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1 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. 4 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 5 • • • 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.