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Chapter 5

- Control Charts for Variables

5-1. Introduction

- Variable - a single quality characteristic that

can be measured on a numerical scale. - When working with variables, we should monitor

both the mean value of the characteristic and the

variability associated with the characteristic.

5-2. Control Charts for and R

- Notation for variables control charts
- n - size of the sample (sometimes called a

subgroup) chosen at a point in time - m - number of samples selected
- average of the observations in the ith

sample (where i 1, 2, ..., m) - grand average or average of the averages

(this value is used as the center line of the

control chart)

5-2. Control Charts for and R

- Notation and values
- Ri range of the values in the ith sample
- Ri xmax - xmin
- average range for all m samples
- is the true process mean
- is the true process standard deviation

5-2. Control Charts for and R

- Statistical Basis of the Charts
- Assume the quality characteristic of interest is

normally distributed with mean , and standard

deviation, . - If x1, x2, , xn is a sample of size n, then he

average of this sample is - is normally distributed with mean, , and

standard deviation,

5-2. Control Charts for and R

- Statistical Basis of the Charts
- The probability is 1 - that any sample mean

will fall between - The above can be used as upper and lower control

limits on a control chart for sample means, if

the process parameters are known.

5-2. Control Charts for and R

- Control Limits for the chart
- A2 is found in Appendix VI for various values of

n.

5-2. Control Charts for and R

- Control Limits for the R chart
- D3 and D4 are found in Appendix VI for various

values of n.

5-2. Control Charts for and R

- Estimating the Process Standard Deviation
- The process standard deviation can be estimated

using a function of the sample average range. - This is an unbiased estimator of

5-2. Control Charts for and R

- Trial Control Limits
- The control limits obtained from equations (5-4)

and (5-5) should be treated as trial control

limits. - If this process is in control for the m samples

collected, then the system was in control in the

past. - If all points plot inside the control limits and

no systematic behavior is identified, then the

process was in control in the past, and the trial

control limits are suitable for controlling

current or future production.

5-2. Control Charts for and R

- Trial control limits and the out-of-control

process - If points plot out of control, then the control

limits must be revised. - Before revising, identify out of control points

and look for assignable causes. - If assignable causes can be found, then discard

the point(s) and recalculate the control limits. - If no assignable causes can be found then 1)

either discard the point(s) as if an assignable

cause had been found or 2) retain the point(s)

considering the trial control limits as

appropriate for current control.

5-2. Control Charts for and R

- Estimating Process Capability
- The x-bar and R charts give information about the

capability of the process relative to its

specification limits. - Assumes a stable process.
- We can estimate the fraction of nonconforming

items for any process where specification limits

are involved. - Assume the process is normally distributed, and x

is normally distributed, the fraction

nonconforming can be found by solving - P(x lt LSL) P(x gt USL)

5-2. Control Charts for and R

- Process-Capability Ratios (Cp)
- Used to express process capability.
- For processes with both upper and lower control

limits, - Use an estimate of if it is unknown.
- If Cp gt 1, then a low of nonconforming items

will be produced. - If Cp 1, (assume norm. dist) then we are

producing about 0.27 nonconforming. - If Cp lt 1, then a large number of nonconforming

items are being produced.

5-2. Control Charts for and R

- Process-Capability Ratios (Cp)
- The percentage of the specification band that the

process uses up is denoted by - The Cp statistic assumes that the process mean

is centered at the midpoint of the specification

band it measures potential capability.

5-2. Control Charts for and R

- Control Limits, Specification Limits, and Natural

Tolerance Limits - Control limits are functions of the natural

variability of the process - Natural tolerance limits represent the natural

variability of the process (usually set at

3-sigma from the mean) - Specification limits are determined by

developers/designers.

5-2. Control Charts for and R

- Control Limits, Specification Limits, and Natural

Tolerance Limits - There is no mathematical relationship between

control limits and specification limits. - Do not plot specification limits on the charts
- Causes confusion between control and capability
- If individual observations are plotted, then

specification limits may be plotted on the chart.

5-2. Control Charts for and R

- Rational Subgroups
- X bar chart monitors the between sample

variability - R chart monitors the within sample variability.

5-2. Control Charts for and R

- Guidelines for the Design of the Control Chart
- Specify sample size, control limit width, and

frequency of sampling - if the main purpose of the x-bar chart is to

detect moderate to large process shifts, then

small sample sizes are sufficient (n 4, 5, or

6) - if the main purpose of the x-bar chart is to

detect small process shifts, larger sample sizes

are needed (as much as 15 to 25)which is often

impracticalalternative types of control charts

are available for this situationsee Chapter 8

5-2. Control Charts for and R

- Guidelines for the Design of the Control Chart
- If increasing the sample size is not an option,

then sensitizing procedures (such as warning

limits) can be used to detect small shiftsbut

this can result in increased false alarms. - R chart is insensitive to shifts in process

standard deviation.(the range method becomes less

effective as the sample size increases) may want

to use S or S2 chart. - The OC curve can be helpful in determining an

appropriate sample size.

5-2. Control Charts for and R

- Guidelines for the Design of the Control Chart
- Allocating Sampling Effort
- Choose a larger sample size and sample less

frequently or, Choose a smaller sample size and

sample more frequently - The method to use will depend on the situation.

In general, small frequent samples are more

desirable.

