FDFOP2015A
Apply Principles of Statistical Process Control
Contents
Introduction and Unit Details ………….………………………………......
4
Element 1: Collect Statistical Information …………………………..…...
7
Collecting Data………………………………………………………………..… 89
Statistical Methods……………………………………………………..……....
10
A Measure of Central Tendency – Arithmetic Mean…………..…….
11
Using Microsoft Excel to Calculate the Mean…………..…….. 13
Calculating Mean from Grouped Data…………………….…..
15
A Measure of Dispersion – Range, Variance & Standard Deviation. 18
Range…………………………………………………………..…. 19
Using Microsoft Excel to Calculate the Range……………..…. 19
Variance………………………………………………………..…. 20
Population Standard Deviation………………………………....
22
Using Microsoft Excel to Calculate the Population SD……...
22
Sample Standard Deviation…………………………………..… 24
Using Microsoft Excel to Calculate the Sample Std Deviation.. 25
The Normal Distribution: Distribution of a continuous random variable26
Element 2: Analyse and Interpret Data ………………………………….…. 28
Statistical Process Control ……….………………………….…………..……
29
Histogram……………………………………………………………...….. 29
Check Sheet…………………………………………………………..…… 37
Pareto………………………………………………………………….…… 39
Cause & Effect (Fishbone) Diagram……………………………….…… 42
Defect Diagram……………………………….....……………………..…. 44
Scatter Diagram (x-y chart)…………………………………………..…. 45
Control Chart………………………………………………………….….. 51
Rules for detecting the presence of Special Causes…….……
Attribute & Variable Data…………………………………..……..
Sampling………………………………………………………..….
Constructing Control Charts …………………………………..…
53
56
60
61
Activity Answers……………………………………………….………………. 73
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Collecting Data
Statisticians select their observations so that all relevant groups are represented in
the data. To determine the potential market for a new product the analysts might
study 100 consumers in a certain geographical area. The analysts must be certain
that this group contains a variety of people representing variables such as income
level, race, education and age.
Data can come from actual observations (primary data) or from records (secondary
data) that are kept for normal purposes. For example, for billing purposes and
doctors’ reports, a hospital will record the number of patients using the x-ray
facilities. But this information can also be organised to produce data that we can
describe and interpret.
Data can assist decision makers in educated guesses about the cause and therefore
the probable effects of certain characteristics in given situations. Also, a knowledge
of trends from past experience can enable those concerned to be aware of potential
outcomes and to plan in advance. For example, if sales records showed that a
product (for example party sausage rolls) is purchased more often in December than
in June, the manufacturer’s personal division should determine if this was an
anomaly or an indication of a trend (and perhaps it should adjust its hiring and
vacation practices accordingly).
When data is arranged in compact usable form, decision makers can take reliable
information from the environment and use it to make intelligent decisions. Today
using computers we can collect enormous volumes of observations and compress
them instantly into tables, graphs and numbers.
Before relying on any interpreted data, test the data by asking these questions
1. Where did the data come from? Is the source biased; that is, is it likely
to have an interest in supplying data points that will lead to one
conclusion rather than another?
2. Do the data support or contradict other evidence we have?
3. Is evidence missing that might cause us to come to a different
conclusion?
4. How many observations do we have? Do they represent all the groups
we wish to study?
5. Is the conclusion logical? Have we made conclusions that the data do
not support?
(Levin, 1978, p 9)
Study your answers to these questions. Are the data worth using? Or should we wait
and collect more information before acting?
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Activity 1
A concentrated sanitiser needs to be mixed with water before use. Each batch of
‘santising solution’ is tested for strength (measured in ppm). What is the average
strength of the sanitising solutions listed below? That is, determine the mean.
16.2
15.4
16.0
16.6
15.9
15.8
16.0
16.8
16.9
16.8
15.7
16.4
15.2
15.8
15.9
16.1
15.6
15.9
15.6
16.0
16.4
15.8
15.7
16.2
15.6
15.9
16.3
16.3
16.0
16.3
Using Microsoft Excel to Calculate the Mean
If you completed activity 1 using a calculator, you would have found that entering 30
values/data points, is a little time consuming and open to the possiblity of error.
By using a spreadsheeting program such as Microsoft Excel, the data would still
need to be entered, but the mean can be easily calculated and as there is a record of
the values entered you could easily check if/where any errors were made.
METHOD 1
Step 1 Enter each of the values/ data points into a separate cell. This can be done
so they are:
 all in a column, or
 that they are all in a row, or
 a table of any dimensions (e.g. 10 columns across by 3 rows high)
Step 2 Select (click on) a cell where you want your answer to be.
