Download Demming`s Funnel Experiment - Institute of Industrial Engineers

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Deming’s Funnel Experiment
Description of the Experiment
Throughout the last half of the 20th century, W. Edwards Deming was an energetic proponent
of statistical methods of quality management in industrial settings. He used a simple
experiment involving a funnel and marbles to illustrate the effects of one strategy to try to
"control" randomness. Here we look at one version of his experiment.
A funnel is suspended with its small opening pointed downward and centered on a target point,
which we will denote as 0. Marbles that are smaller than the diameter of the small opening of
the funnel are dropped into it in succession. They hit in the vicinity of the target, but the exact
locations at which they hit are random. This randomness represents the kind of variability that
is inherent in many industrial production processes.
Two Strategies For “Managing” Variability
An important goal in many industrial processes is to understand and perhaps decrease variation.
Here we look at two approaches for dealing with variability, one passive and one active.
Strategy 1: This strategy is very simple — do nothing to try to control or to compensate for the
randomness. Just accept it and live with it. Imagine a measurement scale that extends in both
directions from the target point 0; points to the left of the target are negative, those to the right
are positive. Suppose that the randomness in our version of the funnel experiment tends to give
the following pattern of hit points along the measurement scale:
•
•
•
•
The pattern is roughly symmetrical with as many marbles hitting to the left of 0
as to the right.
About 2/3 of the marbles hit between –1 and +1 along the scale.
Only about 5% of the marbles hit outside of the interval from –2 to +2.
Only very rarely does a marble hit outside of the interval from –3 to +3.
Later on we will formalize this pattern of randomness as having a "standard normal
distribution."
It is possible to use a computer to simulate such results. Here is a dot plot of 100 simulated
marble hits for the experiment just described.
: .
.
::: :::
:::.::::.:::: :::: .
.
. .::::::::::::::::::::::.: .
-------+---------+---------+---------+---------+-------Strat1
-3.0
-1.5
0.0
1.5
3.0
2
This plot simply puts a dot along the scale to represent each of the 100 marble hits. No two of
the marbles actually hit at exactly the same position, but in the plot above results are rounded
somewhat so that hitting positions that are very close together are taken to be the same, and
their dots are stacked.
Strategy 2: This strategy is only a little more complicated. The funnel is moved after each hit
to try to compensate for random errors. If the first marble hits at position –0.68 then the funnel
is moved to the right by 0.68 units so that the center of the small opening is above 0.68 on the
scale. If the second marble then hits 0.02 units to the left of this new "center," then it hits the
scale at the point 0.68 – 0.02 = 0.66, and the funnel is moved so that its small opening is above
the point +0.02. Similar adjustments are made after each marble is dropped — always in the
direction opposite the last deviation from the "center."
Which strategy is better? Here are dot plots for Strategies 1 and 2 for the same sequence of
marble behaviors. It is clear that Strategy 2 is worse, producing greater variability — not less as
was hoped. Strategy 2 demonstrates the bad effects of "overcontrol."
: .
.
::: :::
:::.::::.:::: :::: .
.
. .::::::::::::::::::::::.: .
-------+---------+---------+---------+---------+---------+-Strat1
:
. :
:
: : .. :
: : .. .:.::..:::::
. . .: .... ::::::.::::::::::::::..... :. .
-------+---------+---------+---------+---------+---------+-Strat2
-3.0
-1.5
0.0
1.5
3.0
4.5
Numerical descriptive statistics of these results are shown below.
Variable
Strat1
Strat2
N
100
100
Mean
0.0015
-0.012
Median
0.0259
0.089
TrMean
-0.0030
-0.003
Variable
Strat1
Strat2
Minimum
-2.6538
-3.250
Maximum
2.2445
3.174
Q1
-0.6831
-1.029
Q3
0.7938
0.922
StDev
0.9816
1.312
SE Mean
0.0982
0.131
The standard deviation 1.3 for Strategy 2 is substantially larger than the standard deviation 1.0
for Strategy 1. Similarly, the range 3.2 – (–3.2) = 6.4 for Strategy 2 is greater than the range
2.2 – (–2.7) = 4.9 for Strategy 1. The mean is not far from the target value 0 with either
strategy. (Note: Of course each sequence of 100 hits will give slightly different results. It can
be shown that the theoretical standard deviations are 1.0 for Strategy 1 and 2 = 1.414 for
Strategy 2.)
This example shows that it is best not to tamper with a process that is "in control" in the sense
that it has a constant mean over time. Here is a time plot of the hit points using Strategy 1.
3
2
Strat1
1
0
-1
-2
-3
Index
10
20
30
40
50
60
70
80
90
100
Notice that the data points appear to vary about the mean value 0 throughout the sequence of
100 marble drops.
When a Process is “Out of Control”
Now consider a process that undergoes a shift in mean value. Here we show the results of a
sudden shift by 1 unit in the negative direction (downward on the plot) beginning when the 51st
marble is dropped. ("SStrat1" is short for shifted process, Strategy 1.)
