Control charts

Goal

The control chart tells whether the process is stable: whether its performance is predictable. It plots each value at its date and compares it with limits that the process itself produced.

In 1931 Walter Shewhart distinguished two sources of variation. Common cause variation comes from the nature of the process: the values fluctuate at random around the mean. Special cause variation comes from an event outside the ordinary running of the process: a breakdown, a power cut, a missing raw material. Shewhart and the CBOK call this cause an "assignable cause". Total variation is the sum of the two. The chart separates one from the other.

The technique supports a decision: intervene or leave the process alone. According to the PMBOK Guide, the project manager and the stakeholders use the calculated limits to set the points at which they take corrective action. The technique delivers the chart and the table of its values. The table states the period over which the team calculated the limits. Each signal gets its date, the cause found and the action taken.

Usage

When to use it

• Indicator recorded every period: decide whether this week's value is a signal before asking for explanations.
• Project monitoring: track cost and schedule variances, a volume or the frequency of scope changes.
• High error rate or erratic results: measure the variation before reducing it.
• Before a Pareto chart or a histogram: check that the process stayed stable over the period analysed.
• Before assessing capability: a process can be judged capable only once it is stable.
• After an improvement: hold the gain in the Control phase of a DMAIC.

When not to use it

• Measurements with no time order: a batch measured in one go has no time order. Describe its distribution with a histogram.
• A question of conformance to the customer's need: the chart tells whether the process is stable. Compare with the specification limits in a process capability study.

Description

The elements of the chart

The horizontal axis shows time. The vertical axis shows the measurement. Each point is the value of one period: a day of production, a week of processing time, a month of cost variance. A line joins the points in the order they were recorded.

The centre line (CL) is the mean of the values in the baseline period. The upper control limit (UCL) and the lower control limit (LCL) sit three standard deviations either side of the centre line. The standard deviation σ is estimated from the data on the chart. NIST gives the reason for three standard deviations: as long as the process stays stable, almost all points fall between the limits. For normally distributed data, the probability that a point exceeds a given limit is about one in a thousand.

Two pairs of dashed lines mark one and two standard deviations on each side of the centre line. They divide the chart into the zones that the detection tests use.

Below the individuals chart sits the moving range chart (mR). The moving range of a period is the absolute difference between its value and the value of the previous period. The centre line of this chart is the mean of the moving ranges. The chart has only an upper limit. It shows the jumps from one period to the next.

Control limits and specification limits

Specification limits come from the requirements: they set the lowest value (LSL, lower specification limit) and the highest value (USL, upper specification limit) that the customer, the contract or the plan accepts. Control limits come from the data: they state what the process produces when nothing external disturbs it. The PMBOK Guide distinguishes the two.

Control limits ask whether the process is stable. Specification limits ask whether the result is acceptable. A process can be stable and produce output outside tolerance: the chart signals nothing because the defect comes from the process itself. Only a change to the process corrects it. A process can also stay within tolerance and give a signal: the result is acceptable today, but nothing guarantees the next period. A capability study compares the control limits with the tolerance.

Building an individuals chart

The XmR chart tracks one individual value (X) per period and its moving range (mR). The XmR chart suits management data: Donald Wheeler observes that such data most often arrive one value at a time.

1. Choose the measurement and the process. Write the definition of the measurement: the unit, the counting rule, the source. When several problems compete for attention, a Pareto chart points to the category to track.
2. Record the values in time order. Each value must be comparable with the previous one: same definition, same source, same process. Wheeler states the rule: whoever builds an XmR chart does not knowingly put data of different kinds on it.
3. Calculate the centre line and the moving ranges. The centre line is the mean of the values. A series of n values gives n − 1 moving ranges. Take their mean.
4. Calculate the limits. UCL and LCL equal X̄ ± 2.66 × mR̄. The moving range limit equals 3.268 × mR̄. Wheeler gives these formulas as the default for the XmR chart.
5. Draw the two charts. The top chart shows the values, the centre line, the two limits and the lines at one and two standard deviations. The bottom chart shares the time axis. It shows the moving ranges, their mean and their upper limit.
6. Read the chart. Apply the detection tests to the individuals chart. Compare each moving range with its limit.
7. Respond. Look for the special cause of each signal. Without a signal, leave the process alone.
8. Keep plotting. Each new value goes on the chart against the same limits. The BPM CBOK ends its section with this instruction: never stop plotting the chart. The team keeps the same limits as long as the process has not changed. After a deliberate change, it calculates new limits from the values that follow.