Step 3 Click fx on the toolbar . The ‘Paste Function’ pop-up box will appear. In the
left ‘Function Category’ column click on ‘Statistical”. Then in the right ‘Function
Name’ column click on ‘AVERAGE’. Click on ‘OK’
Step 4 A grey pop up box will appear which will offer co-ordinates of values/data
points. Either click ‘OK’ or change the co-ordinates to what you want to include.
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THE NORMAL DISTRIBUTION: Distribution of a continuous
random variable.
The normal distribution occupies a prominent place in statistics as it has some
properties that make it applicable to a great many situations (in which it is necessary
to make inferences by taking samples) and it comes close to fitting the actual
observed frequency distributions of many phenomena including human
characteristics (weights, heights and IQ’s), outputs form processes (dimensions and
yields).
The diagram below shows several important features of a normal probability
distribution:
1. the curve has a single peak, and therefore is uni-modal.
2. it has a bell shape
3. the mean of the normally distributed population lies at the centre of the
normal curve
4. Because of its symmetry the mean, median and mode of the distribution are
also at the centre. Therefore for a normal curve the mean median and mode
are the same value
5. the two tails of the normal distribution extend indefinitely and never touch the
horizontal axis (though this is impossible to show)
MEAN,
MEDIAN,
MODE
Normal distribution is
symmetrical around a vertical line
erected at the mean
Left-hand tail extends
indefinitely but never
reaches the horizontal axis
Right-hand tail extend
indefinitely but never
reaches the horizontal axis
Most real life populations do not extend forever in both directions but for such
populations the normal distribution is a convenient approximation. There is no single
normal curve, but rather a family of normal curves. To define a particular normal
probability distribution, we need two parameters: the mean (µ ) and the standard
deviation (δ )
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2. Check Sheet
A Check Sheet allows a team to systematically record and compile data from
historical sources or observations as they happen, so that patterns and trends can
be clearly detected and shown.
Purpose




Creates easy-to-understand data that come from a simple, efficient process
that can be applied to any of the key performance areas
Builds, with each observation, a clearer picture of ‘the facts’ as opposed to
opinions of each team member
Forces agreement on the definition of each condition or event (every person
has to be looking for and recording the same thing)
Makes patterns in the data become obvious quickly.
How to create a Check Sheet
1.


2.
Agree on the definition of the events or conditions being observed.
If you are building a list of events or conditions as the observations are made,
agree on the overall definition of the project: Example If you are looking for
reasons for late payments, agree on the definition of ‘late’.
If you are working from a standard list of events or conditions make sure that
there is agreement of the meaning and application of each one. Example: If
you are tracking sales calls from various regions, make sure everyone knows
which states are in each region.
Decide who will collect the data; over what period; & from what sources.
a)
Who collects the data obviously depends on the product and
resources. The data collection period can range from hours to months.
The data can come from either a sample or an entire population
b)
Make sure the data collector(s) have both the time and knowledge they
need to collect accurate information
c)
Collect the data over a sufficient period to be sure the data represents
‘typical’ results during a ‘typical’ cycle for your business.
d)
Sometimes there may be important differences within a population that
should be reflected by sampling each different subgroup individually.
This is called stratification Example: collect complaint data from
business travelers separately from other types of travelers. Collect
scrap data from each machine separately.
NOTE: It must be safe to record and report ‘bad news’, otherwise the data will
be filtered.
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4.
Cause and Effect Diagram (Fishbone Diagram)
A Cause and Effect diagram allows a team to identify, explore and graphically
display in increasing detail all of the possible causes related to a problem or
condition to discover its root cause(s).
Purpose



Enable a team to focus on the content of the problem, not on the history of the
problem or differing personal interests of team members
Creates a snapshot of the collective knowledge and consensus of a team
around a problem. This builds support for the resulting solutions
Focuses the team on causes not symptoms
How to Create a Cause and Effect Diagram
1. Select the most appropriate cause and effect format. There are two
major formats:
 Dispersion Analysis Type is constructed by placing individual
causes within each ‘major’ cause category and then asking of
each individual cause ‘Why does this cause (dispersion)
happen?’. This question is repeated for the next level of detail
until the team runs out of causes. The graphic examples shown
in Step 3 of this tool section are based on this format.
 Process Classification Type uses the major steps of the
process in place of the major cause categories. The root cause
questioning process is the same as the Dispersion Analysis
Type.
