Statistical Process Control

Statistical Process Control (SPC) is an effective method of monitoring a process through the use of control charts. Control charts enable the use of objective criteria for distinguishing background variation from events of significance based on statistical techniques (Wikipedia 2009).

X-Bar Chart - Average Chart

When population standard deviation (sigma) is known:

  • Central Line = X-bar-bar (average of the averages)
  • UCL = X-bar-bar + z * sigmax
  • LCL = X-bar-bar - z * sigmax
where z = the number of normal standard deviations, and sigmax = sigma / Sqrt[n] n = sample size

When population standard deviation (sigma) is not known:

  • Central Line = X-bar-bar (average of the averages)
  • UCL = X-bar-bar + A2 (from table) * R-bar
  • LCL = X-bar-bar - A2 (from table) * R-bar

R Chart - Range Chart

  • Central Line = R-bar (average of the ranges)
  • UCL = D4 (from table) * R-bar
  • LCL = D3 (from table) * R-bar

Control Limit Factor Table for X-Bar & R-Charts

Control Limit Factor Table
# Observations in Sample (n) X-Chart Factors for Control Limits (A2) R-Chart Factors for Control Limits (D3) R-Chart Factors for Control Limits (D4) d2
2 1.88 0.00 3.27 1.128
3 1.02 0.00 2.57 1.693
4 0.73 0.00 2.28 2.059
5 0.58 0.00 2.11 2.326
6 0.48 0.00 2.00 2.534
7 0.42 0.08 1.92 2.704
8 0.37 0.14 1.86 2.847
9 0.34 0.18 1.82 2.970
10 0.31 0.22 1.78 3.078
11 0.29 0.26 1.74  
12 0.27 0.28 1.72  
13 0.25 0.31 1.69  
14 0.24 0.33 1.67  
15 0.22 0.35 1.65  
16 0.21 0.36 1.64  
17 0.20 0.38 1.62  
18 0.19 0.39 1.61  
19 0.19 0.40 1.60  
20 0.18 0.41 1.59  

P-Chart - Proportion (defective) Chart

  • Central Line = p-bar (average of the proportions)
  • UCL = p-bar + 3 * sp
  • LCL = p-bar - 3 * sp
  • sp = Sqrt [ (p-bar * (1-p-bar))/n ]

c-Chart - Count (defective) Chart

  • Central Line = c-bar (average of the proportions)
  • UCL = c-bar + 3 * Sqrt(c-bar)
  • LCL = p-bar - 3 * Sqrt(c-bar)

Processes In Statistical Control

Four guidelines are used as a basic test for randomness in control charts (Westcott 2006):
  • The points should have no visible pattern
  • Approximately two-thirds of the points should be close to the centerline (within +/- 1 standard deviation)
  • Some points should be close to the outer limits
  • An approximately equal number of points should fall above and below the centerline.

If all four criteria are met, then the process is likely to be in control.

Observed, Product & Measurement Variation

Observed Variation

σo (observed variation) = Sqrt[σp^2 + σm^2], where
  • σo = observed variation
  • σp = product variation
  • σm = measurement variation

Product Variation

σp (product variation) = Sqrt[σo^2 - σm^2]

Measurement Variation

Measurement Variation (σm) = reproducibility(ops)+repeatability(gauge)
  • σm^2 = σrepro^2 + σrepeat^2
  • Measurement Variation estimate (σm) = [R-bar / d2]

Reproducibility Variation

σrepro^2 = σm^2 - σrepeat^2
One method: use most accurate operator's measurement variation (if in control) as an estimate for σrepeat^2 The percent of the measurement error due to operators can be estimated with σrepro^2 / σm^2
Signal to Noise Ratio (SNR) = σo / σm
  • If SNR >= 10 measurement error is negligible
  • If SNR >= 4.5 but <=10 measurement error may affect. May want to take at least two measurements of each sampling unit and average them. Measurement error should be subtracted form the observed variation to estimate process variation
  • If SNR < 4.5 measurement process is inadequate

Process Capability

Process Capability Ratio (Cp)

  • Process capability ratio (Cp) = (UTL - LTL) / 6 * sigma
  • UTL = Upper Tolerance Limit
  • LTL = Lower Tolerance Limit
  • sigma = process standard deviation
  • sigma = Sqrt[(sum(x-bar - xi)^2)/n-1)]
  • x-bar = the sample mean
  • xi = value for the ith observation
  • n = sample size

Process Capability Index (Cpk)

  • Process capability index (Cpk) = Min[(u-LTL)/3sigma, (UTL-u)/3sigma]
  • u = process mean

References

Westcott, 2006 - The Certified Manager of Quality/Organizational Excellence Handbook, Third Edition, ASQ Press.