2
1
SStrat1
0
-1
-2
-3
-4
Index
10
20
30
40
50
60
70
80
90
100
In the case of this process with a shift (out of control), Strategy 2 helps to put the process right
— not necessarily by decreasing the variability, but by helping to get rid of the shift. Notice
that the mean for Strategy 2 is near 0. The ability of Strategy 2 to correct for a change in mean
comes at the cost of a relatively large standard deviation.
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Variable
SStrat1
SStrat2
N
100
100
Mean
-0.499
-0.022
Median
-0.438
0.089
TrMean
-0.493
-0.014
Variable
SStrat1
SStrat2
Minimum
-3.654
-3.250
Maximum
1.933
3.174
Q1
-1.304
-1.104
Q3
0.143
0.922
StDev
1.076
1.319
SE Mean
0.108
0.132
A Control Chart
A control chart can often be used to detect when a process goes out of control (for example,
because of a sudden shift or a gradual drift in the mean). Below is a control chart of the process
where the mean shifts half-way through. In a way we will study later, the data are used to
construct upper and lower control limits. A shift is suspected when the time plot goes outside
one of these boundaries. In this example, the shift is detected at about observation number 67
according to the control-limit criterion. When a shift is detected, measures can be taken to
detect the amount of shift and to correct it. In practice, this is usually superior to Strategy 2,
which continually tampers with the process even when it is OK.
I Chart for SStrat1
3
UCL=2.321
Individual Value
2
1
0
Mean=-0.4985
-1
-2
-3
LCL=-3.318
-4
0
50
100
Observation Number
Minitab Simulation:
Below are the commands for a Minitab session used to produce the numerical examples
presented in this handout. Use Minitab to do a simulation on your own. Your results will not be
the same as those shown here, but should be somewhat similar. Comment on your results.
Some notes on using Minitab:
• To use commands in Minitab, click on the Session (upper) window. Then select the option
to enable commands in the Editor menu. This provides the MTB > prompt. Watch the
Session window (upper) and Worksheet (lower) after each command to see what result is
produced.
• Here all columns are named using commands. When provided, column names may be used
in subsequent commands (included inside single quotes). Names may also be also be
established by typing them directly into the Worksheet without using a command. It is not
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•
•
•
really necessary to supply column names. They are used here to help make the purpose of
each column clear, and also so that the output (text or graphics) will have names as labels
rather than column numbers.
Only the first four letters of a command need to be typed; either lower or upper-case
characters may be used. The extra lines between blocks of commands are for readability in
this handout; you need not leave extra lines when using Minitab.
Some outcomes produce graphs in boxes. These graphics boxes may be minimized for later
reference, cut and pasted into a MS Word document, saved to disk, or simply closed
(discarded). It is best not to keep too many graphs open in Minitab at a time because they
take a lot of RAM.
Menus may be used instead of commands. Explore. You will probably find some "cool"
things not shown here.
You should also look at processes that go out of control with a shift of +2.5 rather than –1.0,
and because of a steadily drifting mean. (See comments below.) In each case, say whether the
control chart detects that the process (Strategy 1, no tampering) is out of control.
MTB >
MTB >
SUBC>
MTB >
MTB >
name c1 'Strat1'
random 100 'Strat1';
normal 0 1.
dotplot c1
tsplot c1
MTB
MTB
MTB
MTB
name c2 'AdjNext'
let c2 = -c1
name c3 'Adj'
stack 0 c2 c3
>
>
>
>
(Note: Semicolon allows subcommand.)
(Period ends subcommand sequence; yields output.)
MTB >
MTB >
MTB >
SUBC>
MTB >
name c4 'Strat2'
let c4 = c1 + c3
dotplot c1 c4;
same.
describe c1 c4
MTB >
MTB >
DATA>
DATA>
name c5 'Shift'
set c5
50(0) 50(-1)
end
MTB
MTB
MTB
MTB
>
>
>
>
name c6 'SStrat1'
let c6 = c1 + c5
tsplot c6
(Does the control chart detect a process "out of control"?)
ichart c6
MTB
MTB
MTB
MTB
>
>
>
>
name c7 'SAdjNext'
let c7 = -c6
name c8 'ShftAdj'
stack 0 c7 c8
MTB
MTB
MTB
MTB
>
>
>
>
name c9 'SStrat2'
let c9 = c6 + c8
(Which strategy has the smaller standard deviation? The mean nearest 0?)
describe c6 c8
ichart c9
(You will get a warning message about unequal column lengths; ignore. it)
(Scroll down the worksheet to look at rows 45 through 55. For alternate
simulations, use data 40(0) 60(2.5) for a sudden shift of +2.5, and
data 0:2.99/0.03 for a steady upward drift.)
Copyright © 2000, 2003 by Bruce E. Trumbo. All rights reserved. This handout is intended primarily for use in statistics
classes at California State University, Hayward. Except that it includes instructions for doing simulations in Minitab, it
covers material somewhat similar to Section 1.5 of Montgomery and Runger: Applied Statistics And Probability For
Engineers (2nd ed), 1999, Wiley.