God, grant me the serenity to accept the things I cannot change, The courage to change the things I can, And the wisdom to know the difference. (Serenity Prayer, Reinhold Niebuhr)

Statistical Process Control (SPC)

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).

The control chart is a graph used to study how a process changes over time. Data are plotted in time order. A control chart always has a central line for the average, an upper line for the upper control limit and a lower line for the lower control limit. These lines are determined from historical data. By comparing current data to these lines, you can draw conclusions about whether the process variation is consistent (in control) or is unpredictable (out of control, affected by special causes of variation) (ASQ).

Control charts for variable data are used in pairs. The top chart monitors the average, or the centering of the distribution of data from the process. The bottom chart monitors the range, or the width of the distribution. If your data were shots in target practice, the average is where the shots are clustering, and the range is how tightly they are clustered. Control charts for attribute data are used singly (ASQ).

When to Use a Control Chart

Control Chart Basic Procedure

  1. Choose the appropriate control chart for your data.
  2. Determine the appropriate time period for collecting and plotting data.
  3. Collect data, construct your chart and analyze the data.
  4. Look for “out-of-control signals” on the control chart. When one is identified, mark it on the chart and investigate the cause. Document how you investigated, what you learned, the cause and how it was corrected. 

Out-of-control signals

Types of Control Charts

X-Bar Chart - Average Chart

When population standard deviation (sigma) is known:

where z = the number of normal standard deviations, and sigmax = sigma / Sqrt[n] n = sample size

When population standard deviation (sigma) is not known:

R Chart - Range Chart

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

c-Chart - Count (defective) Chart

Processes In Statistical Control

Four guidelines are used as a basic test for randomness in control charts (Westcott 2006):

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

Product Variation

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

Measurement Variation

Measurement Variation (σm) = reproducibility(ops)+repeatability(gauge)

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

Process Capability

Process Capability Ratio (Cp)

Process Capability Index (Cpk)