Normal Distribution
The bell curve.
Key idea
The bell curve.
In depth
The bell curve with mean μ and standard deviation σ, symmetric about μ. It appears naturally from the sum of many small effects (central limit theorem). Its density is (1/(σ√(2π)))e^(−(x−μ)²/(2σ²)).
Worked example
Heights with μ = 170 cm and σ = 8 cm: about 68% fall between 162 and 178.
Formula
Real-world uses
- Modelling heights, measurement errors and test scores in psychology and manufacturing
Common mistake
Assuming all data are normal.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Normal Distribution?
The bell curve. It is a topic at the advanced level in Probability & Statistics.
What is the formula for Normal Distribution?
The key formula is f(x) = 1/(σ√2π) e^(−(x−μ)²/2σ²).
Where is Normal Distribution used in real life?
Real-world examples: Modelling heights, measurement errors and test scores in psychology and manufacturing.
What should I learn before Normal Distribution?
Make sure you are comfortable with Uniform Distribution, the previous topic in Probability & Statistics.
What is a common mistake with Normal Distribution?
Assuming all data are normal.
How can I practise Normal Distribution?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Normal Distribution?
Move on to Empirical Rule (68–95–99.7) next.
Related advanced topics in Probability & Statistics
- Quartiles and IQR – Quartiles split data into four parts; IQR = Q3 − Q1.
- Box Plots – Five-number summary visualised.
- Outliers – Values beyond 1.5 IQR from the quartiles.
- Variance – Average squared deviation from the mean.
- Standard Deviation – Square root of variance; typical spread.
- Sample vs Population Variance – Divide by n−1 for sample variance.