←Back to the bookChapter 13 · Frequency Distributions

Part II · Describing Data  |  Chapter 13

Frequency Distributions 🐍 Notebook

Five demos on organizing data by how often it occurs: counting with frequency tables, binning continuous data into classes, the relative and cumulative columns, drawing histograms, polygons and ogives, and the weighted mean hiding inside a grouped table.

Author: John Fisher  ·  Statistics, Data Science and AI: A Visual Handbook  ·  2026

🎯 What you'll build in this notebook

#	Demo	Idea it builds
1	Counting things	an ungrouped frequency table with relative frequency
2	Binning into classes	a grouped table with relative & cumulative columns
3	Three pictures	histogram, frequency polygon, and ogive from one dataset
4	The weighted mean	GPA-style average where weights matter
5	Grouped mean	a weighted mean of midpoints, and its approximation error

---

⚙️ Setup, imports & the book's plotting style

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

rng = np.random.default_rng(13)

NAVY="#0a1230"; INK="#1a2138"; INK_SOFT="#4a5578"
CYAN="#0891b2"; PURPLE="#7c3aed"; AMBER="#d97706"; GREEN="#059669"; PINK="#db2777"; BLUE="#2563eb"; GRID="#e6e9f2"
plt.rcParams.update({
    "figure.facecolor":"white","axes.facecolor":"white","figure.dpi":110,"font.size":11,
    "axes.edgecolor":GRID,"axes.linewidth":1.2,"axes.grid":True,"grid.color":GRID,"axes.axisbelow":True,
    "axes.spines.top":False,"axes.spines.right":False,"axes.titlesize":13,"axes.titleweight":"bold","axes.titlecolor":INK,
    "axes.labelcolor":INK_SOFT,"axes.labelsize":11.5,"xtick.color":INK_SOFT,"ytick.color":INK_SOFT,"legend.frameon":False,
})
print("✅ Environment ready.")

✅ Environment ready.

DEMO 1 · COUNTING THINGS

🧮 The ungrouped frequency table

For a handful of distinct values, a frequency table just pairs each value with its count. Add a relative-frequency column (count / total) and it must sum to 1.

ratings = pd.Series(list("AABBBBCCCABBDA"), name="grade")

freq = ratings.value_counts().sort_index()
table = pd.DataFrame({"frequency": freq})
table["relative"] = (table["frequency"] / table["frequency"].sum()).round(3)
table["percent"]  = (table["relative"] * 100).round(1)
print(table)
print(f"\nrelative frequencies sum to {table['relative'].sum():.3f}  (should be 1.000)")

# --- added visual ---
import matplotlib.pyplot as plt, numpy as np
_CY,_PU,_AM,_GR,_PK,_INK,_GRY,_BL = "#0891b2","#7c3aed","#d97706","#059669","#db2777","#1a2138","#c7ccda","#2563eb"
fig, ax = plt.subplots(figsize=(7, 3.6))
freq.plot.bar(ax=ax, color=_CY, edgecolor="white")
ax.set_xlabel("grade"); ax.set_ylabel("frequency"); ax.set_title("The frequency table as a bar chart")
ax.tick_params(axis="x", rotation=0); ax.grid(axis="x", visible=False)
plt.tight_layout(); plt.show()

       frequency  relative  percent
grade                              
A              4     0.286     28.6
B              6     0.429     42.9
C              3     0.214     21.4
D              1     0.071      7.1

relative frequencies sum to 1.000  (should be 1.000)

DEMO 2 · BINNING INTO CLASSES

📦 A grouped frequency table

Continuous data with many distinct values gets binned into equal-width classes. pandas pd.cut makes the classes; from there the relative, cumulative, and cumulative-relative columns are one line each.

