Empirical Rule Calculator
Apply the 68-95-99.7 rule for normal distribution analysis
Calculate Empirical Rule Intervals
The average value of the normal distribution
The measure of spread in the distribution (must be positive)
Empirical Rule Results
One Standard Deviation
68% of data falls between 85.00 and 115.00
Two Standard Deviations
95% of data falls between 70.00 and 130.00
Three Standard Deviations
99.7% of data falls between 55.00 and 145.00
Formulas Used
68% Rule:
[μ - σ, μ + σ]
95% Rule:
[μ - 2σ, μ + 2σ]
99.7% Rule:
[μ - 3σ, μ + 3σ]
Normal Distribution Visualization
Interpretation:
- • 68% of values lie within 1 standard deviation of the mean
- • 95% of values lie within 2 standard deviations of the mean
- • 99.7% of values lie within 3 standard deviations of the mean
Example: IQ Scores
Problem Setup
Scenario: Intelligence Quotient (IQ) scores are normally distributed
Mean (μ): 100
Standard Deviation (σ): 15
Empirical Rule Application
68% of people have IQ between: 100 - 15 = 85 and 100 + 15 = 115
95% of people have IQ between: 100 - 30 = 70 and 100 + 30 = 130
99.7% of people have IQ between: 100 - 45 = 55 and 100 + 45 = 145
Quick Reference
1 Standard Deviation
μ ± σ contains ~68% of data
2 Standard Deviations
μ ± 2σ contains ~95% of data
3 Standard Deviations
μ ± 3σ contains ~99.7% of data
Common Applications
Quality control in manufacturing
Identifying outliers in datasets
Risk assessment and prediction
Educational testing and scoring
Financial modeling and analysis
Understanding the Empirical Rule
What is the Empirical Rule?
The empirical rule, also known as the 68-95-99.7 rule or three-sigma rule, is a statistical principle that applies to normally distributed data. It provides a quick way to understand how data is distributed around the mean.
Key Assumptions
- •Data follows a normal (bell-shaped) distribution
- •Mean and standard deviation are known
- •Distribution is symmetric about the mean
- •Sample size is sufficiently large
Mathematical Foundation
Mean Formula: μ = Σxᵢ / n
Standard Deviation: σ = √[Σ(xᵢ - μ)² / (n-1)]
Empirical Rule Intervals:
- • 68%: [μ - σ, μ + σ]
- • 95%: [μ - 2σ, μ + 2σ]
- • 99.7%: [μ - 3σ, μ + 3σ]
Practical Benefits
The empirical rule provides a quick assessment of data distribution without complex calculations. It's especially useful for quality control, outlier detection, and risk assessment in various fields.