Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 14.1 - 1 Lecture Slides...

Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 14.1 - 1

Lecture Slides

Elementary Statistics Eleventh Edition

and the Triola Statistics Series

by Mario F. Triola


14-1 Review and Preview

14-2 Control Charts for Variation and Mean

14-3 Control Charts for Attributes

Chapter 14Statistical Process Control


Section 14-1Review and Preview


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.


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.


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.


Section 14-2 Control Charts for

Variation and Mean


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).

x


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.


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.


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.



Following is the Minitab-generated run chart for the data in Table 14-1. 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.



Run Chart of Individual Temperatures in Table 14-1.



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.


Definition

A process is statistically stable (or within statistical control) if it has natural variation, with no patterns, cycles, or any unusual points.

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.


Figure 14-2 Processes That Are Not Statistically Stable

Figure 14-2(a): There is an obvious upward trend that corresponds to values that are increasing over time.



Figure 14-2(b): There is an obvious downward trend that corresponds to steadily decreasing values.

Minitab



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.

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.

Minitab



Figure 14-2(e): The process is stable except for one exceptionally high value.

Minitab



Figure 14-2(f): There is an exceptionally low value.

Minitab



Figure 14-2(g): There is a cyclical pattern (or repeating cycle). This pattern is clearly nonrandom and therefore reveals a statistically unstable process.

Minitab



Figure 14-2(h): The variation is increasing over time. This is a common problem in quality control.

Minitab


A common goal of many different methods of quality

control is this:

Reduce variation in a product or a service.


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).


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.

Control Chart for Monitoring Variation: The R Chart - Definition


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.

Control Chart for Monitoring Variation: The R Chart

R


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.

Monitoring Process Variation: Control Chart for R: Objective


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.

Requirements


n = size of each sample, or subgroup

Notation

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


Points plotted: Sample ranges

Graphs

Lower Control Limit (LCL):

(where D3 is found in Table 14-2) D3

R

Upper Control Limit (UCL):

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

R

Centerline: (mean of sample ranges) R


Table 14-2 Control Chart

Constants


Refer to the temperatures of the earth listed in Table 14-1. Using the samples of size n = 10 for each decade, construct a control chart for R.

Example: R Chart of Earth’s Temperatures

R

0.49 0.41 ... 0.36

130.377

D30.223 and D

41.777 from Table 14-2

UCL: D4R (1.777)(0.3777) 0.6712

LCL: D3R (0.223)(0.3777) 0.0842


Using a centerline value of and control limits of 0.6712 and 0.0842, proceed to plot the 13 sample ranges as 13 individual points. The result is shown in the Minitab display.

Example: R Chart of Earth’s Temperatures

R 0.377


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.

Caution


1. Based on the current behavior of the process, can we conclude that the process is within statistical control?

2. Do the process goods or services meet design specifications?

When investigating the quality of some process, there are typically two key questions that need to be addressed:

The methods of this chapter are intended to address the first question, but not the second.

Interpreting Control Charts


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.


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.


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.


Example: Interpreting R Chart of Earth’s Temperatures

Apply the three criteria:

1. There is no obvious trend, or pattern that is not random.

2. 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.


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.

Control Chart for Monitoring Means: The x Chart


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.

Monitoring Process Variation: Control Chart for R: Objective



2. The distribution of the process data is essentially normal.


Requirements


n = size of each sample, or subgroup

Notation

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


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

where A2 is found in Table 14-2

Control Chart for Monitoring Means: The x Chart


Example: x Chart of Earth’s Temperatures

Refer to the earth’s temperatures in Table 14-1. 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.



From Table 14-2, with n = 10, we get A2 = 0.308

x

13.819 13.692 ...14.636

1314.019

R

0.49 0.41 ... 0.36

13 0.377

Upper control limit:

2 14.019 (0.308)(0.377) 14.135 x A R

Lower control limit:

x A2R 14.019 (0.308)(0.377) 13.903



Minitab display:



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.


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.


Section 14-3 Control Charts for

Attributes


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.


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.


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).


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.

Monitoring a Process Attribute: Control Chart for p:

Objective



2. Each sample item belongs to one of two categories.


Requirements


p = pooled estimate of proportion of defective items in the process

= total number of defects found among all items sampledtotal 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

Notation


Center line:



Graph

(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.)

p

3pq

pn

3pq

pn


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.

Caution


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.

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:15 12 14 14 11 16 17 11 167 8 5 7 6 6 8 5 7 8 7



Defects:

15 12 14 14 11 16 17 11 16

7 8 5 7 6 6 8 5 7 8 7


p

1512 14 ... 7

2010,000

200

200,0000.001

p total number of defects from all samples combinedtotal number of altimeters sampled



q 1 p 1 0.001 0.999


p 3

pq

n 0.001 3

(0.001)(0.999)

10,000 0.001948

p 3

pq

n 0.001 3

(0.001)(0.999)

100 0.000052


The Minitab control chart for p is on the next slide.



We can interpret the control chart for p by considering the three out-of-control criteria listed in Section 14-2. 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.


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.

Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 14.1 - 1 Lecture Slides...

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