Matplotlib Histograms, Pie Charts & Machine Learning Statistics

Before we begin, here is a quick checkpoint of the tools we will use.

What is Matplotlib?

Matplotlib is a Python library that lets you draw charts and graphs. Think of it as graph paper + coloured pens, but controlled entirely by code.

Install it if needed:

pip install matplotlib

What is NumPy?

NumPy is a Python library for fast number crunching. It can store lists of numbers in special structures called arrays and calculate things like averages instantly.

pip install numpy

What is SciPy?

SciPy builds on NumPy and adds advanced scientific tools · including the ability to find the mode of a dataset.

pip install scipy

What is a Dataset?

A dataset is simply a collection of values · for example, the ages of 100 people, or the scores of 50 students.

What Is a Histogram?

A histogram is a type of bar chart that shows how frequently values appear within ranges (called bins).

Analogy: Imagine you asked 100 classmates how many hours they sleep per night. Some said 5, some said 7, some said 9. A histogram groups those answers into ranges · "5 · 6 hours", "7 · 8 hours", "9 · 10 hours" · and draws a bar showing how many people fall in each range.

Why do we need histograms?

• To understand the distribution (spread) of data
• To spot patterns: Is data bunched together? Spread out? Skewed to one side?
• Used constantly in data science, quality control, medical research, and finance

A histogram is different from a regular bar chart:

Your First Histogram

import matplotlib.pyplot as plt
import numpy as np

# Create a list of 250 random ages between 0 and 130
x = np.random.normal(170, 10, 250)

plt.hist(x)
plt.show()

What does each line do?

Expected output: A bell-shaped histogram centred around 170. Most bars will be near 170, with fewer bars further away on either side.

💡 Thinking prompt: What would happen if you changed 10 (the spread) to 30? Try it! The histogram would become much wider because values would be more spread out.

Understanding np.random.normal()

This function creates normally distributed (bell-curve) data. Three arguments control it:

np.random.normal(mean, standard_deviation, count)

Quick demo · narrow vs wide spread:

import matplotlib.pyplot as plt
import numpy as np

narrow = np.random.normal(50, 2, 200)   # tightly packed around 50
wide   = np.random.normal(50, 20, 200)  # spread far around 50

plt.figure(figsize=(12, 4))

plt.subplot(1, 2, 1)
plt.hist(narrow, color='steelblue')
plt.title("Narrow Spread (std=2)")

plt.subplot(1, 2, 2)
plt.hist(wide, color='tomato')
plt.title("Wide Spread (std=20)")

plt.tight_layout()
plt.show()

Expected output: Two side-by-side histograms. The left one has a tall, narrow peak. The right one is short and wide.

💡 Thinking prompt: Which spread better represents a consistent factory product? The narrow one · meaning almost all items are very similar in size.

Controlling the Number of Bins

Bins are the ranges that your data is grouped into. More bins = more detail. Fewer bins = simpler picture.

import matplotlib.pyplot as plt
import numpy as np

x = np.random.normal(170, 10, 250)

plt.hist(x, bins=5)   # only 5 groups
plt.title("5 Bins")
plt.show()

plt.hist(x, bins=30)  # 30 groups — much more detail
plt.title("30 Bins")
plt.show()

Expected output (5 bins): Just 5 wide bars · you can see the general shape but not the fine detail.

Expected output (30 bins): 30 narrow bars · the bell curve shape is very visible.

💡 Rule of thumb: A good starting point for bin count is roughly the square root of your data count. For 250 data points → about 15 · 16 bins.

Common Beginner Mistake: Forgetting plt.show()

# WRONG — chart never appears
import matplotlib.pyplot as plt
import numpy as np
x = np.random.normal(100, 15, 100)
plt.hist(x)
# Missing plt.show()!

# CORRECT
import matplotlib.pyplot as plt
import numpy as np
x = np.random.normal(100, 15, 100)
plt.hist(x)
plt.show()  # ✅ This line opens the chart window

Real-World Use of Histograms

What Is a Pie Chart?

A pie chart is a circle divided into slices. Each slice represents a proportion (share) of the whole.

Analogy: Imagine a pizza cut into 4 slices · one person gets 2 slices, another gets 1, and two others get half a slice each. A pie chart shows exactly that kind of proportional split.

Why use pie charts?

• Perfect for showing part-to-whole relationships (e.g., what percentage of sales came from each product)
• Easy to understand at a glance
• Used in business reports, surveys, budgets, and research

Your First Pie Chart

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]

plt.pie(y)
plt.show()

What does each line do?

Expected output: A circle divided into 4 coloured slices · the largest slice takes up 35% of the circle.

