c hapter 6 control charts for variables. 6-1. i ntroduction variable - a single quality...

CHAPTER 6

Control Charts for Variables

6-1. INTRODUCTION

Variable - a single quality characteristic that can be measured on a numerical scale.

We monitor both the mean value of the characteristic and the variability associated with the characteristic.

VARIABLES DATA

Examples Length, width, height Weight Temperature Volume

BOTH MEAN AND VARIABILITY MUST BE CONTROLLED

6-2. 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)

x

ix

x


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

x

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 the average of this sample is

= (X1 + X2 + … + Xn)/n

is normally distributed with mean, , and standard deviation,

x

n/x x

x


Statistical Basis of the Charts If n samples are taken and their average is

computed, the probability is 1- α 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.

x

nZZ

andn

ZZ

2/x2/

2/x2/

CONTROL CHARTS FOR When the parameters are not known

Compute Compute R(bar)

Now UCL =

LCL =

x

x

nX

3

nX

3

CONTROL CHARTS FOR But, est =

So, can be written as:

Let A2 =

Then,

x

RAnd

R2

2

3

2/ dR

n/3nd

R

2

3

nd2

3


Control Limits for the chart

A2 is found in Appendix VI for various values of n.

x

x

RAxLCL

xLineCenter

RAxUCL

2

2

CONTROL CHART FOR R

We need an estimate of R

We will use relative range W = R/R = WLet d3 be the standard deviation of W

StdDev [R] = StdDev [WRd

Use R to estimate R

CONTROL CHART FOR R

est,R = d3(R(bar)/d2)

UCL = R(bar) + 3 est,R = R(bar) + 3d3(R(bar)/d2)

CL = R(bar)LCL = R(bar) - 3 est,R = R(bar) - 3d3(R(bar)/d2)

Let D3 = 1 – 3(d3/d2)

And, D4 = 1 + 3(d3/d2)


Control Limits for the R chart

D3 and D4 are found in Appendix VI for various values of n.

x

UCL D R

Center Line R

LCL D R

4

3


Trial Control Limits The control limits obtained from equations (6-4) and

(6-5) 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.

x


Trial control limits and the out-of-control process

Eqns 6.4 and 6.5 are trial control limits Determined from m initial samples

Typically 20-25 subgroups of size n between 3 and 5 Any out-of-control points should be examined for

assignable causes If assignable causes are found, discard points from

calculations and revise the trial control limits Continue examination until all points plot in control If no assignable cause is found, there are two options

1. Eliminate point as if an assignable cause were found and revise limits

2. Retain point and consider limits appropriate for control If there are many out-of-control points they should

be examined for patterns that may identify underlying process problems

x

EXAMPLE 6.1 THE HARD BAKE PROCESS

6-2. CONTROL CHARTS FOR AND R Estimating Process Capability

The x-bar and R charts give information about the performance or Process Capability of the process.

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 < LSL) + P(x > USL)

x

EXAMPLE 6-1: ESTIMATING PROCESS CAPABILITY

Determine est = R(bar)/d2 = .32521/2.326=0.1398 Where d2 values are given in App. Table VI

Specification limits are 1.5 + .5 microns. (1.00, 2.00)

Assuming N(1.5056, 0.13982) Compute p = P(x < 1.00) + P(x > 2.00)

EXAMPLE 6-1: ESTIMATING PROCESS CAPABILITY

P = ([1.00–1.5056]/0.1398) +1 – ([2.00-1.5056]/.1398) = (-3.6166) + 1 – (3.53648) = .00035

These values from the APPENDIX 2 / 693 So, about 350 ppm will be out of specification


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.

Cest,p = (2.0 – 1.0)/[6(.1398)] = 1.192

x

6

LSLUSLCp


If Cp > 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 < 1, then a large number of nonconforming items are being produced.

x


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

the process uses up is denoted by

**The CThe Cpp statistic assumes that the process statistic assumes that the process mean is centered at the midpoint of the mean is centered at the midpoint of the specification band – it measures potential specification band – it measures potential capability.capability.

x

%100C

1P̂

p

EXAMPLE 6-1: ESTIMATING BANDWIDTH USED

Pest = (1/1.192)100% = 83.89% The process uses about 84% of the specification

band.

PHASE II OPERATION OF CHARTS

Use of control chart for monitoring future production, once a set of reliable limits are established, is called phase II of control chart usage (Figure 6.4)

A run chart showing individuals observations in each sample, called a tolerance chart or tier diagram (Figure 6.5), may reveal patterns or unusual observations in the data


Control Limits, Specification Limits, and Natural Tolerance Limits (See Fig 6.6)

Control limits are functions of the natural variability of the process (LCL, UCL)

Natural tolerance limits represent the natural variability of the process (usually set at 3-sigma from the mean) (LNTL, UNTL)

Specification limits are determined by developers/designers. (LSL, USL)

x

6-2. CONTROL CHARTS FOR AND R x


Control Limits, Specification Limits, and Natural Tolerance Limits

There is no mathematical relationship between control limits and specification limits.

x

RATIONAL SUBGROUPS

charts monitor between-sample variability (variability in the process over time)

