11-15-2025, 04:43 PM
Chapter 14 — The Normal Distribution
The normal distribution is one of the MOST important ideas in probability and statistics.
It appears everywhere in nature, science, exams, psychology, economics — almost everywhere.
It is the famous “bell curve.”
---
14.1 What Is the Normal Distribution?
It is a symmetrical, bell-shaped curve that describes data where:
• most values are near the mean
• fewer values appear far from the mean
• extreme values are rare
Examples that follow a normal distribution:
• heights
• test scores
• measurement errors
• reaction times
• IQ scores
• scientific variations
---
14.2 Why It Happens
The “bell curve” appears when MANY small, random factors add together.
Example:
Your height depends on:
• genetics
• nutrition
• sleep
• childhood health
• hormones
Each factor makes a tiny difference — add them up, and you get a normal distribution.
This is called the Central Limit Theorem, a key concept later in more advanced courses.
---
14.3 Key Features of the Normal Curve
1. Symmetrical
Left side mirrors the right.
2. Mean = Median = Mode
All three are the same at the centre.
3. Most data lies close to the mean
The curve rises sharply at the centre.
4. Tails extend forever (but get very thin)
Outliers are rare but possible.
---
14.4 The 68–95–99.7 Rule
This is the most famous rule in statistics.
For a perfect normal distribution:
• 68% of values lie within 1 standard deviation (SD) of the mean
• 95% lie within 2 SD
• 99.7% lie within 3 SD
This means:
Most people are close to average.
Few are far from average.
Almost nobody is extremely far from average.
Example:
Mean height = 170 cm
SD = 7 cm
68% of people are between 163 and 177 cm
95% are between 156 and 184 cm
99.7% are between 149 and 191 cm
---
14.5 Z-Scores (Beginner Level)
A z-score tells you:
How many standard deviations a value is from the mean.
Formula (you don’t need to memorise yet):
z = (value − mean) ÷ SD
Meaning:
• z = 0 → exactly average
• z = 1 → 1 SD above average
• z = −2 → 2 SD below average
Example:
Mean = 60
SD = 5
Value = 70
z = (70 − 60) / 5 = +2
→ this score is 2 SD above the mean
---
14.6 Example — Exam Scores
Exam marks follow a normal distribution:
Mean = 52
SD = 10
1. A student scores 52 → z = 0 → average
2. A student scores 62 → z = +1 → top ~16%
3. A student scores 42 → z = −1 → bottom ~16%
4. A student scores 72 → z = +2 → top ~2%
This is why universities use standard deviation.
---
14.7 Not All Bell Curves Are Perfect
Real data may be slightly:
• skewed
• bumpy
• stretched
But the normal model is usually close enough for:
• predictions
• grading
• scientific measurement
• manufacturing tolerances
---
14.8 Why Standard Deviation Matters Here
In a normal distribution, SD isn’t just “spread” —
It literally shapes the curve.
Small SD → tall, narrow bell
Large SD → wide, flat bell
---
14.9 Real-World Uses
The normal distribution is used in:
• exam grade boundaries
• scientific error analysis
• quality control
• medical diagnostics
• finance and risk modelling
• IQ tests
• sports performance analysis
• machine learning
It is one of the most practical tools in mathematics.
---
14.10 Exam-Style Questions
1. Heights of trees follow a normal distribution:
mean = 8 m, SD = 0.5 m
Estimate the percentage of trees between 7.5 m and 8.5 m.
2. Test scores:
mean = 45, SD = 12
A student scores 69
(a) Find the z-score
(b) Explain what it means
3. Shoe sizes (normal):
mean = 7, SD = 1
How many SD above the mean is size 9?
4. A machine produces rods that are normally distributed:
mean = 50 mm, SD = 3 mm
What percentage fall between 44 and 56 mm?
5. Reaction times:
mean = 280 ms, SD = 30 ms
Someone responds in 340 ms.
Find their z-score.
---
14.11 Chapter Summary
• The normal distribution is the famous bell curve
• Mean, median, and mode are equal
• 68–95–99.7 rule describes almost all of the data
• Z-scores measure how far from the mean a value is
• Normal distribution appears everywhere
• Standard deviation controls the shape
You now understand one of the most important concepts in all of maths and science.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
The normal distribution is one of the MOST important ideas in probability and statistics.
