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**Lecture Slides Elementary Statistics Eleventh Edition**

and the Triola Statistics Series by Mario F. Triola

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**Statistical Process Control**

Chapter 14 Statistical Process Control 14-1 Review and Preview 14-2 Control Charts for Variation and Mean 14-3 Control Charts for Attributes

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Section 14-1 Review and Preview

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Review In Section 2-1 we noted that an important characteristic of data is a changing pattern over time. Some populations change over time so that values of parameters change.

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Preview The main objective of this chapter is to learn how to construct and interpret control charts that can be used to monitor changing characteristics of data over time. That knowledge will better prepare us for work with businesses trying to improve the quality of their goods and services.

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Preview Control charts are good examples of visual tools that allow us to see and understand some property of data that would be difficult or impossible to understand without graphs.

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**Control Charts for Variation and Mean**

Section 14-2 Control Charts for Variation and Mean

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Key Concept The main objective of this section is to construct run charts, R charts, and charts so that we can monitor important characteristics of data over time. We will use such charts to determine whether some process is statistically stable (or within statistical control).

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Definition Process data are data arranged according to some time sequence. They are measurements of a characteristic of goods or services that result from some combination of equipment, people, materials, methods, and conditions. Important characteristics of process data can change over time.

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Definition A run chart is a sequential plot of individual data values over time. One axis (usually vertical) is used for the data values, and the other axis (usually horizontal) is used for the time sequence.

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**Example: Run Chart of Earth’s Temperatures**

Treating the 130 mean temperatures of the earth in Table 14-1 as a string of consecutive measurements, construct a run chart using a vertical axis for the temperatures and a horizontal axis to identify the chronological order of the sample data, beginning with the first year of 1880.

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**Example: Run Chart of Earth’s Temperatures**

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**Example: Run Chart of Earth’s Temperatures**

Following is the Minitab-generated run chart for the data in Table The vertical scale ranges from 13.0 to 15.0 to accommodate the minimum and maximum temperature values of 13.44ºC and 14.77ºC, respectively. The horizontal scale is designed to include the 130 values arranged in sequence by year. The first point represents the first value of 13.88ºC, and so on.

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**Example: Run Chart of Earth’s Temperatures**

Run Chart of Individual Temperatures in Table 14-1.

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**Example: Run Chart of Earth’s Temperatures**

We see that as time progresses from left to right, the heights of the points appear to increase in value. If this pattern continues, rising temperatures will cause melting of large ice formations and widespread flooding, as well as substantial climate changes. This figure is evidence of global warming, which threatens us in many different ways.

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**Control Charts for Variation and Mean**

Only when a process is statistically stable can its data be treated as if they came from a population with a constant mean, standard deviation, distribution, and other characteristics. Definition A process is statistically stable (or within statistical control) if it has natural variation, with no patterns, cycles, or any unusual points. page 736 of Elementary Statistics, 10th Edition

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**Figure 14-2 Processes That Are Not Statistically Stable**

page 718 of Elementary Statistics, 11th Edition Figure 14-2(a): There is an obvious upward trend that corresponds to values that are increasing over time.

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**Figure 14-2 Processes That Are Not Statistically Stable**

Minitab page 718 of Elementary Statistics, 11th Edition Figure 14-2(b): There is an obvious downward trend that corresponds to steadily decreasing values.

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**Figure 14-2 Processes That Are Not Statistically Stable**

Minitab page 718 of Elementary Statistics, 11th Edition Figure 14-2(c): There is an upward shift. A run chart such as this one might result from an adjustment to the filling process, making all subsequent values higher.

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**Figure 14-2 Processes That Are Not Statistically Stable**

Minitab Figure 14-2(d): There is a downward shift-the first few values are relatively stable, and then something happened so that the last several values are relatively stable, but at a much lower level. page 718 of Elementary Statistics, 11th Edition

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**Figure 14-2 Processes That Are Not Statistically Stable**

Minitab page 718 of Elementary Statistics, 11th Edition Figure 14-2(e): The process is stable except for one exceptionally high value.

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**Figure 14-2 Processes That Are Not Statistically Stable**

Minitab page 718 of Elementary Statistics, 11th Edition Figure 14-2(f): There is an exceptionally low value.

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**Figure 14-2 Processes That Are Not Statistically Stable**

Minitab page 718 of Elementary Statistics, 11th Edition Figure 14-2(g): There is a cyclical pattern (or repeating cycle). This pattern is clearly nonrandom and therefore reveals a statistically unstable process.

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**Figure 14-2 Processes That Are Not Statistically Stable**

Minitab page 718 of Elementary Statistics, 11th Edition Figure 14-2(h): The variation is increasing over time. This is a common problem in quality control.

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**A common goal of many different methods of quality control is this:**

Reduce variation in a product or a service.

