Newcastle University School of Chemical Engineering and Advanced Materials
CONTENTS
Foreword
What is SPC ?
Tools for SPC
Flow charts
  Run charts
  Pareto charts
  Cause and effect diagrams
  Frequency histograms
  Control charts
common cause variations
    special cause variations
    how they work
    types of charts
    assumptions
    properties of the normal PDF
    interpretation
    hypothesis testing
    run rules for Shewhart charts
    CUSUM charts
    relative merits
Process Capability
concepts
  relative capability
  capability index
  performance index
  the message
Summary
References
Copyright Information

 

 
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 (R-chart)
    • Shewhart sample (X-chart)
    • Cumulative sum (CUSUM)
    • Exponentially Weighted Moving Average (EWMA) chart
    • Moving-average and range charts
  • for discrete (attributes and countable) data
    • sample proportion defective (p-chart)
    • sample number of defectives (np-chart)
    • sample number of defects (c-chart)
    • sample number of defects per unit (u-chart 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,s2) has several distinct properties:
  • The normal distribution is bell-shaped 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 X-chart 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 out-of-statistical control
Control charts: run rules for Shewhart charts
Run rules are rules that are used to indicate out-of-statistical control situations. Typical run rules for Shewhart X-charts 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 ( 0.57x100 = 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 Xi
  • difference between the raw variable and a target  Xi - Xtarget
  • difference between the raw variable and its mean Xi - m
  • difference between successive variables Xi - Xi-1
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.

Cause of Change

Chart Type

Mean

Range

Standard Deviation

CUSUM

Gross Error tickticktick ticktick   tick
Shifts in Mean ticktick   tick tickticktick
Shifts in Variability   tickticktick    
Slow Fluctuation ticktick     tickticktick
Rapid Fluctuation   tickticktick ticktick  
    Written by: M. Tham