scores = pd.Series(rng.normal(68, 14, 80).clip(20, 100), name="score")

edges = [20, 35, 50, 65, 80, 95, 100]
classes = pd.cut(scores, bins=edges, right=False)         # equal-width-ish classes
f = classes.value_counts(sort=False)

tbl = pd.DataFrame({"frequency": f})
tbl["relative"]     = (tbl["frequency"] / tbl["frequency"].sum()).round(3)
tbl["cumulative"]   = tbl["frequency"].cumsum()
tbl["cum_relative"] = (tbl["cumulative"] / tbl["frequency"].sum()).round(3)
print(tbl)
print(f"\nlast cumulative = {tbl['cumulative'].iloc[-1]} = n,  last cum_relative = {tbl['cum_relative'].iloc[-1]:.3f} = 1.0")

           frequency  relative  cumulative  cum_relative
score                                                   
[20, 35)           2     0.025           2         0.025
[35, 50)           3     0.038           5         0.062
[50, 65)          27     0.338          32         0.400
[65, 80)          29     0.362          61         0.762
[80, 95)          18     0.225          79         0.988
[95, 100)          1     0.012          80         1.000

last cumulative = 80 = n,  last cum_relative = 1.000 = 1.0

DEMO 3 · THREE PICTURES

📊 Histogram, polygon, ogive

The same frequency counts drive three classic graphs. The histogram and polygon use class MIDPOINTS; the ogive plots cumulative frequency at class BOUNDARIES. That boundary-vs-midpoint distinction is the detail people miss.

counts, bin_edges = np.histogram(scores, bins=edges)
mids = (bin_edges[:-1] + bin_edges[1:]) / 2
cum  = np.cumsum(counts)

fig, (a1, a2, a3) = plt.subplots(1, 3, figsize=(14, 4))

a1.hist(scores, bins=edges, color=CYAN, alpha=0.85, edgecolor="white")
a1.set_title("Histogram"); a1.set_xlabel("score"); a1.set_ylabel("frequency")

# frequency polygon: midpoints, anchored to zero at both ends
px = np.concatenate([[mids[0]-15], mids, [mids[-1]+15]])
py = np.concatenate([[0], counts, [0]])
a2.plot(px, py, "-o", color=PURPLE, lw=2, markersize=5)
a2.set_title("Frequency polygon (midpoints)"); a2.set_xlabel("score"); a2.set_ylabel("frequency")

# ogive: cumulative frequency at upper boundaries, starting at (first boundary, 0)
ox = np.concatenate([[bin_edges[0]], bin_edges[1:]])
oy = np.concatenate([[0], cum])
a3.plot(ox, oy, "-o", color=AMBER, lw=2, markersize=5)
a3.axhline(len(scores)/2, color=GREEN, ls="--", lw=1.5, label="n/2 → median")
a3.set_title("Ogive (boundaries)"); a3.set_xlabel("score"); a3.set_ylabel("cumulative frequency"); a3.legend()
plt.tight_layout(); plt.show()
print("Histogram bars touch (continuous). Ogive rises to n. Read the median where the ogive crosses n/2.")

Histogram bars touch (continuous). Ogive rises to n. Read the median where the ogive crosses n/2.

DEMO 4 · THE WEIGHTED MEAN

🏫 When some values count more

A weighted mean is sum(w*x)/sum(w). GPA is the everyday example: grades weighted by credit hours. The plain average ignores that a 4-credit course should pull harder than a 1-credit one.

grades  = np.array([4.0, 3.0, 4.0, 2.0])   # A, B, A, C as grade points
credits = np.array([4,   4,   3,   1  ])   # weights

weighted = np.average(grades, weights=credits)
plain    = grades.mean()
print(f"weighted GPA (by credits) = {weighted:.2f}")
print(f"plain average of grades   = {plain:.2f}")
print("\nThe heavier (4-credit) courses pull the weighted GPA toward their grades.")

weighted GPA (by credits) = 3.50
plain average of grades   = 3.25

The heavier (4-credit) courses pull the weighted GPA toward their grades.

DEMO 5 · THE GROUPED MEAN

📐 A weighted mean of midpoints

Once data is grouped you no longer have the exact values, so the mean is estimated by treating every value in a class as its midpoint: mean ≈ sum(f*m)/sum(f). That is just a weighted mean, and it differs slightly from the true mean.

true_mean = scores.mean()

# estimate from the grouped table alone, using midpoints weighted by frequency
grouped_mean = np.average(mids, weights=counts)

print(f"true mean (from raw data)      = {true_mean:.2f}")
print(f"grouped mean (from the table)  = {grouped_mean:.2f}")
print(f"approximation error            = {abs(true_mean - grouped_mean):.2f}")
print("\nGrouping trades exact values for a tidy summary: the grouped mean is close, not exact.")