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Definitions Random variation is due to chance; it is the type of variation inherent in any process that is not capable of producing every good or service exactly the same way every time. Assignable variation results from causes that can be identified (such factors as defective machinery, untrained employees, and so on).

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**Control Chart for Monitoring Variation: The R Chart - Definition**

A control chart of a process characteristic (such as mean or variation) consists of values plotted sequentially over time, and it includes a center line as well as a lower control limit (LCL) and an upper control limit (UCL). The centerline represents a central value of the characteristic measurements, whereas the control limits are boundaries used to separate and identify any points considered to be unusual.

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**Control Chart for Monitoring Variation: The R Chart**

An R chart (or range chart) is a plot of the sample ranges instead of individual sample values, and it is used to monitor the variation in a process. In addition to plotting the range values, it includes a centerline located at , which denotes the mean of all sample ranges, as well as another line for the lower control limit and a third line for the upper control limit.

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**Monitoring Process Variation: Control Chart for R: Objective**

Construct a control chart for R (or an “R chart”) that can be used to determine whether the variation of process data is within statistical control.

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Requirements 1. The data are process data consisting of a sequence of samples all of the same size n. 2. The distribution of the process data is essentially normal. 3. The individual sample data values are independent.

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**Notation n = size of each sample, or subgroup**

= mean of the sample ranges (that is, the sum of the sample ranges divided by the number of samples)

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**(where D4 is found in Table 14-2)**

Graphs Points plotted: Sample ranges Centerline: (mean of sample ranges) Upper Control Limit (UCL): (where D4 is found in Table 14-2) Lower Control Limit (LCL): (where D3 is found in Table 14-2)

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**Table 14-2 Control Chart Constants**

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**Example: R Chart of Earth’s Temperatures**

Refer to the temperatures of the earth listed in Table Using the samples of size n = 10 for each decade, construct a control chart for R.

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**Example: R Chart of Earth’s Temperatures**

Using a centerline value of and control limits of and , proceed to plot the 13 sample ranges as 13 individual points. The result is shown in the Minitab display.

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Caution Upper and lower control limits of a control chart are based on the actual behavior of the process, not the desired behavior. Upper and lower control limits are totally unrelated to any process specifications that may have been decreed by the manufacturer.

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**Interpreting Control Charts**

When investigating the quality of some process, there are typically two key questions that need to be addressed: Based on the current behavior of the process, can we conclude that the process is within statistical control? Do the process goods or services meet design specifications? The methods of this chapter are intended to address the first question, but not the second.

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**Criteria for Determining When a Process Is Not Statistically Stable (Out of Statistical Control)**

1. There is a pattern, trend, or cycle that is obviously not random. 2. There is a point lying beyond the upper or lower control limits. 3. Run of 8 Rule: There are eight consecutive points all above or all below the center line.

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**Additional Criteria Used**

by Some Businesses There are 6 consecutive points all increasing or all decreasing. There are 14 consecutive points all alternating between up and down (such as up, down, up, down, and so on). Two out of three consecutive points are beyond control limits that are 2 standard deviations away from centerline. Four out of five consecutive points are beyond control limits that are 1 standard deviation away from the centerline.

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**Example: Interpreting R Chart of Earth’s Temperatures**

Examine the R chart shown in the Minitab display for the preceding example and determine whether the process variation is within statistical control.

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**Example: Interpreting R Chart of Earth’s Temperatures**

Apply the three criteria: 1. There is no obvious trend, or pattern that is not random. No point lies outside of the region between the upper and lower control limits. 3. There are not eight consecutive points all above or all below the centerline. We conclude that the variation (not necessarily the mean) of the process is within statistical control.

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**Control Chart for Monitoring**

Means: The x Chart The x chart is a plot of the sample means and is used to monitor the center in a process. In addition to plotting the sample means, we include a centerline located at x, which denotes the mean of all sample means, as well as another line for the lower control limit and a third line for the upper control limit.

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**Monitoring Process Variation: Control Chart for R: Objective**

Construct a control chart for x (or an “x chart”) that can be used to determine whether the center of process data is within statistical control.

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Requirements 1. The data are process data consisting of a sequence of samples all of the same size n. 2. The distribution of the process data is essentially normal. 3. The individual sample data values are independent.

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**Notation n = size of each sample, or subgroup**

= mean of the sample means (equal to the mean of all sample values combined)

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**Control Chart for Monitoring where A2 is found in Table 14-2**

Means: The x Chart Points plotted: Sample means Center line: x = mean of all sample means Upper Control Limit (UCL): x + A2R where A2 is found in Table 14-2 Lower Control Limit (LCL): x – A2R

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**Example: x Chart of Earth’s Temperatures**

Refer to the earth’s temperatures in Table Using samples of size n = 10 for each decade, construct a control chart for x. Based on the control chart for x, only, determine whether the process mean is within statistical control. Before plotting the 13 points corresponding to the 13 values of x, we must first find the value for the centerline and the values for the control limits.

