Normal Distribution Basics
In many real-world situations, data follows a predictable pattern. Heights, exam scores, test results, and measurement errors often cluster around an average value.
The normal distribution is a probability distribution that describes this common pattern.
What Is a Normal Distribution?
A normal distribution is a continuous probability distribution that is symmetric and centered around its mean.
Most values lie close to the average, and fewer values appear as we move away from the center.
The shape of a normal distribution is commonly called a bell-shaped curve.
Key Characteristics of the Normal Distribution
- The curve is symmetric about the mean
- Mean, median, and mode are equal
- The total area under the curve is 1
- Values closer to the mean are more frequent
Real-World Examples
Normal distribution appears naturally in many situations, such as:
- Heights of people in a population
- Scores on standardized exams
- Measurement errors in experiments
- Manufacturing variations
Understanding the Bell Curve
The center of the bell curve represents the mean. As we move away from the mean, the frequency of values decreases.
Because of symmetry:
- 50% of the data lies on each side of the mean
- Extreme values are rare
The Empirical Rule (68–95–99.7 Rule)
One of the most important properties of the normal distribution is the Empirical Rule.
It tells us how data is spread around the mean:
| Range | Percentage of Data |
|---|---|
| Within 1 standard deviation | About 68% |
| Within 2 standard deviations | About 95% |
| Within 3 standard deviations | About 99.7% |
Numerical Example
Suppose the average score on a test is 70 with a standard deviation of 10.
- About 68% of students scored between 60 and 80
- About 95% scored between 50 and 90
- About 99.7% scored between 40 and 100
This helps us quickly estimate where most values lie without complex calculations.
Why the Normal Distribution Is Important
- Many natural processes follow this pattern
- It simplifies probability calculations
- It forms the foundation for statistical inference
- Many tests assume normality
Quick Check
In a normal distribution, what is the relationship between mean, median, and mode?
Practice Quiz
Question 1:
What percentage of data lies within one standard deviation of the mean?
Question 2:
Is the normal distribution symmetric?
Question 3:
What happens to frequency as we move away from the mean?
Mini Practice
A dataset follows a normal distribution with a mean of 50 and a standard deviation of 5.
- Between which values does about 68% of the data lie?
What’s Next
In the next lesson, we will study Z-Scores and Standardization, which allow us to compare values from different distributions.