💡 Note: The values don't have to add up to 100. Matplotlib automatically converts them into percentages. If you wrote [1, 1, 1, 1], each slice would be 25%.

Adding Labels to a Pie Chart

Without labels, a pie chart is hard to interpret. Let's add them:

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]
labels = ["Apples", "Bananas", "Cherries", "Dates"]

plt.pie(y, labels=labels)
plt.show()

Expected output: A pie chart where each slice is labelled with the fruit name.

Starting Angle: startangle

By default, the first slice starts at the 3 o'clock position (right side). You can rotate the start:

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]
labels = ["Apples", "Bananas", "Cherries", "Dates"]

plt.pie(y, labels=labels, startangle=90)
plt.show()

Expected output: The first slice ("Apples") now starts at the 12 o'clock position (top of the circle).

💡 startangle=90 rotates the chart 90° counter-clockwise from the default starting position.

Exploding a Slice

You can "pull out" one slice to highlight it:

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]
labels = ["Apples", "Bananas", "Cherries", "Dates"]
explode = [0.2, 0, 0, 0]   # Pull the first slice (Apples) 0.2 outward

plt.pie(y, labels=labels, explode=explode, startangle=90)
plt.show()

Expected output: The "Apples" slice is visually separated from the rest of the pie, drawing the viewer's attention to it.

Understanding the explode list:

explode = [0.2, 0, 0, 0]
#          ^    ^  ^  ^
#     Apples  Bananas  Cherries  Dates
# 0.2 = pull out, 0 = stay in place

💡 Try explode = [0.2, 0.1, 0, 0] · both Apples and Bananas will be pulled out!

Adding Shadows

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]
labels = ["Apples", "Bananas", "Cherries", "Dates"]
explode = [0.2, 0, 0, 0]

plt.pie(y, labels=labels, explode=explode, startangle=90, shadow=True)
plt.show()

Expected output: The pie chart now has a subtle shadow beneath it, giving it a 3D-like appearance.

Custom Colors

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]
labels = ["Apples", "Bananas", "Cherries", "Dates"]
colors = ["red", "yellow", "pink", "orange"]

plt.pie(y, labels=labels, colors=colors, startangle=90)
plt.show()

Expected output: Each slice now uses your specified colours: red for Apples, yellow for Bananas, etc.

Showing Percentages: autopct

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]
labels = ["Apples", "Bananas", "Cherries", "Dates"]

plt.pie(y, labels=labels, autopct="%1.1f%%")
plt.show()

What is autopct="%1.1f%%"?

Expected output: Each slice now shows its percentage, e.g., 35.0%, 25.0%, 25.0%, 15.0%.

Adding a Legend

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]
labels = ["Apples", "Bananas", "Cherries", "Dates"]

plt.pie(y, labels=labels, autopct="%1.1f%%", startangle=90)
plt.legend(title="Fruits")
plt.show()

Expected output: A coloured legend box appears beside the chart, labelled "Fruits".

Full Polished Pie Chart Example

import matplotlib.pyplot as plt

y = [35, 25, 25, 15]
labels = ["Apples", "Bananas", "Cherries", "Dates"]
explode = [0.2, 0, 0, 0]
colors = ["#e74c3c", "#f1c40f", "#e91e63", "#ff9800"]

plt.pie(
    y,
    labels=labels,
    explode=explode,
    colors=colors,
    autopct="%1.1f%%",
    shadow=True,
    startangle=90
)

plt.legend(title="Fruits", loc="lower right")
plt.title("Fruit Sales Distribution")
plt.show()

Expected output: A professional-looking pie chart with:

• Highlighted "Apples" slice (pulled out)
• Custom colours
• Percentage labels on each slice
• Drop shadow
• Legend in the lower right
• Chart title at the top

Real-World Use of Pie Charts

What Is Machine Learning?

Machine Learning (ML) is a branch of Artificial Intelligence where computers learn patterns from data · without being explicitly programmed for every situation.

Analogy: Imagine teaching a child to recognise cats. You don't write rules like "cats have pointy ears, whiskers, and fur." Instead, you show the child hundreds of cat pictures, and they learn the pattern. ML works exactly the same way · it learns from data examples.

Why does ML matter?

The Core of ML: Data and Statistics

Before a machine can learn, it needs to understand the data it's working with. This is where statistics become essential.

The key statistical concepts used in ML are:

Let's learn each one deeply.

What Is the Mean?