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**Example: x Chart of Earth’s Temperatures**

From Table 14-2, with n = 10, we get A2 = 0.308 Upper control limit: page 743 of Elementary Statistics, 10th Edition. Lower control limit:

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**Example: x Chart of Earth’s Temperatures**

Minitab display:

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**Example: x Chart of Earth’s Temperatures**

Examination of the x chart shows that the process mean is out of statistical control because at least one of the three out-of-control criteria is not satisfied. Specifically, the first criterion is violated because there is a trend of values that are increasing over time, and the second criterion is violated because there are points lying beyond the control limits. page 743 of Elementary Statistics, 10th Edition.

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**Recap In this section we have discussed:**

Control charts for variation and mean. Run charts determine if characteristics of a process have changed. R charts (or range charts) monitor the variation in a process. x charts monitor the center in a process.

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**Control Charts for Attributes**

Section 14-3 Control Charts for Attributes

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Key Concept This section presents a method for constructing a control chart to monitor the proportion p for some attribute, such as whether a service or manufactured item is defective or nonconforming. The control chart is interpreted by using the same three criteria from Section 14-2 to determine whether the process is statistically stable.

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**Control Charts for Attributes**

These charts monitor the qualitative attributes of whether an item has some particular characteristic. In the previous section, the charts monitored the quantitative characteristics. The control chart for p (or p chart) is used to monitor the proportion p for some attribute. Samples must have the same size. There are ways to deal with samples of different sizes, but they are not dealt with in this text.

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Definition A control chart for p (or p chart) is a graph of proportions of some attribute (such as whether products are defective) plotted sequentially over time, and it includes a centerline, a lower control limit (LCL), and an upper control limit (UCL). Samples must have the same size. There are ways to deal with samples of different sizes, but they are not dealt with in this text.

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**Monitoring a Process Attribute: Control Chart for p: Objective**

Construct a control chart for p (or a “p chart”) that can be used to determine whether the proportion of some attribute (such as whether products are defective) from process data is within statistical control.

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Requirements 1. The data are process data consisting of a sequence of samples all of the same size n. 2. Each sample item belongs to one of two categories. 3. The individual sample data values are independent.

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Notation p = pooled estimate of proportion of defective items in the process = total number of defects found among all items sampled total number of items sampled q = pooled estimate of the proportion of process items that are not defective = 1 – p n = size of each sample or subgroup

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**Graph Center line: Upper control limit: Lower control limit:**

(If the calculation for the lower control limit results in a negative value, use 0 instead. If the calculation for the upper control limit exceeds 1, use 1 instead.)

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Caution Upper and lower control limits of a control chart for a proportion p are based on the actual behavior of the process, not the desired behavior. Upper and lower control limits are totally unrelated to any process specifications that may have been decreed by the manufacturer.

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**Example: Defective Heart Defibrillators**

The Guidant Corporation manufactures implantable heart defibrillators. Families of people who have died using these devices are suing the company. According to USA Today, “Guidant did not alert doctors when it knew 150 of every 100,000 Prizm 2DR defibrillators might malfunction each year.” Because lives could be lost, it is important to monitor the manufacturing process of implantable heart defibrillators.

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**Example: Defective Heart Defibrillators**

Consider a manufacturing process that includes careful testing of each defibrillator. Listed below are the numbers of defective defibrillators in successive batches of 10,000. Construct a control chart for the proportion p of defective defibrillators and determine whether the process is within statistical control. If not, identify which of the three out-of-control criteria apply. Defects:

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**Example: Defective Heart Defibrillators**

Defects: total number of defects from all samples combined total number of altimeters sampled

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**Example: Defective Heart Defibrillators**

Upper control limit: Lower control limit: The Minitab control chart for p is on the next slide.

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**Example: Defective Heart Defibrillators**

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**Example: Defective Heart Defibrillators**

We can interpret the control chart for p by considering the three out-of-control criteria listed in Section Using those criteria, we conclude that this process is out of statistical control for this reason: There appears to be a downward trend. Also, there are 8 consecutive points lying above the centerline, and there are also 8 consecutive points lying below the centerline. Although the process is out of statistical control, it appears to have been somehow improved, because the proportion of defects has dropped. The company would be wise to investigate the process so that the cause of the lowered rate of defects can be understood and continued in the future.

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**Recap In this section we have discussed:**

A control chart for attributes is a graph of proportions plotted sequentially over time. It includes a centerline, a lower control limit, and an upper control limit. The same three out-of-control criteria listed in Section 14-2 can be used.

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