2. Generate the causes needed to build a Cause & Effect Diagram.
Choose one method:
 Brainstorming without previous preparation
 Check sheets based on data collected by team members before
the meeting.
3. Construct the cause & effect diagram
 Place a problem statement in a box on the right-hand side of the
writing surface.
NOTE: Make sure everyone agrees on the problem statement. Include as much
information as possible on the ‘what’, ‘where’, ‘when’ and ‘how much’ of the problem.
Use data to specify the problem.
 Draw major cause categories or steps in the production or
service process. Connect them to the ‘backbone’ of the chart
 Place the brainstormed or data-based caused in the appropriate
category
 In brainstorming, possible causes can be placed in a
major cause category as each is generated or only after
the entire list has been created. Either works well but
brainstorming the whole list first maintain the creative flow
of ideas without being constrained by the major cause
categories or where the ideas fit in each ‘bone’.
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1. Positive Correlation. An increase in y may depend on an increase in x.
For example: course ratings are likely to increase as trainees receive more training.
5
Rating
4
3
2
1
0
0
5
10
15
20
25
30
35
Contact Hours
2. Possible Positive Correlation. An increase in y may depend on an increase in x.
For example: course ratings are likely to increase as trainees receive more
training.
5
Rating
4
3
2
1
0
0
5
10
15
20
25
30
35
Contact Hours
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7 Control Chart
A control chart is used monitor, control and improve process performance over time
by studying variation and its source.
Purpose







A graphical display of a quality characteristic that has been measured
or computed from a sample versus the sample number or time.
Focuses attention on detecting and monitoring process variation over
time
Monitoring and surveillance of a process
Helps improve a process to perform consistently and predictably for
higher quality, lower cost and higher effective capacity.
Prevents unnecessary process adjustments distinguishing between
background noise and abnormal variation
Provides diagnostic information
Provides information about process capability by proving information
about process parameters and their stability over time.
An example of a simple control chart (n=1) is pictured below. The chart contains a
center line that represents the average value of the quality characteristic
corresponding to the in-control state. The other two horizontal lines, called the upper
control limit (UCL) and lower control limit (LCL) are also on the chart. These
control limits are chosen so that if the process is in control, nearly all of the sample
points will fall between them. As long as the points plot within the control limits, the
process is assumed to be in control and no action is necessary. However if a point
plots outside the control limits it is interpreted as evidence that the process is out of
control and investigation and corrective action is required to find and eliminate the
assignable cause or causes.
Quality Characteristic
A simple control chart
18
16
14
12
10
8
6
4
2
0
UCL
Value
Avg
LCL
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Sample number or time
NOTE: A run chart is a simpler version of a control chart as it will only track the trends and
does not have statistically based control limits displayed. The danger of using a run chart is
that there is a tendency to see every variation in the data as being important. A mistake
commonly made is that specification limits will displayed, creating confusion with control
limits.
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The revised centre line and control limits are for activity 10 are shown on the
control chart below.
Revised Control Limits for Fraction Non-conforming chart
Sample fraction nonconforming
0.6
New
material
0.5
New
operator
Revised UCL
0.4
Fraction Nonconforming
0.3
Revised centre line
0.2
Revised LCL
0.1
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Sample number
Note that we have not dropped samples 15 and 23 from the chart but they have
been excluded from the control limit calculations and we have noted this directly on
the control chart.
This notation on the control chart is used to indicate process adjustments, or the type
of investigation made at a particular point in time. It forms a useful record for future
process analysis and should become standard practice in control chart usage.
Also note that the fraction nonconforming from sample 21 now exceeds the upper
control limit. However, analysis of the data does not produce any reasonable or
logical assignable cause for this and we decide to retain this point.
We can now conclude that the new control limits can be used for future samples and
therefore we have concluded the control limit estimation phase of control chart
usage.
Sometimes examination of control chart data reveals information that affects other
points that are not necessarily outside the control limits. For example if we had found
that the temporary operator working when sample 23 was obtained was actually
working during the entire two-hour period in which samples 21 – 24 were obtained,
then we should discard all four samples, even if only sample 21 exceeded the control
limits.
We should also examine the remaining 28 samples for runs and other nonrandom
patterns.
We conclude that the process is in control at the level p=0.2150 and that the revised
control limits should be adopted for monitoring current production. However we note
that although the process is in control the fraction nonconforming is much too high.
That is the process is operating in a stable manner and no unusual operatorcontrollable problems are present. It is unlikely that the process quality can be
improved by action at the work-force level.
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