The mean (also called the average) is calculated by:

1. Adding up all values
2. Dividing by how many values there are

Formula:

Mean = (Sum of all values) ÷ (Number of values)

Simple example:

Values: 10, 20, 30
Sum = 10 + 20 + 30 = 60
Count = 3
Mean = 60 ÷ 3 = 20

In Python with NumPy:

import numpy as np

speed = [99, 86, 87, 88, 111, 86, 103, 87, 94, 78, 77, 85, 86]
x = np.mean(speed)
print(x)

Expected output:

89.77

💡 This means the average speed in this dataset is about 89.77 km/h.

Why is mean useful in ML?

The mean is used to fill in missing data, normalise features, and understand the central tendency of a dataset.

What Is the Median?

The median is the middle value when all values are sorted in order.

Why not always use the mean?

The mean is pulled by extreme values. Consider this scenario:

Salaries: 30000, 32000, 35000, 38000, 1000000
Mean = (30000 + 32000 + 35000 + 38000 + 1000000) ÷ 5 = 227000

A mean of £227,000 is misleading because 4 out of 5 people earn between £30,000 · £38,000. The billionaire distorts the mean.

Median = 35000  ← the actual middle value

The median is more honest here.

How to find the median manually:

Odd count of values:

Values: 77, 86, 86, 86, 87, 87, 88, 94, 99, 103, 111
Count: 11 values (odd)
Middle position: (11+1) ÷ 2 = position 6
6th value = 87
Median = 87

Even count of values:

Values: 77, 86, 86, 87, 87, 88, 94, 99, 103, 111
Count: 10 values (even)
Two middle values: position 5 and 6 → 87 and 88
Median = (87 + 88) ÷ 2 = 87.5

In Python with NumPy:

import numpy as np

speed = [99, 86, 87, 88, 111, 86, 103, 87, 94, 78, 77, 85, 86]
x = np.median(speed)
print(x)

Expected output:

87.0

What Is the Mode?

The mode is the value that appears most often in a dataset.

Simple example:

Values: 4, 7, 7, 7, 9, 11
Mode = 7  (appears 3 times — more than any other value)

Why is mode useful?

In Python with SciPy:

from scipy import stats

speed = [99, 86, 87, 88, 111, 86, 103, 87, 94, 78, 77, 85, 86]
x = stats.mode(speed)
print(x)

Expected output:

ModeResult(mode=86, count=3)

This tells us: 86 is the most common value, and it appears 3 times.

Comparing Mean, Median, and Mode

Let's compare all three on the same dataset:

import numpy as np
from scipy import stats

data = [99, 86, 87, 88, 111, 86, 103, 87, 94, 78, 77, 85, 86]

print("Mean:  ", np.mean(data))
print("Median:", np.median(data))
print("Mode:  ", stats.mode(data).mode)

Expected output:

Mean:   89.77
Median: 87.0
Mode:   86

💡 Thinking prompt: The mean (89.77) is higher than the median (87). Why? Because the value 111 pulls the mean upward. The median ignores this.

Common Beginner Mistake: Using Mean When Outliers Exist

# WRONG approach for skewed data with outliers
salaries = [30000, 32000, 35000, 38000, 2000000]
average = sum(salaries) / len(salaries)
print("Average salary:", average)
# Output: 427000 ← misleading!

import numpy as np
# BETTER — use median for skewed data
salaries = [30000, 32000, 35000, 38000, 2000000]
print("Median salary:", np.median(salaries))
# Output: 35000.0 ← realistic picture

Rule: Use mean when data is evenly spread. Use median when there are outliers or skew.

What Is Standard Deviation?

Standard deviation measures how spread out the values in a dataset are from the mean.

Analogy: Imagine two archery teams both hitting an average score of 80. Team A's scores are 79, 80, 81, 80, 80. Team B's scores are 50, 95, 100, 65, 90. Both have the same mean · but Team B is wildly inconsistent. Standard deviation captures that inconsistency.

• Low standard deviation → values are close to the mean → consistent, predictable
• High standard deviation → values are spread far from the mean → varied, unpredictable

Two Example Datasets Side by Side

import numpy as np

speed1 = [86, 87, 88, 86, 87, 85, 86]       # similar values — low spread
speed2 = [32, 111, 138, 28, 59, 77, 97]     # very different values — high spread

print("Speed1 Mean:", np.mean(speed1))
print("Speed2 Mean:", np.mean(speed2))
print()
print("Speed1 Std Dev:", np.std(speed1))
print("Speed2 Std Dev:", np.std(speed2))

Expected output:

Speed1 Mean: 85.57
Speed2 Mean: 77.43

Speed1 Std Dev: 0.90
Speed2 Std Dev: 37.85

💡 Both datasets have similar means (~85 vs ~77), but Speed1 has a tiny standard deviation (0.90) while Speed2 has a huge one (37.85). Speed1 values barely deviate from their mean; Speed2 values vary wildly.

