the Number of Bins

The bin count changes what you see, so it matters:

Sturges' Rule:        k = 1 + 3.322 × log₁₀(n)
Square-root rule:     k = √n
Rice rule:            k = 2 × n^(1/3)

Freedman-Diaconis (bin WIDTH, robust to outliers):
                                2 × IQR
                      width = ───────────
                                 n^(1/3)

Worked example. For n = 100 observations:

Sturges:      k = 1 + 3.322 × log₁₀(100) = 1 + 3.322(2) = 7.64  ≈  8 bins
Square-root:  k = √100 = 10 bins
Rice:         k = 2 × 100^(1/3) = 2 × 4.64 = 9.28  ≈  9 bins

Python

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

np.random.seed(42)
marks = np.random.normal(loc=65, scale=12, size=300).clip(0, 100)

fig, axes = plt.subplots(1, 3, figsize=(16, 4))

# Effect of bin count
for ax, bins in zip(axes, [5, 20, 60]):
    ax.hist(marks, bins=bins, color="#168B99", edgecolor="white")
    ax.set_title(f"{bins} bins")
    ax.set_xlabel("Marks")
axes[0].set_ylabel("Frequency")
plt.tight_layout()
plt.show()

# Full-featured histogram with statistical reference lines
fig, ax = plt.subplots(figsize=(9, 5))

n, bins, patches = ax.hist(marks, bins=20, color="#168B99",
                           edgecolor="white", alpha=0.85)

ax.axvline(marks.mean(), color="#ef4444", linestyle="--", linewidth=2,
           label=f"Mean = {marks.mean():.1f}")
ax.axvline(np.median(marks), color="#10b981", linestyle="-", linewidth=2,
           label=f"Median = {np.median(marks):.1f}")

ax.set_title("Distribution of Student Marks (n = 300)", fontsize=13, fontweight="bold")
ax.set_xlabel("Marks")
ax.set_ylabel("Number of Students")
ax.legend()
ax.spines[["top", "right"]].set_visible(False)
plt.tight_layout()
plt.show()

print(f"Mean {marks.mean():.2f} | Median {np.median(marks):.2f} | Skew {pd.Series(marks).skew():.3f}")
# Mean ≈ Median and skew ≈ 0  ->  symmetric distribution

# Histogram + KDE (smoothed density curve) with Seaborn
plt.figure(figsize=(9, 5))
sns.histplot(marks, bins=20, kde=True, color="#168B99", edgecolor="white")
plt.title("Marks Distribution with KDE Overlay")
plt.xlabel("Marks"); plt.ylabel("Count")
plt.show()

# Comparing distributions across a group
tips = sns.load_dataset("tips")
plt.figure(figsize=(9, 5))
sns.histplot(data=tips, x="total_bill", hue="time", bins=25,
             kde=True, alpha=0.6, element="step")
plt.title("Total Bill Distribution — Lunch vs Dinner")
plt.show()

# Skewed data — histogram before and after a log transform
income = np.random.lognormal(mean=10.5, sigma=0.8, size=1000)

fig, axes = plt.subplots(1, 2, figsize=(13, 4))

axes[0].hist(income, bins=40, color="#f59e0b", edgecolor="white")
axes[0].set_title(f"Raw Income — skew = {pd.Series(income).skew():.2f}")

axes[1].hist(np.log(income), bins=40, color="#10b981", edgecolor="white")
axes[1].set_title(f"Log(Income) — skew = {pd.Series(np.log(income)).skew():.2f}")

plt.tight_layout(); plt.show()
# The log transform turns a heavily right-skewed distribution into a near-normal one

Frequency Table — The Histogram in Numbers

freq_table = pd.cut(pd.Series(marks), bins=[0, 40, 50, 60, 70, 80, 90, 100]).value_counts().sort_index()
freq_df = pd.DataFrame({
    "Frequency": freq_table,
    "Relative Freq %": (freq_table / len(marks) * 100).round(2),
    "Cumulative Freq": freq_table.cumsum(),
})
print(freq_df)
#            Frequency  Relative Freq %  Cumulative Freq
# (0, 40]           10             3.33               10
# (40, 50]          28             9.33               38
# (50, 60]          64            21.33              102
# (60, 70]          98            32.67              200
# (70, 80]          70            23.33              270
# (80, 90]          27             9.00              297
# (90, 100]          3             1.00              300

Related Chart Types

The histogram shows the full shape of a distribution. The next chart — the box plot — sacrifices shape detail in exchange for a compact five-number summary that makes group comparison and outlier detection trivial.