5-2. Control Charts for and R

- Changing Sample Size on the and R Charts
- In some situations, it may be of interest to know

the effect of changing the sample size on the

x-bar and R charts. Needed information - average range for the old sample

size - average range for the new sample

size - nold old sample size
- nnew new sample size
- d2(old) factor d2 for the old sample size
- d2(new) factor d2 for the new sample size

5-2. Control Charts for and R

- Changing Sample Size on the and R Charts
- Control Limits

5-2.3 Charts Based on Standard Values

- If the process mean and variance are known or can

be specified, then control limits can be

developed using these values - Constants are tabulated in Appendix VI

5-2.4 Interpretation of and R Charts

- Patterns of the plotted points will provide

useful diagnostic information on the process, and

this information can be used to make process

modifications that reduce variability. - Cyclic Patterns
- Mixture
- Shift in process level
- Trend
- Stratification

5-2.5 The Effects of Nonnormality

and R

- In general, the chart is insensitive

(robust) to small departures from normality. - The R chart is more sensitive to nonnormality

than the chart - For 3-sigma limits, the probability of committing

a type I error is 0.00461on the R-chart. (Recall

that for , the probability is only 0.0027).

5-2.6 The Operating Characteristic

Function

- How well the and R charts can detect process

shifts is described by operating characteristic

(OC) curves. - Consider a process whose mean has shifted from an

in-control value by k standard deviations. If

the next sample after the shift plots in-control,

then you will not detect the shift in the mean.

The probability of this occurring is called the

-risk.

5-2.6 The Operating Characteristic

Function

- The probability of not detecting a shift in the

process mean on the first sample is - L multiple of standard error in the control

limits - k shift in process mean (of standard

deviations).

5-2.6 The Operating Characteristic

Function

- The operating characteristic curves are plots of

the value against k for various sample sizes.

5-2.6 The Operating Characteristic

Function

- If is the probability of not detecting the

shift on the next sample, then 1 - is the

probability of correctly detecting the shift on

the next sample.

5-3.1 Construction and Operation of

and S Charts

- First, S2 is an unbiased estimator of 2
- Second, S is NOT an unbiased estimator of
- S is an unbiased estimator of c4
- where c4 is a constant
- The standard deviation of S is

5-3.1 Construction and Operation of

and S Charts

- If a standard is given the control limits for

the S chart are - B5, B6, and c4 are found in the Appendix for

various values of n.

5-3.1 Construction and Operation of

and S Charts

- No Standard Given
- If is unknown, we can use an average
- sample standard deviation,

5-3.1 Construction and Operation of

and S Charts

- Chart when Using S
- The upper and lower control limits for the

chart are given as - where A3 is found in the Appendix

5-3.1 Construction and Operation of

and S Charts

- Estimating Process Standard Deviation
- The process standard deviation, can be

estimated by

5-3.2 The and S Control Charts with

Variable Sample Size

- The and S charts can be adjusted to account

for samples of various sizes. - A weighted average is used in the calculations

of the statistics. - m the number of samples selected.
- ni size of the ith sample

5-3.2 The and S Control Charts with

Variable Sample Size

- The grand average can be estimated as
- The average sample standard deviation is

5-3.2 The and S Control Charts with

Variable Sample Size

- Control Limits
- If the sample sizes are not equivalent for each

sample, then - there can be control limits for each point

(control limits may differ for each point

plotted)

5-3.3 The S2 Control Chart

- There may be situations where the process

variance itself is monitored. An S2 chart is - where and are

points found from the chi-square distribution.

5-4. The Shewhart Control Chart for

Individual Measurements

- What if you could not get a sample size greater

than 1 (n 1) Examples include - Automated inspection and measurement technology

is used, and every unit manufactured is analyzed. - The production rate is very slow, and it is

inconvenient to allow samples sizes of N gt 1 to

accumulate before analysis - Repeat measurements on the process differ only

because of laboratory or analysis error, as in

many chemical processes. - The X and MR charts are useful for samples of

sizes - n 1.

5-4. The Shewhart Control Chart for

Individual Measurements

- Moving Range Chart
- The moving range (MR) is defined as the absolute

difference between two successive observations - MRi xi - xi-1
- which will indicate possible shifts or

changes in the process from one observation to

the next.

5-4. The Shewhart Control Chart for

Individual Measurements

- X and Moving Range Charts
- The X chart is the plot of the individual

observations. The control limits are - where

5-4. The Shewhart Control Chart for

Individual Measurements

- X and Moving Range Charts
- The control limits on the moving range chart are

5-4. The Shewhart Control Chart for

Individual Measurements

- Example
- Ten successive heats of a steel alloy are

tested for hardness. The resulting data are - Heat Hardness Heat Hardness
- 1 52 6 52
- 2 51 7 50
- 3 54 8 51
- 4 55 9 58
- 5 50 10 51

5-4. The Shewhart Control Chart for

Individual Measurements

- Example

5-4. The Shewhart Control Chart for

Individual Measurements

- Interpretation of the Charts
- X Charts can be interpreted similar to charts.

MR charts cannot be interpreted the same as

or R charts. - Since the MR chart plots data that are

correlated with one another, then looking for

patterns on the chart does not make sense. - MR chart cannot really supply useful information

about process variability. - More emphasis should be placed on interpretation

of the X chart.

5-4. The Shewhart Control Chart for

Individual Measurements

- The normality assumption is often taken for

granted. - When using the individuals chart, the normality

assumption is very important to chart

performance.

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