Calculating Standard Deviation Step by Step

Let's manually understand how it's calculated with a tiny dataset: [4, 8, 6, 5, 3]

Step 1 · Find the mean:

Mean = (4 + 8 + 6 + 5 + 3) ÷ 5 = 26 ÷ 5 = 5.2

Step 2 · Find each value's distance from the mean (deviation):

4 - 5.2 = -1.2
8 - 5.2 =  2.8
6 - 5.2 =  0.8
5 - 5.2 = -0.2
3 - 5.2 = -2.2

Step 3 · Square each deviation (to remove negatives):

(-1.2)² = 1.44
( 2.8)² = 7.84
( 0.8)² = 0.64
(-0.2)² = 0.04
(-2.2)² = 4.84

Step 4 · Find the mean of those squares (this is the VARIANCE):

Variance = (1.44 + 7.84 + 0.64 + 0.04 + 4.84) ÷ 5
         = 14.80 ÷ 5
         = 2.96

Step 5 · Square root of the variance = Standard Deviation:

Standard Deviation = √2.96 ≈ 1.72

In Python with NumPy (does all 5 steps instantly):

import numpy as np

data = [4, 8, 6, 5, 3]

variance = np.var(data)
std_dev  = np.std(data)

print("Variance:          ", variance)
print("Standard Deviation:", std_dev)

Expected output:

Variance:           2.96
Standard Deviation: 1.7204650534085253

✅ NumPy's result matches our manual calculation exactly.

What Is Variance?

Variance is simply the square of the standard deviation · or equivalently, the mean of the squared deviations. It's an intermediate step in calculating standard deviation.

Variance = (Standard Deviation)²
Standard Deviation = √Variance

Standard deviation is more practical because it's in the same unit as your data.

What Are Percentiles?

A percentile tells you what percentage of values fall below a specific value.

Analogy: If you scored in the 90th percentile on an exam, it means 90% of students scored lower than you.

import numpy as np

ages = [5, 31, 43, 48, 50, 41, 7, 11, 15, 39, 80, 82, 32, 2, 8, 6, 25, 36, 27, 61, 31]

p75 = np.percentile(ages, 75)
print("75th percentile:", p75)

Expected output:

75th percentile: 43.0

This means: 75% of people in this dataset are younger than 43 years old.

Percentiles in Action

import numpy as np

ages = [5, 31, 43, 48, 50, 41, 7, 11, 15, 39, 80, 82, 32, 2, 8, 6, 25, 36, 27, 61, 31]

for p in [25, 50, 75, 90]:
    val = np.percentile(ages, p)
    print(f"{p}th percentile: {val}")

Expected output:

25th percentile: 11.0
50th percentile: 31.0
75th percentile: 43.0
90th percentile: 61.0

💡 The 50th percentile is always equal to the median. Notice above: 50th percentile = 31, and np.median(ages) would also give 31.

Why Standard Deviation and Percentiles Matter in ML

Mistake 1 · Confusing Histogram with Bar Chart

# WRONG — using bar() for continuous numerical distribution
categories = ["60-70", "70-80", "80-90"]
counts = [10, 25, 15]
plt.bar(categories, counts)  # ← this works but is semantically wrong for distributions

# CORRECT — use hist() for continuous numerical data
data = [65, 72, 74, 78, 81, 85, 88, 63, 71]
plt.hist(data)  # ← Matplotlib handles the binning automatically

Mistake 2 · Forgetting to Import NumPy or SciPy

# WRONG
x = mean([1, 2, 3])  # mean() is not a built-in Python function!

# CORRECT
import numpy as np
x = np.mean([1, 2, 3])

Mistake 3 · Using Mean for Skewed Data

import numpy as np

house_prices = [200000, 210000, 195000, 205000, 1500000]  # one outlier

print("Mean:  ", np.mean(house_prices))    # 462000 — misleading!
print("Median:", np.median(house_prices))  # 205000 — realistic

Output:

Mean:   462000.0
Median: 205000.0

Mistake 4 · Misreading stats.mode() Output

from scipy import stats

data = [4, 4, 7, 9]
result = stats.mode(data)

# WRONG — printing the whole object
print(result)  # Prints: ModeResult(mode=4, count=2)

# CORRECT — access .mode and .count attributes
print("Mode:", result.mode)   # 4
print("Count:", result.count) # 2

Mistake 5 · Not Setting bins Intentionally

import matplotlib.pyplot as plt
import numpy as np

data = np.random.normal(50, 10, 1000)

# Default bins might hide patterns
plt.hist(data)  # Uses 10 bins by default — may not be ideal

# Better — set bins deliberately
plt.hist(data, bins=30)  # More detail visible
plt.show()