Lean Six Sigma Matters
๐Ÿ“ Statistics

Statistics โ€” The Quantitative
Backbone of Six Sigma

Six Sigma is built on data and statistical thinking. This reference covers the key statistical concepts, formulas, and tools every Green Belt practitioner needs โ€” from basic descriptive statistics through to process capability and control charts.

Key Formulas & Concepts

These foundational statistical measures describe the central tendency and spread of your process data โ€” the starting point for any Six Sigma analysis.

๐Ÿ“Š Mean (Average)

xฬ„ = ฮฃx / n

The arithmetic average of a data set. The most common measure of central tendency.

ฮฃx = sum of all values  |  n = number of values

๐Ÿ“ Standard Deviation

ฯƒ = โˆš[ ฮฃ(x - xฬ„)ยฒ / n ]

Measures the spread or dispersion of data around the mean. Small ฯƒ = consistent process; large ฯƒ = high variation.

x = each value  |  xฬ„ = mean  |  n = count

๐Ÿ“ Variance

ฯƒยฒ = ฮฃ(x - xฬ„)ยฒ / n

The square of the standard deviation. Used in many statistical calculations. Standard deviation is more intuitive as it shares the same units as the data.

ฯƒยฒ = variance  |  ฯƒ = โˆšvariance

๐ŸŽฏ Process Capability โ€” Cp

Cp = (USL - LSL) / 6ฯƒ

Measures the potential capability of a process โ€” how well it could perform if perfectly centred within specification limits.

USL = upper spec limit  |  LSL = lower spec limit  |  Target: Cp โ‰ฅ 1.33

๐ŸŽฏ Process Capability โ€” Cpk

Cpk = min[(USL-xฬ„)/3ฯƒ, (xฬ„-LSL)/3ฯƒ]

Accounts for both process spread AND centering. Always โ‰ค Cp. The most important capability index for real process performance.

Target: Cpk โ‰ฅ 1.33  |  World class: Cpk โ‰ฅ 1.67

๐Ÿ“‰ Z-Score

Z = (x - ฮผ) / ฯƒ

Measures how many standard deviations a data point is from the mean. Used to calculate probabilities and convert to sigma levels.

x = value  |  ฮผ = population mean  |  ฯƒ = std deviation

Understanding the Normal Curve

The Empirical Rule (68-95-99.7)

For normally distributed data, a predictable proportion of values fall within each standard deviation band from the mean.

68.3%
within ยฑ1ฯƒ
95.4%
within ยฑ2ฯƒ
99.7%
within ยฑ3ฯƒ
99.9997%
within ยฑ6ฯƒ
1

Why the Normal Distribution Matters

Many natural and process measurements are approximately normally distributed, making it the most important distribution in Six Sigma statistics.

2

Central Limit Theorem

Sample means tend toward a normal distribution as sample size increases โ€” even if the underlying population is not normal. This underpins control charts.

3

Testing for Normality

Before applying many statistical tests, check whether your data is normally distributed using a histogram, normal probability plot, or Anderson-Darling test.

4

Non-Normal Data

Not all process data is normal โ€” consider Weibull, Poisson, or binomial distributions for reliability, count, or attribute data respectively.

Control Chart Selection Guide

Choosing the right control chart depends on whether your data is continuous (variable) or discrete (attribute), and your sample size. Use this guide to select the appropriate chart for your process.

๐Ÿ“ˆ

X-bar & R Chart

Monitors process mean and range for continuous data with subgroup sizes 2โ€“10.

Variable data
๐Ÿ“‰

X-bar & S Chart

Monitors process mean and standard deviation. Preferred for larger subgroup sizes (n > 10).

Variable data
ใ€ฐ๏ธ

I-MR Chart

Individuals and Moving Range chart for continuous data with subgroup size of 1 โ€” common in process industries.

Variable data
๐Ÿ”ข

p Chart

Monitors the proportion of defective items in a sample. Used when subgroup sizes vary.

Attribute data
๐Ÿ”ข

np Chart

Monitors the number of defective items. Used when subgroup sizes are constant.

Attribute data
๐Ÿ”ข

c Chart

Monitors the number of defects per unit when sample size is constant.

Attribute data
๐Ÿ”ข

u Chart

Monitors defects per unit when sample sizes vary. Most flexible attribute chart.

Attribute data
๐Ÿ“Š

CUSUM / EWMA

Advanced charts that detect small, sustained process shifts more quickly than traditional Shewhart charts.

Advanced SPC

Statistics Resources & Further Reading

Curated links from the site and trusted external sources to support your statistical learning.

Need Help Applying Statistics to Your Project?

From Green Belt training to hands-on statistical analysis โ€” I can help your team understand and apply the right tools for your data.