Exercises
7. MORE ON GRAPHICS
The importance of graphing your data cannot be overemphasized. If a feature you expect to see is not present in the plots, statistical analyses will be of no avail. Moreover, creative graphics can often highlight features in the data and even give new insights.
The devastation of Napoleon’s Grand Army during his ill-fated attempt to capture Russia was vividly depicted by Charles Minard. The 422,000 troops that entered Russia near Kaunas are shown as a wide (shaded) river flowing toward Moscow and the retreating army as a small (black) stream. The width of the band indicates the size of the army at each location on the map. Even the simplified version of the original graphic, appearing in Figure 17, dramatically conveys the losses that reduced the army of 422,000 men to 10,000 returning members. The temperature scale at the bottom, pertaining to the retreat, helps to explain the loss of life, including the incident where thousands died trying to cross the Berezina River in subzero temperatures. (For a copy of Minard’s more detailed map and additional discussion, see E. R. Tufte, The Visual Display of Quantitative Information, Cheshire, CT: Graphics Press, 1983.) Kaunas 422,000 10,000 Niemen R. Berezina R. Smolensk Moscow 100,000 0˚ C –15˚ –30˚ –9˚ –21˚ –11˚ –20˚ –30˚
Dec. 6 Nov. 28 Nov. 14 Oct. 9 Temperature
Figure 17 The demise of Napoleon’s army in Russia, 1812 – 1813, based on Charles Minard.
Another informative graphic, made possible with modern software, is the display of ozone by city in Figure 18. This figure illustrates improved air qual- ity relative to the previous year and the ten year average.
Graphs can give a vivid overall picture.
Dave Carpenter/Cartoon Stock
8.
STATISTICS IN CONTEXT
The importance of visually inspecting data cannot be overemphasized. We present a mini-case study2 that shows the importance of first appropriately
plotting and then monitoring manufacturing data. This statistical application con- cerns a ceramic part used in a popular brand of coffee makers. To make this ce- ramic part, a mixture of clay, water, and oil is poured into the cavity between two dies of a pressing machine. After pressing but before the part is dried to a hard- ened state, critical dimensions are measured. The depth of a slot is of interest here. Sources of variation in the slot depth abound: the natural uncontrolled variation in the clay–water–oil mixture, the condition of the press, differences in operators, and so on. Some variation in the depth of the slot is inevitable. Even so, for the part to fit when assembled, the slot depth needs to be controlled within certain limits.
Every half hour during the first shift, slot depth is measured on three ce- ramic parts selected from production. Table 12 gives the data obtained on a Fri- day. The sample mean for the first sample of 218, 217, and 219 (thousandths of inch) is ( 218 217 219 )/ 3 654/ 3 218, and so on.
TABLE 12 Slot Depth (thousandths of an inch)
Time 7:00 7:30 8:00 8:30 9:00 9:30 10:00 1 218 218 216 217 218 218 219 2 217 218 218 220 219 217 219 3 219 217 219 221 216 217 218 SUM 654 653 653 658 653 652 656 218.0 217.7 217.7 219.3 217.7 217.3 218.7 Time 10:30 11:00 11:30 12:30 1:00 1:30 2:00 2:30 1 216 216 218 219 217 219 217 215 2 219 218 219 220 220 219 220 215 3 218 217 220 221 216 220 218 214 SUM 653 651 657 660 653 658 655 644 217.7 217.0 219.0 220.0 217.7 219.3 218.3 214.7 x x
An x-bar chart will indicate when changes have occurred and there is a need for corrective actions. Because there are 3 slot measurements at each time, it is the 15 sample means that are plotted versus time order. We will take the centerline to be the mean of the 15 sample means, or
Centerline:x 218.0 214.7
15 218.0
When each plotted mean is based on several observations, the variance can be estimated by combining the variances from each sample. The first sample has vari-
ance (3 1 )
1.000 and so on. The details are not important, but a computer calculation of the variance used to set control limits first determines the average of the 15 individual sample variances,
and, for reasons given in Chapter 7, divides by 3 to give the variance of a single sample mean. That is, 1.58/ 3 .527 is the appropriate s2. The control limits are set at three times the estimated standard deviation .726, or 3 .726 2.2 units from the centerline.
Lower control limit: LCL 218.0 2.2 215.8 Upper control limit: UCL 218.0 2.2 220.2 The x-bar chart is shown in Figure 19. What does the chart tell us?
s
√
.527 1.000 0.333 0.333 15 1.58 s12 [(218 218)2 (217 218)2 (219 218)2]/ 220 219 218 217 216 215 214 0 5 Sample number Sample mean 10 15 221 UCL 220.2 Mean 218 LCL 215.8 • • • • • • • • • • • • • • •Figure 19 X-bar chart for depth.
The x-bar chart shows that the process was stable throughout the day and no points were out of control except the last sample. It was then that an unfor- tunate oversight occurred. Because it was near the end of her shift and the start of the weekend, the operator did not report the out-of-control value to either the setup person or the foreman. She knew the setup person was already clean- ing up for the end of the shift and that the foreman was likely thinking about going across the street to the Legion Bar for some refreshments as soon as the shift ended. The operator did not want to ruin anyone’s weekend plans so she kept quiet.
When the pressing machine was started up on Monday morning, one of the dies broke. The cost of the die was over a thousand dollars. But, when a cus- tomer was called and told there would be a delay in delivering the ceramic parts, he canceled the order. Certainly the loss of a customer is an even more ex- pensive item.
Later, it was concluded that the clay had likely dried and stuck to the die leading to the break. A problem was predicted by the chart on Friday. Although the chart correctly indicated a problem at that time, someone had to act for the monitoring procedure to work.
USING STATISTICS WISELY
1. As a first step, always graph the data as a dot diagram or histogram to as-