Normal Distribution
A normal distribution is a common pattern that shows how numerical data is distributed around its average. It is often shaped like a bell and appears in many real-world datasets.
Most values are near the mean, while fewer values are far away from it.
A normal distribution is centered around the mean and becomes less common as values move farther away from the center.
What Is a Normal Distribution?
A normal distribution describes a particular way that data can be spread out.
In a normal distribution, most values are close to the mean. Values become less common as we move farther away from the mean.
Most values
↓
███████████
█████████████████
███████████████████████
█████████████████████████████
────────────────────────────────────
Low Mean High
When we draw this distribution as a graph, it looks similar to a bell. That is why it is also called a bell curve.
A Simple Real-World Example
Imagine we record the heights of a large group of people.
We might see something like:
150 cm 160 cm 165 cm 168 cm 170 cm 171 cm 172 cm 173 cm 175 cm 180 cm 190 cm
We would usually expect many people to have heights close to the average, while very short and very tall values would occur less frequently.
This type of pattern can be represented using a normal distribution.
The Bell Curve
A normal distribution is often visualized using a bell-shaped curve.
▲
│
███
███████
███████████
███████████████
█████████████████████
███████████████████████████
────────────────────────────────────────────
Low Mean High
The highest point is around the mean because that is where values are most concentrated.
As we move away from the mean, the curve becomes lower because fewer values occur there.
Mean and Normal Distribution
The mean is located at the center of a normal distribution.
For example, suppose test scores have:
Mean = 70
Many students may have scores around 70.
60, 65, 68, 69, 70, 71, 72, 75, 80
Scores very far from 70 would generally be less common in a normally distributed dataset.
Standard Deviation and Normal Distribution
Standard deviation tells us how spread out the values are around the mean.
In a normal distribution, we can use standard deviation to understand how far a value is from the mean.
Mean = 70 Standard Deviation = 10
Then:
60 = 1 standard deviation below the mean 70 = Mean 80 = 1 standard deviation above the mean
So standard deviation gives us a useful measurement of distance from the center.
The 68–95–99.7 Rule
One of the most important ideas about a normal distribution is the 68–95–99.7 rule.
For data that follows a normal distribution:
About 68% → within 1 standard deviation of the mean About 95% → within 2 standard deviations of the mean About 99.7% → within 3 standard deviations of the mean
This gives us a quick way to understand where most of the data is located.
Example of the 68–95–99.7 Rule
Suppose exam scores follow a normal distribution with:
Mean = 70 Standard Deviation = 10
One standard deviation from the mean gives:
70 - 10 = 60 70 + 10 = 80 Range = 60 to 80
Approximately 68% of the values would be expected to fall between 60 and 80.
Two standard deviations give:
70 - 20 = 50 70 + 20 = 90 Range = 50 to 90
Approximately 95% of the values would be expected to fall between 50 and 90.
What Does This Mean?
The rule gives us a simple way to understand how unusual a value is.
Mean = 70 Standard Deviation = 10 Score = 72 → Very close to the mean → Common value
Mean = 70 Standard Deviation = 10 Score = 100 → 3 standard deviations above the mean → Much less common
Therefore, standard deviation helps us understand whether a value is close to or far from the typical value.
Z-Score
A z-score tells us how many standard deviations a value is away from the mean.
The basic formula is:
z = (x - mean) / standard deviation
For example:
Score = 80 Mean = 70 Standard Deviation = 10 z = (80 - 70) / 10 z = 1
A z-score of 1 means the value is exactly one standard deviation above the mean.
Negative Z-Score
A z-score can also be negative. This simply means the value is below the mean.
Score = 50 Mean = 70 Standard Deviation = 10 z = (50 - 70) / 10 z = -2
This means the score is two standard deviations below the mean.
Positive z-score → Above the mean Negative z-score → Below the mean z = 0 → Exactly at the mean
Why Is Normal Distribution Important in AI?
Normal distributions appear frequently in statistics and machine learning.
Understanding them helps us reason about numerical data, probabilities, unusual values, and statistical models.
For example, suppose a machine learning dataset contains measurements such as:
Age Height Weight Test Scores Measurement Errors
Some of these types of measurements can approximately follow a normal distribution, depending on how the data was collected.
This allows statistical techniques to make useful assumptions about the data.
Normal Distribution in Machine Learning
Normal distributions are especially useful when working with statistical models and data preprocessing.
For example, after standardizing a feature, we can describe values using their distance from the mean in terms of standard deviations.
Original value
↓
Find mean
↓
Find standard deviation
↓
Calculate z-score
↓
Understand position relative to the data
This gives an AI system a standardized way to represent how far a value is from the center.
Detecting Unusual Values
Normal distributions can also help us identify values that are unusually far from the mean.
For example:
Mean = 100 Standard Deviation = 10 Value = 102 z = 0.2 → Very close to the mean
Mean = 100 Standard Deviation = 10 Value = 160 z = 6 → Extremely far from the mean
A value that is many standard deviations away from the mean may deserve further investigation.
Normal Distribution Is Not Every Dataset
An important point: not every dataset follows a normal distribution.
Real-world data can have different shapes.
Normal distribution → Bell-shaped Skewed distribution → One side has a longer tail Uniform distribution → Values are more evenly spread
Therefore, we should not automatically assume that every dataset is normally distributed.
We first need to understand the data and check whether the assumption makes sense.
Normal Distribution in Python
Python libraries can be used to work with normal distributions. For example, NumPy can generate normally distributed random values.
import numpy as np
numbers = np.random.normal(
loc=70,
scale=10,
size=10
)
print(numbers)
Here:
loc=70 → Mean scale=10 → Standard deviation size=10 → Generate 10 values
This is useful when experimenting with statistical concepts and building machine-learning examples.
The Complete Picture
You can connect everything we have learned in this lesson:
Mean ↓ Center of the data Standard Deviation ↓ Spread around the mean Normal Distribution ↓ Bell-shaped pattern around the mean Z-Score ↓ How many standard deviations a value is from the mean
These concepts work together and will appear repeatedly in statistics and machine learning.
A normal distribution is a bell-shaped pattern where most values are close to the mean.
The mean tells us the center, standard deviation tells us the spread, and a z-score tells us how many standard deviations a value is from the mean. The 68–95–99.7 rule gives us a useful way to understand how values are distributed around the mean. Normal distributions are important in AI because they provide a useful statistical model for understanding numerical data, standardized values, and unusual observations.