On the XmR chart, the standard deviation is estimated from the moving ranges. NIST estimates it as mR̄ / 1.128. Three standard deviations then equal 3 / 1.128 × mR̄ = 2.66 × mR̄: the factor 2.66 places the limits three standard deviations from the centre line. The factor 3.268 of the moving range limit is a tabulated constant. The standard deviation calculated over all the values is the wrong estimate: it absorbs the shifts that the chart must detect, widens the limits and hides the signals.

Reading the chart

Each test in the BPM CBOK detects a pattern that ordinary variation rarely produces.

1. Test 1. One point falls beyond a control limit: more than three standard deviations from the centre line.
2. Test 2. At least two of three successive values fall more than two standard deviations from the centre line on the same side.
3. Test 3. At least four of five successive values fall more than one standard deviation from the centre line on the same side.
4. Test 4. At least eight successive values fall on the same side of the centre line.

Tests 2 to 4 detect a process that has shifted without any point crossing a limit. According to NIST, a non-random pattern signals a problem even within the limits.

According to the CBOK, the tests require statistically independent successive values so that ordinary variation spreads symmetrically around the mean. Cost incurred to date contains the value of the previous period by construction: a cumulative series breaks this condition. The team then plots the cost of each period.

On the moving range chart, a point above the upper limit signals a jump between two periods larger than any the process produces on its own.

Responding to a signal

A signal says when something happened. It does not say what. The team looks for the special cause around the date of the signal: the operations log, changes of staff or supplier, incidents, maintenance work. The CBOK gives as examples an equipment breakdown, a power surge, a raw material shortage, a strike or weather that stops work.

The CBOK distinguishes two kinds of special cause. A transient cause occurs rarely and unexpectedly: a power cut in an urban area where cuts are rare. The team treats it as a risk and plans its response. A persistent cause recurs often and predictably in a process that makes no allowance for it: the same cut in a remote region where cuts are common. The team may need to adjust its forecasting models and the expected capability of the process. Once the special causes are eliminated and prevented from recurring, only common cause variation remains: the process is stable and predictable.

Without a signal, the point belongs to ordinary variation. An adjustment based on that point alone targets no identified cause. Reducing common cause variation means changing the process: the method, the tooling, the rules. An improvement project takes this on, for example a DMAIC. Its control plan then sets the planned response to each signal.

The other charts

The BPM CBOK lists the types of control chart: average and range (X̄-R), average and standard deviation (X̄-S), individual values and moving range (XmR), individual values and median moving range, moving average and moving range (MAMR), c charts, u charts and Z charts. It gives no rule for choosing between them.

Common mistakes

Plotting the tolerance instead of the control limits. A chart that shows only the specification limits answers the conformance question. It says nothing about stability. A process that drifts within tolerance goes unnoticed on it.

Reacting to every point. A low value without a signal triggers a meeting, an explanation and sometimes an adjustment. The team spends its time on ordinary variation. The chart loses its purpose.

Calculating the limits with the overall standard deviation. The limits widen by as much as the process has moved. Wheeler calls this practice an error. On the thirty values of the well, the spreadsheet function STDEV.S returns 7.32. The lower limit then falls to 60.73 − 3 × 7.32 = 38.8. Day 15 (39) stays above it and the only signal in the series disappears. With σ = mR̄ / 1.128 = 6.91, the signal remains.

Mixing sources. Two teams, two sites or two types of request on one chart inflate the moving ranges. The limits move apart and the signals of each source get lost.

Recalculating the limits every period. Limits that follow the data also follow the drift. The chart stops detecting it.

AI considerations

An assistant writes the spreadsheet formulas or the Python script that calculates the moving ranges, the limits and the two charts. The analyst states the method in the prompt. The analyst checks that the script takes the moving ranges as absolute values, estimates the standard deviation from the mean moving range and plots the chart in date order. A standard deviation function called on the whole column returns the overall standard deviation. The XmR chart does not use it.

A language model given thirty values pasted into the conversation can return a wrong mean or a wrong limit without saying so. The analyst calculates in the spreadsheet or in code and keeps the conversation for writing the formulas.

The model readily proposes causes for a signal. It does not know that a supplier was replaced on the 14th or that a team was short-staffed that day. Its causes remain leads: the facts recorded on the day of the signal confirm or rule them out.

Processing times or volumes linked to named employees are personal data under the Federal Act on Data Protection (FADP). Before passing them to an external AI service, the analyst removes the identifiers: the chart needs only the date and the value.

Examples

The BPM CBOK illustrates the XmR chart with the daily output of an oil well over thirty days.

The thirty values have a mean of 60.73. The 29 moving ranges have a mean of 7.79. The estimated standard deviation is 7.79 / 1.128 = 6.91. The two means give 60.73 ± 2.66 × 7.79: limits of 81.5 and 40.0. The moving range limit is 3.268 × 7.79 = 25.5.

Only day 15 fires a test on the individuals chart: 39 falls below the lower limit. Days 16 and 17 give 70 and 73. These values stay below the line at two standard deviations (74.6): test 2 does not fire. From day 13 to day 17, only two values out of five exceed the line at one standard deviation (67.6). Test 3 needs four. On day 16, the moving range of 31 exceeds its limit of 25.5. This gap of 31 separates 39 from 70. The value 70 lies within ordinary variation, so the jump comes from day 15. Both charts point to the same event on day 15. The team looks for the cause there.

Visualisations

The control chart is itself a figure: points in time against horizontal lines. The diagrams show the XmR chart of the well, the difference between control limits and specification limits, the patterns of the detection tests and the response to a signal. The formulas, the table of the well's values and the cost are made of rows and columns. They are tables.

Cost

Tooling

A spreadsheet is enough for an XmR chart: a column of values, a column of moving ranges and a few calculation cells. A line chart plots the values with the centre line and the limits as constant series.

Minitab and JMP draw control charts in one command and apply the detection tests. They are a better fit than a spreadsheet when the team keeps several charts or uses a chart type other than XmR. In R, the qcc package draws the chart. In Python, a script with matplotlib draws it. The team reruns the script every period.

At the workstation, a paper chart posted next to the process is enough: the operator enters the day's value and sees at once whether it falls outside the limits.

Sources

• PMI, A Guide to the Project Management Body of Knowledge (PMBOK Guide), 8th ed., §2.6.2.5 "Monitor and Control Resourcing" and Figure 2-45: the control chart among the data analysis tools. The "Control charts" entry in Section 5: the definition, the difference between specification limits and control limits, the use on cost and schedule variances, on volumes and on the frequency of scope changes.
• ABPMP, Guide to the Business Process Management Common Body of Knowledge (BPM CBOK), version 3.0, §6.11.4 "SPC" and §6.12 "The voice of the process": common and assignable causes according to Shewhart, the chart types, the XmR example of the oil well, the detection tests and their independence condition.
• Donald J. Wheeler, "Individual Charts Done Right and Wrong", Quality Digest, 2010: the XmR formulas, the error of the overall standard deviation and the rule against mixing data on one chart.
• NIST/SEMATECH, e-Handbook of Statistical Methods, §6.3.1 "What are Control Charts?": the elements of the chart, the choice of limits at three standard deviations and the non-random pattern within the limits.
• NIST/SEMATECH, e-Handbook of Statistical Methods, §6.3.2.2 "Individuals Control Charts": the individuals chart, the moving range and the estimate of the standard deviation as mR̄ / 1.128.