Principle 6: Variability (Item 4)
All systems and processes exhibit variability, which impacts on
predictability and performance.
Control charts are by far the best way of presenting data so that it
can be easily interpreted. The charting procedure makes the statistical
analysis easy. They are easy to interpret and use.
If you do not use control charts, you will probably waste time and
effort over and under-reacting to events.
To proceed further with example we began in question 47, you need to
get into more detail about the numbers. Unfortunately, no one can tell
if 111 and 112 are different without more information.
- First, you need the `caring point'. This can be a specification,
a break even, an answer to an important `what if' question.
- Second, you need to know the `scale'. Dollars or cents or
millions of dollars. Meters or millimeters or kilometers or light
years. Seconds or thousands of a second or hours.
- Third, you need to know the `accuracy'. Accuracy becomes
the critical issue.
All measurement, whether of time or distance, can only be within a
certain degree of accuracy. Everything you measure or count is only
an approximation.
The accountants among you all cry "not true, our accounting is
dead accurate". Think about that for a little longer. It does usually
take long to think of examples where the dollars are only approximate.
Is there any foreign exchange in the calculations? The exchange rate
is never exact. Any variable rate borrowing? Do you know exactly
how much remains to be repaid? Do you and the lender agree, exactly?
Any share holdings in other companies? Do you know exactly what
they are worth? How much did you make today on your shares holding?
Exactly? By exactly, we mean to the cent not to the nearest thousand
dollars. (Usually, the `caring point' is set a little way out so that
`exact' accuracy is not required. So long as everything balances.)
Often companies (even quite small ones) do not even know exactly how
many staff they have. How do you count Mrs Smith who works Tuesdays
and Thursdays from 9:30 to 4pm? What about Mrs Jones who shares work
with Mrs Smith and works Wednesdays and Fridays? Is that one employee
or two?
If you order a piece of steel six millimeters
long you can have it made accurately to within the finest margin of
which the best instruments are capable – millionths of a millimeter.
But where the exact point of six millimeters lies within that tiny margin
is something you do not know. It may be that your piece of steel is
exactly six millimeters long, but you will not know it. All you know
is that the length is accurate to within such and such a fraction of
a millimeter. With the next improvement in measuring tools you would
get a piece of steel an even closer margin. But you will never get one
exactly six millimeters except by chance. And you can never know when
you do.
(From Bryan Magee's book Popper.)
Let us stay with accuracy for a while longer and work with a couple
of examples.
Consider the following two sets of data that each contain 111 and 112.
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Example 1
111.0 111.1 111.5 111.2 111.0 111.2 112.0 111.4 111.3
111.4 111.3 111.5 111.3 111.1 111.2 111.4 111.2 111.2 111.4 111.2
111.3 111.3 111.1 111.0 111.3 111.2 111.2 111.5111.0 111.1
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Example 2
111.0 112.4 112.9 111.6 112.0 111.8 112.9 111.6 111.7
112.7 111.7 112.3 111.0 111.5 111.8 112.9 112.9 112.6 111.3 1
12.4111.3 111.9 111.9 111.4 111.6 111.8 111.0 1 11.7112.6 111.7
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We probably cannot tell very much from the table. So let us draw graphs.
[In this analysis, you need to know the `average' and the `standard
deviation'. The graph shows the average and three `standard deviations'
above the average and three `standard deviations' below the average.
A good spreadsheet package will let you do this elementary calculation.]
What you are asking is "does the number 112 belong to the same
set of numbers as 111". Or, "does it belong to the same population".
In Example 1, the answer is "No it does not". Because
112 is more than three standard deviations away from the average of
the points, in the terminology from above, 112 is beyond the `caring
point'.
Three standard deviations are used because the probability of saying
that 112 is different from 111 when they are really the same has fallen
to 1%. That is, one chance in 100 of making that error.
This graph is a simple `Control Chart' and you have found that
point 112 was "out of control". By "out of control"
we mean it does not belong to the same data set as the other points.
In terms of Principle 4 (`To Improve the Outcome, Improve the System'),
think of your data points as having been generated from a process. Then
the process that gave the value 112 was not the same as the process
that gave the other 29 points. That is a very important finding. You
now know that the 112 point really is different. It is significantly
different. The reasons for the difference might be worth exploring.
To use this simple tool, you need at least 30 numbers that come from
the set of numbers that you think are all the same. When you are calculating
the average and standard deviation, leave out the number(s) that you
think are different. If the numbers are themselves averages (ie, if
111 is actually the average of a few numbers), then you have a more
robust tool.
If these points are all measurements from a process, your process is
"out of control" if when the Control Chart is divided into
zones, as shown below, and any of the following are true. Each zone
is one standard deviation wide.
- One or more points fall outside the control limits (ie three standard
deviations away from the average)
- Two points, out of three consecutive points, are on the same side
of the average in Zone A or beyond
- Four points, out of five consecutive points, are on the same side
of the average in Zone B or beyond
- Nine consecutive points are on the same side of the average
In example 2, 112 is the same as 111 because all the
points are within the three standard deviations.
The lines at three standard deviations are called the `Control Limits'
of the Control Chart.
When systems are `in control', do not react to the up and down fluctuations.
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