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

•  When controlling ongoing processes by finding and correcting problems as they occur.
•  When predicting the expected range of outcomes from a process.
•  When determining whether a process is stable (in statistical control).
•  When analyzing patterns of process variation from special causes (non-routine events) or common causes (built into the process).
•  When determining whether your quality improvement project should aim to prevent specific problems or to make fundamental changes to the process.

## 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

• A single point outside the control limits. In Figure 1, point sixteen is above the UCL (upper control limit).
• Two out of three successive points are on the same side of the centerline and farther than 2 σ from it. In Figure 1, point 4 sends that signal.
• Four out of five successive points are on the same side of the centerline and farther than 1 σ from it. In Figure 1, point 11 sends that signal.
• A run of eight in a row are on the same side of the centerline. Or 10 out of 11, 12 out of 14 or 16 out of 20. In Figure 1, point 21 is eighth in a row above the centerline. • Continue to plot data as they are generated. As each new data point is plotted, check for new out-of-control signals.
• When you start a new control chart, the process may be out of control. If so, the control limits calculated from the first 20 points are conditional limits. When you have at least 20 sequential points from a period when the process is operating in control, recalculate control limits.

## Types of Control Charts

### 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