R charts measure within-sample variability (the instantaneous process variability at a

given time) Standard deviation estimate of used to

construct control limits is calculated from within-sample variability

It is not correct to estimate using

x


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).

x



R chart is insensitive to shifts in process standard deviation when n is small The range method becomes less effective as the

sample size increases May want to use S or S2 chart for larger values of n

> 10

x



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 In general, small frequent samples are more desirableare more desirable.

x

CHANGING SAMPLE SIZE OF X-BAR AND R CHARTS

1- variable sample sizevariable sample size: the center line changes continuously and the chart is difficult to interpret. (X-bar and s is more appropriate)

2- Permanent or semi-permanent change: Permanent or semi-permanent change:

Introduction to Statistical Q

uality Control, 4th

Edition

old

old

Roldd

newdAXLCL

XCL

Roldd

newdAXUCL

chartXfor

])(

)([

])(

)([

2

22

2

22

old

oldnew

old

Roldd

newdDLCL

Roldd

newdRCL

Roldd

newdDUCL

chartRfor

])(

)([

])(

)([

])(

)([

2

23

2

2

2

24

6-2.3 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

ALCL

CL

AUCL

chartX

1

2

2

DLCL

dCL

DUCL

chartR

6-2.4 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 ( systematic environmental changes ) Mixture ( tend to fall near or slightly outside the control

limits – over control and adjustment too often) Shift in process level ( new workers, raw material,…) Trend ( continuous movement in one direction- wear out

) Stratification ( a tendency for the point to cluster

artificially around the center line- e.g. Error in control limits computations )

x

6-2.6 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.

x


The probability of not detecting a shift in the process mean on the first sample is = P{LCL < Xbar < UCL = 1 = 0 + k}

L= multiple of standard error in the control limits

k = shift in process mean (# of standard deviations).

)nkL()nkL(

EXAMPLE We are using a 3 limit Xbar chart with

sample size equal to 5 L = 3 n = 5

Determine the probability of detecting a shift to 1 = 0 + 2 on the first sample following the shift k = 2

EXAMPLE, CONTINUED

So, =[3–2 SQRT(5)]-[-3–2 SQRT(5)]= (-1.47) – (-7.37) = .0708

P(not detecting on first sample) = .0708P(detecting on first sample) =1- .0708=.9292


The operating characteristic curves are plots of the value against k for various sample sizes

If the shift is 1.0σ and the sample size is n = 5, then

β = 0.75.

USE OF FIGURE 6-13 Let k = 1.5

When n = 5, = .35 P(detection on 2nd sample) = .35(.65) P(detection on 3rd sample) = .352(.65)

P(detection on rth sample) = r-1(1-) ARL = 1/(1-)

1/(1-.35) = 1/.65 = 1.54

OC CURVE FOR THE R CHART

Use the ratio of new to old process standard deviation )in Fig. 6.14

R chart is insensitive when n is small But, when n is large, W = R/ loses efficiency, and S chart is better

6-3.1 CONSTRUCTION AND OPERATION OF AND S CHARTS

S is an estimator of c4

where c4 is a constant depends on sample size n.

The standard deviation of S is

x

24c1


If a standard is given the control limits for the S chart are:

B5, B6, and c4 are found in the Appendix VI for various values of n.

x

5

4

6

BLCL

cCL

BUCL


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

standard deviation

x

m

1iiS

m

1S

SBLCL

SCL

SBUCL

3

4


Chart when Using SThe upper and lower control limits for the chart

are given as

where A3 is found in the Appendix VI

x

x

UCL x A S

CL x

LCL x A S

3

3

x

6-3.2 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

x

x


The grand average can be estimated as:

The average sample standard deviation is:

x

m

1ii

m

1iii

n

xnx

m

1ii

m

1i

2ii

mn

S)1n(S


Control Limits

x

SAxLCL

xCL

SAxUCL

3

3

SBLCL

SCL

SBUCL

3

4

6-4. 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 > 1 to accumulate before analysis

The X and MR charts are useful for samples of sizes n = 1.


Moving Range Control 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.


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

observations. The control limits are

where 2

2

d

MR3xLCL

xCL

d

MR3xUCL

m

MRMR

m

1ii


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

are:

0LCL

MRCL

MRDUCL 4


Example Ten successive heats of a steel alloy are tested

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


Example

109876543210

62

52

42

Observation

Indi

vid

uals

X=52.40

3.0SL=60.97

-3.0SL=43.83

10

5

0Mov

ing

Ran

ge

R=3.222

3.0SL=10.53

-3.0SL=0.000

I and MR Chart for hardness


Interpretation of the Charts MR charts cannot be interpreted the same as

R or 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.

x


The normality assumption is often taken for granted.

When using the individuals chart, the normality assumption is very important to chart performance.

P-Value: 0.063A-Squared: 0.648

Anderson-Darling Normality Test

N: 10StDev: 2.54733Average: 52.4

585756555453525150

.999

.99

.95

.80

.50

.20

.05

.01

.001P

roba

bilit

y

hardness

Normal Probability Plot

c hapter 6 control charts for variables. 6-1. i ntroduction variable - a single quality...

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