Three-Sigma Limits

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Three-Sigma Limits

What is a three‑sigma limit?

A three‑sigma limit defines the range of values within three standard deviations (σ) of a process mean. For a normally distributed process, about 99.73% of observations lie within ±3σ of the mean. Three‑sigma limits are commonly used as upper and lower control limits on control charts to judge whether a process is in statistical control.

Why it matters

  • Provides a practical threshold to detect unusual or special‑cause variation.
  • Helps distinguish normal, random variation (in control) from signals that require investigation.
  • Widely applied in manufacturing, quality control, finance, healthcare, and anomaly detection.

Control charts and variation

Control charts (Shewhart charts) plot process measurements over time with a center line (the mean) and control limits, typically set at mean ±3σ. The theory assumes some inherent random variability even in well‑designed processes:
– Points within ±3σ → likely due to common (random) causes.
– Points outside ±3σ → indicate potential special causes that warrant investigation.

How to calculate three‑sigma limits (example)

Suppose a manufacturing process yields the following 10 measurements:
8.4, 8.5, 9.1, 9.3, 9.4, 9.5, 9.7, 9.7, 9.9, 9.9

  1. Compute the mean:

  2. Sum = 93.4 → mean = 93.4 / 10 = 9.34

  3. Compute variance (population variance, dividing by N):

  4. Sum of squared deviations = 2.564

  5. Variance = 2.564 / 10 = 0.2564

  6. Standard deviation:

  7. σ = sqrt(0.2564) ≈ 0.5064

  8. Three‑sigma band:

  9. 3σ = 3 × 0.5064 ≈ 1.5192
  10. Upper control limit (UCL) = mean + 3σ ≈ 9.34 + 1.5192 = 10.86
  11. Lower control limit (LCL) = mean − 3σ ≈ 9.34 − 1.5192 = 7.82

In this example, all observations (max 9.9) lie inside the ±3σ limits, suggesting the process is within expected variation.

When to use three‑sigma limits

Use three‑sigma limits to:
– Set control limits on process control charts
– Identify and analyze outliers
– Distinguish normal vs. unusual variation
– Monitor quality and detect anomalies
– Estimate probabilities of extreme outcomes in normally distributed data

Three‑sigma vs. Six‑sigma

  • Three‑sigma: mean ±3σ, covers ≈99.73% of a normal distribution. Useful for general control and monitoring where some variation is acceptable.
  • Six‑sigma: aims for much tighter performance (±6σ in the idealized normal model) and far lower defect rates. In industry practice, Six Sigma methodology combines statistical rigor, process improvement tools, and often an assumed process shift to pursue near‑zero defects. Six Sigma is typically applied where extremely high accuracy and low error rates are required.

Key definitions

  • Standard deviation (σ): a measure of spread; the positive square root of the variance. It quantifies how much observations typically differ from the mean.
  • Bell curve (normal distribution): a symmetric, bell‑shaped distribution where probabilities are defined by standard deviations from the mean. The probability mass within ±1σ, ±2σ, and ±3σ follows the empirical rule (≈68.27%, 95.45%, 99.73%).

Bottom line

Three‑sigma limits (mean ±3σ) provide a practical, widely accepted threshold for monitoring process stability. Data points outside this band are rare under normal variation and should trigger investigation for special causes. Use three‑sigma control limits to maintain process quality, detect anomalies, and guide corrective actions.