Control chart for Iindustrial Statistics

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    Statistical basis of the control chart

    1. Basic principles

    2. Choice of control limits

    3. Sample size and sampling frequency4. Rational subgroups

    5. Analysis of patterns on control charts

    6. Discussion of sensitizing rules for control chart

    7. Phase I and Phase II of control chart application

    1. Basic principles

    The control chart contains a center line (CL) that represents the average value of the

    quality characteristic. Two other horizontal lines, called the upper control limit (UCL) and

    the lower control limit (LCL) are also shown on the chart.

    If all of the sample points fall between the UCL and LCL, the process is assumed to be in

    control and no action is necessary.

    If a point falls outside of the UCL and LCL, the process is assumed to be out of control

    and investigation and corrective action are required.

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    It is customary to connect the sample points on the control chart with straight line.

    If all the points plot inside the control limits and if they behave in a systematic or

    nonrandom manner, then this could be an indication that the process is out of control.

    General model of a control chart

    Ifx is a sample statistic that measures some quality characteristic of interest, then the

    center line (CL), the upper control limit (UCL) and the lower control limit (LCL) become:

    xx

    x

    xx

    LLCL

    CL

    LUCL

    Where L is the distance of the control limits from the center line expressed in standard

    deviation units.Classification of control chart

    Control chart may be classified into two general types:

    (a) Variable control chart

    (b) Attribute control chart

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    If the quality characteristic can be measured and expressed as a number on some

    continuous scale of measurement, then a variable control chart is used.

    Many quality characteristics are not measured on a continuous scale, In these cases, we

    may judge each unit of product as either conforming or non-conforming on the basis ofwhether or not it possesses certain attributes, then an attribute control chart is used.

    2. Choice of control limits

    By moving the control limits farther from the center line, we decrease the risk of a type I

    error, that is, the risk of a point falling beyond the control limits, indicating an out ofcontrol condition when no assignable cause is present.

    Widening the control limits also increase the risk of a type II error, that is, the risk of a

    point falling between the control limits when the process is really out of control.

    By moving the control limits closer to the center line, the opposite effect is obtained, therisk of type I error is increased while the risk of type II error is decreased.

    Regardless of the distribution of the quality characteristic, it is standard practice in the

    United States to use the three sigma control limits.

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    Warning limits on control charts

    Some analysts suggest using two sets of limits on control charts. The outer limits usually

    at three sigma are known as action limits. The inner limits usually at two are known as

    warning limits.

    If one or more points fall between the warning limits and the action limits or very close

    to the warning limits, we should be suspicious that the process may not be operating

    properly.

    In this case it is necessary to increase the sampling frequency and/or the sample size sothat more information about the process can be obtained quickly.

    3. Sample size and sampling frequency

    In general, larger samples will make it easier to detect small shifts in the process.

    If the process shift is relatively large, then we use smaller sample size.

    Current industry practice tends to favor smaller, more frequent samples, particularly in

    high-volume manufacturing processes, or where a great many types of assignable causes

    can occur.

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    4. Rational subgroups

    The rational subgroup means that subgroups or samples should be selected so that if

    assignable causes are present, the chance for differences between subgroups will be

    maximized, while the chance for differences due to these assignable causes within a

    subgroup will be minimized.

    Two general approaches to constructing rational subgroups are used. In the first

    approach, each sample consists of units that are produced at the same time (or as closely

    together as possible).

    In the second approach, each sample consists of units of product that are representative of

    all units that have been produced since the last sample was taken.

    If a process consists of several machines that pool their output into a common stream. A

    logical approach to rational subgrouping here is to apply control chart techniques to theoutput for each individual machine.

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    5. Analysis of patterns on control charts

    A control chart may indicate an out of control condition when:

    (a) One or more points fall beyond the control limits,

    (b) Plotted points exhibit some nonrandom pattern of behavior,

    (c) Plotted points show an unusual run up or run down,

    (d) Plotted points show a cyclic behavior, a trend and a sudden jump.

    The Western Electric Handbook (1956) suggests a set of decision rules for detecting

    nonrandom patterns on control charts. The process is out of control if either:

    (a) One point plots outside the three-sigma control limits,

    (b) Two out of three consecutive points plot beyond the two-sigma warning limits,

    (c) Four out of five consecutive points plot at a distance of one-sigma or beyond from the

    center line,

    (d) Eight consecutive points plot on one side of the center line.

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    The process in the figure shows out of control since four of five consecutive points (last

    four points) fall in zone B or beyond.

    6. Discussion of sensitizing rules for control charts

    Several criteria may be applied simultaneously to a control chart to determine whether

    the process is out of control. These criteria are known as sensitizing rule:

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    (a) One or more points outside of the control limits,

    (b) Two of three consecutive points outside the two-sigma warning limits but still inside

    the control limits,

    (c) Four of five consecutive points beyond the one-sigma limits,

    (d) A run of eight consecutive points on one side of the center line,(e) Six points in a row steadily increasing or decreasing,

    (f) Fifteen points in a row in zone C (both above and below the center line),

    (g) Fourteen points in a row alternating up and down,

    (h) Eight points in a row on both sides of the center line with none in zone C,

    (i) An unusual or nonrandom pattern in the data,

    (j) One or more points near a warning or control limit.

    7. Phase I and Phase II of control application

    In Phase I, a set of process data is gathered and analyzed. Trial control limits are

    determined based on the data and a reliable control limits are established to monitor futureproduction.

    In Phase II, we use the control chart to monitor the process for each successive sample.