# --- added visual ---
import matplotlib.pyplot as plt, numpy as np
_CY,_PU,_AM,_GR,_PK,_INK,_GRY,_BL = "#0891b2","#7c3aed","#d97706","#059669","#db2777","#1a2138","#c7ccda","#2563eb"
fig, ax = plt.subplots(figsize=(6, 3.8))
ax.bar(["true mean","grouped mean"], [true_mean, grouped_mean], color=[_CY,_AM], edgecolor="white", width=0.55)
ax.set_ylabel("mean"); ax.set_ylim(0, max(true_mean, grouped_mean)*1.15)
for i, v in enumerate([true_mean, grouped_mean]): ax.text(i, v+0.4, f"{v:.2f}", ha="center", fontweight="bold")
ax.set_title(f"Grouped mean approximates the true mean (gap {abs(true_mean-grouped_mean):.2f})")
ax.grid(axis="x", visible=False)
plt.tight_layout(); plt.show()

true mean (from raw data)      = 69.92
grouped mean (from the table)  = 68.88
approximation error            = 1.05

Grouping trades exact values for a tidy summary: the grouped mean is close, not exact.

👥 Real-World Example: Binning Customer Ages

Here is a frequency distribution on real-shaped data: 500 customer ages grouped into ten-year bands. We build the full table, the count, relative frequency, and cumulative frequency for each band, then plot the histogram. The cumulative column answers questions like what share of customers are under 50.

# --- Real-World beat: a full frequency table from raw ages ---
BASE = "https://raw.githubusercontent.com/johnfisher-ai/Statistics-Data-Science-AI-Visual-Book/main/data/"
try:    cust = pd.read_excel("../../data/frequency-distributions--customer_ages.xlsx", sheet_name="Data")
except FileNotFoundError: cust = pd.read_excel(BASE+"frequency-distributions--customer_ages.xlsx", sheet_name="Data")
bins = [18, 30, 40, 50, 60, 70, 90]
freq = pd.cut(cust.age, bins=bins, right=False).value_counts().sort_index()
table = pd.DataFrame({"frequency": freq,
                      "relative_%": (freq / len(cust) * 100).round(1),
                      "cumulative": freq.cumsum()})
print(table.to_string())
print(f"\nmodal age band: {freq.idxmax()}   |   share under 50: {(cust.age < 50).mean()*100:.0f}%")
fig, ax = plt.subplots(1, 2, figsize=(11,4))
ax[0].hist(cust.age, bins=bins, color=CYAN, alpha=0.85, edgecolor="white")
ax[0].set_title("Age distribution (10-year bands)"); ax[0].set_xlabel("age"); ax[0].set_ylabel("customers")
mc = cust.membership.value_counts().reindex(["Basic","Plus","Premium"])
ax[1].bar(mc.index, mc.values, color=[CYAN, PURPLE, AMBER], alpha=0.85, edgecolor="white")
ax[1].set_title("Membership tier (categorical frequency)"); ax[1].set_ylabel("customers")
plt.tight_layout(); plt.show()

          frequency  relative_%  cumulative
age                                        
[18, 30)         70        14.0          70
[30, 40)        152        30.4         222
[40, 50)        116        23.2         338
[50, 60)         78        15.6         416
[60, 70)         55        11.0         471
[70, 90)         29         5.8         500

modal age band: [30, 40)   |   share under 50: 68%

---

🎓 Recap

• A frequency table pairs values (or classes) with counts; relative frequency = count / total and sums to 1.
• Grouped tables bin continuous data into classes, then add cumulative and cumulative-relative columns.
• Histogram and polygon use midpoints; the ogive plots cumulative frequency at boundaries and crosses n/2 at the median.
• A weighted mean = sum(w*x)/sum(w); GPA is the classic case.
• The grouped mean is a weighted mean of class midpoints, close to but not equal to the true mean.

Statistics, Data Science and AI: A Visual Handbook · © 2026 John Fisher

←Back to the bookChapter 13 · Frequency Distributions