Normal Distribution
The foundation of statistical analysis - a symmetric, bell-shaped distribution that forms the basis for most statistical methods and quality control applications.
Distribution Overview
What is the Normal Distribution:
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric around its mean. It forms the classic bell-shaped curve and serves as the foundation for most statistical methods due to the Central Limit Theorem. This distribution occurs naturally in many phenomena where multiple independent factors contribute to the outcome.
Key Properties:
- • Bell-shaped and perfectly symmetric
- • Mean, median, and mode are equal
- • Defined by two parameters: μ (mean) and σ (standard deviation)
- • Total area under the curve equals 1
- • Follows the 68-95-99.7 rule
Applications:
- • Quality control and Six Sigma
- • Process capability analysis
- • Measurement errors and tolerances
- • Human characteristics (height, weight)
- • Financial modeling and risk assessment
Mathematical Properties:
Normal Probability Density Function
= population mean (location parameter)
= population standard deviation (scale parameter)
Statistical Characteristics
Key Statistics:
- • Mean: μ
- • Median: μ
- • Mode: μ
- • Variance: σ²
- • Skewness: 0 (perfectly symmetric)
- • Kurtosis: 3 (mesokurtic)
68-95-99.7 Rule:
- • 68% within ±1σ of mean
- • 95% within ±2σ of mean
- • 99.7% within ±3σ of mean
- • Forms basis for quality control limits
- • Critical for process capability calculations
Reading the Normal Density Curve
Example Distribution Fit & Signal Interpretation
Reading a Normality Check:
A sample of 12 measurements yields x=24.6 and ˆσ=1.1. Standardizing an observed value of 26.2 gives:
Z-Score Standardization
= observed value
= sample mean
= sample standard deviation
Normal-Looking Signal:
- • Histogram is symmetric and bell-shaped
- • Skewness near 0, kurtosis near 3
- • Shapiro-Wilk p-value > 0.05
- • Q-Q plot points fall close to the line
Non-Normal Signal:
- • Heavy skew or bimodal shape
- • Extreme kurtosis (heavy/light tails)
- • Formal test p-value ≤ 0.05
- • Points curve away from Q-Q reference line
sixsigmacenter.com Implementation Guide
Example Data Layout for Normal Distribution Analysis
Sample data with Z-scores and cumulative probabilities for normal distribution analysis. Used for testing normality assumptions and calculating probabilities under the normal curve. Sample Size Recommendation: Minimum 30-50 observations for reliable normal distribution parameter estimation and probability calculations.
This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.