Standard Deviation and the Normal Distribution: The Empirical Rule

How variance, standard deviation (σ), and the 68-95-99.7 rule measure risk and spread.

Measuring Data Dispersion

The mean (μ) tells you the central average; the standard deviation (σ) tells you how tightly or broadly data points cluster around that mean.

The Empirical 68-95-99.7 Rule

For any symmetrical, bell-shaped normal distribution: • 68.27% of all data lies within ±1 standard deviation of the mean • 95.45% of all data lies within ±2 standard deviations • 99.73% of all data lies within ±3 standard deviations

Calculating Z-Scores for Standardization

A Z-score measures exactly how many standard deviations a raw observation is above or below the population mean.

Real-World Quality Control (Six Sigma)

Industrial manufacturers use Six Sigma quality standards, allowing only 3.4 defective parts per million by maintaining process variance within ±6σ.