Normal Distribution
Normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve.
What is normal distribution?
The normal distribution, also called the Gaussian distribution, describes how values of a continuous random variable are distributed around a mean. Most values cluster near the mean, with probabilities decreasing symmetrically on both sides.
Key properties of normal distribution
- The distribution is symmetric around the mean
- Mean, median, and mode are equal
- The curve is bell-shaped
- The total area under the curve equals 1
Normal distribution formula
The probability density function of the normal distribution is:
f(x) = (1 / (σ √(2π))) × e−(x − μ)² / (2σ²)
Example
Suppose test scores follow a normal distribution:
Mean (μ) = 70
Standard deviation (σ) = 10
Most students score between 60 and 80.
Mean (μ) = 70
Standard deviation (σ) = 10
Most students score between 60 and 80.
How to interpret normal distribution
Normal distribution allows us to calculate probabilities for ranges of values. Using standard deviation intervals:
- ≈ 68% of values fall within ±1σ
- ≈ 95% within ±2σ
- ≈ 99.7% within ±3σ
Why is normal distribution important?
Normal distribution is fundamental in statistics, data science, finance, quality control, and natural sciences. It underpins hypothesis testing, confidence intervals, and many statistical models.