Normal Distribution Formula
Understand the normal distribution (bell curve) with f(x) = (1/σ√(2π)) × e^(-(x-μ)²/(2σ²)).
The most important distribution in statistics.
The Formula
The normal distribution, also called the bell curve, describes how data is distributed around the mean. Most values cluster near the center, with fewer values at the extremes. It is the foundation of many statistical methods.
Variables
| Symbol | Meaning |
|---|---|
| f(x) | Probability density at value x |
| μ (mu) | Mean (center of the curve) |
| σ (sigma) | Standard deviation (width of the curve) |
| π (pi) | Pi, approximately 3.14159 |
| e | Euler's number, approximately 2.71828 |
The 68-95-99.7 Rule
For any normal distribution:
- 68% of data falls within 1 standard deviation of the mean (μ ± σ)
- 95% of data falls within 2 standard deviations (μ ± 2σ)
- 99.7% of data falls within 3 standard deviations (μ ± 3σ)
Example 1
Adult IQ scores are normally distributed with μ = 100 and σ = 15. What range covers 95% of all scores?
Using the 68-95-99.7 rule, 95% falls within μ ± 2σ
Lower bound = 100 - (2 × 15) = 100 - 30 = 70
Upper bound = 100 + (2 × 15) = 100 + 30 = 130
95% of IQ scores fall between 70 and 130.
Example 2
A factory produces bolts with a mean length of 50 mm and standard deviation of 0.5 mm. What percentage of bolts are between 49.5 mm and 50.5 mm?
49.5 mm is μ - 1σ = 50 - 0.5
50.5 mm is μ + 1σ = 50 + 0.5
This range covers μ ± 1σ
About 68% of bolts fall between 49.5 mm and 50.5 mm.
When to Use It
Use the normal distribution when:
- Analyzing naturally occurring data (heights, weights, test scores, measurement errors)
- Determining the probability of a value occurring within a range
- Setting quality control limits in manufacturing
- Understanding the basis for z-scores, confidence intervals, and hypothesis testing