It appears everywhere in nature, science, exams, psychology, economics — almost everywhere.
It is the famous “bell curve.”
---
14.1 What Is the Normal Distribution?
It is a symmetrical, bell-shaped curve that describes data where:
• most values are near the mean
• fewer values appear far from the mean
• extreme values are rare
Examples that follow a normal distribution:
• heights
• test scores
• measurement errors
• reaction times
• IQ scores
• scientific variations
---
14.2 Why It Happens
The “bell curve” appears when MANY small, random factors add together.
Example:
Your height depends on:
• genetics
• nutrition
• sleep
• childhood health
• hormones
Each factor makes a tiny difference — add them up, and you get a normal distribution.
This is called the Central Limit Theorem, a key concept later in more advanced courses.
---
14.3 Key Features of the Normal Curve
1. Symmetrical
Left side mirrors the right.
2. Mean = Median = Mode
All three are the same at the centre.
3. Most data lies close to the mean
The curve rises sharply at the centre.
4. Tails extend forever (but get very thin)
Outliers are rare but possible.
---
14.4 The 68–95–99.7 Rule
This is the most famous rule in statistics.
For a perfect normal distribution:
• 68% of values lie within 1 standard deviation (SD) of the mean
• 95% lie within 2 SD
• 99.7% lie within 3 SD
This means:
Most people are close to average.
Few are far from average.
Almost nobody is extremely far from average.
Example:
Mean height = 170 cm
SD = 7 cm
68% of people are between 163 and 177 cm
95% are between 156 and 184 cm
99.7% are between 149 and 191 cm
---
14.5 Z-Scores (Beginner Level)
A z-score tells you:
How many standard deviations a value is from the mean.
Formula (you don’t need to memorise yet):
z = (value − mean) ÷ SD
Meaning:
• z = 0 → exactly average
• z = 1 → 1 SD above average
• z = −2 → 2 SD below average
Example:
Mean = 60
SD = 5
Value = 70
z = (70 − 60) / 5 = +2
→ this score is 2 SD above the mean
---
14.6 Example — Exam Scores
Exam marks follow a normal distribution:
Mean = 52
SD = 10
1. A student scores 52 → z = 0 → average
2. A student scores 62 → z = +1 → top ~16%
3. A student scores 42 → z = −1 → bottom ~16%
4. A student scores 72 → z = +2 → top ~2%
This is why universities use standard deviation.
---
14.7 Not All Bell Curves Are Perfect
Real data may be slightly:
• skewed
• bumpy
• stretched
But the normal model is usually close enough for:
• predictions
• grading
• scientific measurement
• manufacturing tolerances
---
14.8 Why Standard Deviation Matters Here
In a normal distribution, SD isn’t just “spread” —
It literally shapes the curve.
Small SD → tall, narrow bell
Large SD → wide, flat bell
---
14.9 Real-World Uses
The normal distribution is used in:
• exam grade boundaries
• scientific error analysis
• quality control
• medical diagnostics
• finance and risk modelling
• IQ tests
• sports performance analysis
• machine learning
It is one of the most practical tools in mathematics.
---
14.10 Exam-Style Questions
1. Heights of trees follow a normal distribution:
mean = 8 m, SD = 0.5 m
Estimate the percentage of trees between 7.5 m and 8.5 m.
2. Test scores:
mean = 45, SD = 12
A student scores 69
(a) Find the z-score
(b) Explain what it means
3. Shoe sizes (normal):
mean = 7, SD = 1
How many SD above the mean is size 9?
4. A machine produces rods that are normally distributed:
mean = 50 mm, SD = 3 mm
What percentage fall between 44 and 56 mm?
5. Reaction times:
mean = 280 ms, SD = 30 ms
Someone responds in 340 ms.
Find their z-score.
---
14.11 Chapter Summary
• The normal distribution is the famous bell curve
• Mean, median, and mode are equal
• 68–95–99.7 rule describes almost all of the data
• Z-scores measure how far from the mean a value is
• Normal distribution appears everywhere
• Standard deviation controls the shape
You now understand one of the most important concepts in all of maths and science.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive