Methodology Glossary Tier 2
Statistical Process Control
Introduction
Statistical Process Control (SPC) is the application of statistical methods to monitor
and analyse the variation present within a system or process. SPC assists in
differentiating between ‘common cause’ and ‘special cause’ variation. These terms
along with the concept of a ‘state of statistical control’ were used by Walter A.
Shewhart, who pioneered SPC and the use of control charts in the early 1920s.
‘Common cause’ variation is that variation which is expected to exist within a stable
process and is usually due to errors such as recording or measurement error. These
sources of error will exist regardless of external factors, and will result in slight
differences between measurements. Even two simultaneous measures of the same
item are unlikely to give exactly the same result.
‘Special cause’ variation is the term given to variations caused by a factor other than
those attributed to common causes. In other words, something has happened that has
had an effect on the process and this should be investigated ‘Special cause’ variation
could be the result of actions such as the implementation of a new policy, using new
or damaged machinery or changes in staff numbers or skill.
SPC has most frequently been applied to manufacturing lines. However, SPC also
applies well to any process with a measureable output. For example, SPC has been
used in the health service to identify possible outbreaks of infectious disease.
SPC can assist with implementing continuous quality improvement and for this reason
it is best used where the user has direct ‘control’ or a good understanding of the
process. If you are unable to make changes in the process or don’t fully understand
the causes of variation within a given system then it may be difficult to utilise the full
potential of SPC.
For an example of the theory behind statistical process control see Deming's famous
Red Bead Experiment (http://www.maaw.info/DemingsRedbeads.htm), which helps
to explain common cause variation.
Charts used in SPC
Central to SPC is the use of charts to monitor the variation within a process; there are
a number of different charts that can be used.
Run charts, also known as run-sequence plots, are the basic charts used in SPC, they
consist of a central line (usually a historic average; see the describing numerical data
Methodology Glossary paper for details on calculating averages) and the current data
points, usually plotted against time.
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Methodology Glossary Tier 2
Example Run chart
A simple run chart showing data collected over time. The median of the observed data
(73) is also shown on the chart as the ‘central line’. This chart shows a stable process
with only common cause variation.
Identifying ‘special cause’ variation using Run charts
Run charts can be used to identify changes in a process which may be the result of a
special cause. Typical indicators of a shift in the natural variability include unusually
long "runs" of data points above or below the average line, repeating patterns in the
data, and unusually long series of consecutively increasing or decreasing points.
There exists a set of ‘rules’ with which run charts can be assessed. Where a chart is
seen to follow the conditions of a certain rule the process is said to be ‘out of control’,
or under the influence of ‘special cause’ variation. Three common identifiers of
special causes are presented below along with examples:

Shifts: If there are eight or more consecutive points on the same side of the
centre line, that indicates that a special cause has influenced the process.
Points on the centre line don't count; they neither break the string, nor add to
it.

Trends: Six consecutive points moving in the same direction indicate that a
special cause is acting on the process to cause a trend. Flat line segments don't
count; they neither break a trend, nor count towards it.

Pattern: If there is a repeating arrangement of points that recurs eight or more
times in a row, this may indicator a special cause has influenced the process..
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Example - Trend
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Example - Shift
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Example - Pattern
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To be able to fairly assess a run chart against these rules it is important that you have
sufficient data points (usually said to be around 25 or more) and that the ordering of
the points on the chart and any natural orderings are considered. If for example a chart
was used to assess the number of cases of infection within a number of hospitals, then
the ordering of the hospitals would affect whether the criteria for identifying special
causes were met or not.
It may be possible to detect special causes more accurately and quickly by using
control charts.
Control Charts, also known as Shewhart charts or process-behaviour charts are the
most common tool used under SPC. Standard control charts are produced by
calculating an average result for a time series of data, plotting this as the central line,
as in run charts above, and then calculating control limits either side of this mean.
These control limits are usually set at plus and minus three standard deviations from
the central line. This range will account for approximately 99.7% of all natural,
‘common cause’, variation.
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Example standard control chart
This control chart shows a central line, upper and lower control limits and warning
limits (the thin dashed red lines). Warning limits are usually set at plus and minus two
standard deviations and indicate points where further investigation may be necessary,
especially should a number of consecutive points fall between the warning and control
limits.
Producing Control Charts
An important part of the control chart is obviously the setting of the limits used to
identify points where the system is ‘out of control’. There are a number of formulae
that are used to produce these limits, and each is dependant on the type of chart being
used and certain characteristics of the data.
Below are some of the most commonly seen formulae for calculating control limits:
For “p-charts”, where the data are proportions the following formula is used:
Control Limits = p  3
p (1  p )
n
Where: p = historical average proportion
n = number of opportunities
A “c-chart” control chart is produced for count data with n observations each with a
constant ‘area of opportunity’.
Where counts are involved we the following formula is used:
Control Limits = c  3 c
Where: c = historical average count
This formula assumes that the data follows a Poisson distribution, as the standard
deviation of a Poisson distribution can be estimated by the square root of the mean.
Where the data does not follow a Poisson distribution the following formula is used
for counts:
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Methodology Glossary Tier 2
Control Limits =   3
Where:  = Mean value of the data
 = Standard Deviation of the data
Identifying special cause variation using control charts
If the process is in a ‘state of control’ then the sample means will fall within the
control limits about 299 times out of 300 (where three standard deviations have been
used for the limits). Any points where the sample means fall outside the control limits
should be investigated further, as something may be affecting the system and the
quality of outputs may be affected.
Other Chart Types
Run charts and Control charts are the most commonly used charts in SPC however
there are a number of others that can be used under certain circumstances. For
example where you are comparing performance in different areas with differing
population sizes, a funnel plot may be most suitable. A funnel plot gets its name from
the shape of the control limits; as they narrow as the population increases. This is the
case because as the underlying population increases the estimates made should be
more accurate and so less fluctuations and natural variation is expected to occur.
Choosing which type of control chart to use
(APHO, Technical Briefing – Statistical process control methods in public health intelligence, December 2007)
Using SPC
Statistical Process Control may be broadly broken down into three sets of activities:
understanding the process; understanding the causes of variation; and elimination of
the sources of special cause variation.
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Methodology Glossary Tier 2
In understanding a process, the process is typically mapped out and the variations in
the process are monitored using control charts, this is a continuous, ongoing activity.
When excessive variation is identified by the control chart, additional effort is exerted
to determine the causes of that variance.
Once the causes of variation are understood, if it is wished for the process to return to
a state of control then the cause of variation can be investigated further and removed.
Once the process is back under control, the points outwith the limits should be
disregarded in any future calculation of the central line and / or limits.
It is important when using SPC to be aware that not every point identified by a control
chart as ‘out of control’ will be the result of a special cause. As indicated above, by
chance alone you would expect one in twenty points to fall outwith the two sigma
limits and about one in three hundred points to fall outwith control limits based on
three standard deviations. It is therefore important that when a point falls outwith the
limits it is investigated in detail to decide whether it is the result of a ‘special cause’
or just a chance occurrence. This emphasises the need to have a good understanding
of the process being measured under SPC to make full use of the technique.
If you have any questions on using Statistical Process Control in the Public Sector
please feel free to contact the Office of the Chief Statistician, Scottish government,
who may be able to provide you with some support and / or advice:
Analysts.Network@Scotland.gsi.gov.uk or 0300 244 1015
Related Paper
Tier 1 Describing Numerical data
www.scotland.gov.uk/Topics/Statistics/About/Methodology/SPC