SPC Tools  Control charts 
Processes that are not in
a state of statistical control
 show excessive variations
 exhibit variations that change
with time
A process in a state of statistical
control is said to be statistically stable.
Control charts are used to detect whether a process is statistically
stable. Control charts differentiates between variations
 that is normally expected of the
process due chance or common causes
 that change over time due to assignable
or special causes

Control
charts: common cause variations 
Variations due to common causes
 have small effect on the process
 are inherent to the process because
of:
 the nature of the system
 the way the system is managed
 the way the process is organised
and operated
 can only be removed by
 making modifications to the
process
 changing the process
 are the responsibility of higher
management

Control
charts: special cause variations 
Variations due to special causes are
 localised in nature
 exceptions to the system
 considered abnormalities
 often specific to a
 certain operator
 certain machine
 certain batch of material,
etc.
Investigation and removal of variations
due to special causes are key to process improvement
Note: Sometimes the
delineation between common and special causes may not be very clear

Control
charts: how they work 
The principles behind the application
of control charts are very simple and are based on the combined use
of
 run charts
 hypothesis testing
The procedure is
 sample the process at regular
intervals
 plot the statistic (or
some measure of performance), e.g.
 mean
 range
 variable
 number of defects, etc.
 check (graphically) if the process
is under statistical control
 if the process is not under statistical
control, do something about it

Control
charts: types of charts 
Different charts are used depending
on the nature of the charted data Commonly used charts are:
 for continuous (variables)
data
 Shewhart sample mean (chart)
 Shewhart sample range (Rchart)
 Shewhart sample (Xchart)
 Cumulative sum (CUSUM)
 Exponentially Weighted Moving
Average (EWMA) chart
 Movingaverage and range charts
 for discrete (attributes
and countable) data
 sample proportion defective
(pchart)
 sample number of defectives
(npchart)
 sample number of defects (cchart)
 sample number of defects per
unit (uchart or chart)

Control
charts: assumptions 
Control charts make assumptions about
the plotted statistic, namely
 it is independent, i.e.
a value is not influenced by its past value and will not affect
future values
 it is normally distributed,
i.e. the data has a normal probability density function
Normal Probability
Density Function
The assumptions of normality and
independence enable predictions to be made about the data.

Control
charts: properties of the normal distribution

The normal distribution N(m,s^{2})
has several distinct properties:
 The normal distribution is bellshaped
and is symmetric
 The mean, m,
is located at the centre
 The probabilities that a point,
x, lies a certain distance beyond the mean are:
 Pr(x > m + 1.96s) = Pr(x > m  1.96s) = 0.025
 Pr(x > m + 3.09s) = Pr(x > m  3.09s) = 0.001
s
is the standard deviation of the data

Control
charts: interpretation 
 Control charts are normal
distributions with an added time dimension
 Control charts are run charts
with superimposed normal distributions

Control
charts: a graphical means for hypothesis testing

Control charts provide a graphical
means for testing hypotheses about the data being monitored.
Consider the commonly used Shewhart Chart as an example.
Shewhart Xchart with
control and warning limits
The probability of a sample having
a particular value is given by its location on the chart. Assuming
that the plotted statistic is normally distributed, the probability
of a value lying beyond the:
 warning limits is approximately
0.025 or 2.5% chance
 control limits is approximately
0.001 or 0.1% chance, this is rare and indicates that
 the variation is due to an
assignable cause
 the process is outofstatistical
control

Control
charts: run rules for Shewhart charts 
Run rules are rules that
are used to indicate outofstatistical control situations. Typical
run rules for Shewhart Xcharts with control and warning limits are:
 a point lying beyond the control
limits
 2 consecutive points lying beyond
the warning limits (0.025x0.025x100 = 0.06% chance of
occurring)
 7 or more consecutive points lying
on one side of the mean ( 0.5^{7}x100 = 0.8%
chance of occurring and indicates a shift in the mean of the process)
 5 or 6 consecutive points going
in the same direction (indicates a trend)
 Other run rules can be formulated
using similar principles

Control
charts: CuSum charts 
CUSUM Charts are excellent for detecting
changes in means. A CUSUM Chart is simply a plot of the sum of some
process characteristic against time. Examples of typical characteristics
that are plotted are:
 the raw variable X_{i}
 difference between the raw variable
and a target X_{i}  X_{target}
 difference between the raw variable
and its mean X_{i}  m
 difference between successive
variables X_{i}  X_{i1}

Control
charts: examples 

Control
charts: relative merits 
Different control charts have different
capabilities. The table below shows the relative merits of different
chart types when applied to detect the changes listed in the first
column. 