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    Control charts for variable

    When dealing with a quality characteristic that is a variable, it is usually necessary to

    monitor both the mean value of the quality characteristic and its variability.

    Control of the process mean is usually done with control chart for mean or the chart.

    Process variability can be monitored with either a control chart for the standard deviation,

    called the s chart, or a control chart for the range, called anR chart.

    The R chart is more widely used than s chart.

    x

    Control charts for and Rx

    Charts based on standard values

    When it is possible to specify standard values for the process mean and standard

    deviation, we may use these standards to establish the control chart without analysis of past

    data.

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    Suppose that the standards given are and . Then the control limits of the chart are x

    nLCL

    CL

    nUCL

    3

    3

    The quantity is a constant that depends on n, which is obtained from the given

    table.A

    n

    3

    1

    2

    2

    DLCL

    dCL

    DUCL

    The control limits of the R chart can be written as

    ALCL

    CL

    AUCL

    So the control limits can be written as

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    Charts based on standard values not given

    Suppose that a quality characteristic is normally distributed with mean and standard

    deviation .

    In practice, we do not know and . Therefore they must be estimated from samples or

    subgroups.

    Suppose that m samples are available, each containing n observations on the quality

    characteristics. Typically,m should be at least 20 to 25 samples and n should be 4, 5 or 6.

    Let be the average of each sample. Then the best estimator of ismxxx ,,,

    21

    m

    xxxx

    m

    21

    Let be the ranges of the m samples. The average range ism

    RRR ,,,21

    m

    RRRR

    m

    21

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    Example: Suppose in an industry we want to establish control chart of the process using

    mean and range charts. We have 25 samples each of 5 size. The data are given as follows:

    Then the control limits of the chart arex

    RAxLCL

    xCL

    RAxUCL

    2

    2

    The quantity is a constant that depends on n, which is obtained from the given table.2A

    The control limits of the R chart can be written as

    RDLCL

    RCL

    RDUCL

    3

    4

    Process standard deviation can be estimated as

    2

    d

    R

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    When setting up and R control chart, it is best to begin with the R chart. Because the

    control limits on the chart depend on the process variability, unless process variability is

    in control, these limits will not have much meaning.

    xx

    The center line for the R chart is

    For samples ofn = 5, we find that and . Therefore, the limits for the R chart

    are:

    03D 114.24 D

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    Since the R chart indicates that process variability is in control, we may now construct the

    mean chart.

    The center line is

    The upper and lower control limits are

    Since both charts exhibit control, we would conclude that the process is in control.

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    The process standard deviation can be estimated as

    Revision of control limits and center lines

    The effective use of any control chart will require periodic revision of the control limits

    and center lines.

    Some practitioners establish regular periods for review and revision of control chartlimits, such as every week, every month or every 25, 50 or 100 samples.

    When the R chart is out of control, we eliminate the out of control points and recompute a

    revised value ofR. This value is then used to determine new limits and center line on the R

    chart and new limits on the chart.x

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    Example: Establish and R control charts for the following data. Is the process in

    statistical control?x

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    0 2 4 6 8 10 12 14 16 18 20

    0

    5

    10

    15

    20

    0 2 4 6 8 10 12 14 16 18 20

    72

    74

    76

    78

    80

    82

    84

    86

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    function y=controlchart(data);

    s=size(data);

    x=1:s(1); % value of the x axis

    meandata=mean(data'); % mean of the observations

    rangedata=range(data'); % range of the observations

    gmean=mean(meandata); % grand mean

    grange=mean(rangedata); % grand range

    % control limits of the mean chart

    uclx=gmean+.577*grange;

    clx=gmean;lclx=gmean-.577*grange;

    % control limits of the range chart

    uclr=2.114*grange;

    clr=grange;

    lclr=0*grange;

    % mamimum and minimum value of the mean and range data

    maxx=max(meandata);

    maxr=max(rangedata);

    minx=min(meandata);

    minr=min(rangedata);

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    % range chart

    figure(1);

    hold on;

    axis([0 s(1) minr-5 maxr+5]); %axis([xmin xmax ymin ymax])line([0 s(1)],[clr clr], 'color','m');

    line([0 s(1)],[lclr lclr], 'color','k');

    line([0 s(1)],[uclr uclr], 'color','k');

    plot(x,rangedata,'-*b');

    % mean chart

    figure(2);

    hold on;

    axis([0 s(1) minx-5 maxx+5]);

    line([0 s(1)],[clx clx], 'color','m');

    line([0 s(1)],[lclx lclx], 'color','k');

    line([0 s(1)],[uclx uclx], 'color','k');

    plot(x,meandata,'-*b');

    y=[uclr clr lclr;uclx clx lclx];

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    Guidelines for the design of and R chart

    If the chart is being used primarily to detect moderate to large process shifts say, on the

    order of or larger, then relatively small samples of size n = 4, 5 or 6 are reasonably

    effective. On the other hand, if we are trying to detect small shifts, than larger sample sizesof possibly n = 15 to n = 25 are needed.

    The R chart is relatively insensitive to shifts in the process standard deviation for small

    samples. For large n say, n > 10 or 12, it is probably best to use a control chart for or s

    instead of the R chart.

    From economic consideration, if the cost associated with producing defective items is

    high, smaller, more frequent samples are better that larger, less frequent ones.

    The use of 3 sigma control limits on the andR control charts is a widespread practice. If

    type I errors (an out of signal is generated when the process is really in control) are very

    expensive to investigate, then it may be best to use wider control limits, perhaps at 3.5

    sigma. However, if the process is such that out of signals are quickly and easily investigated

    with a minimum of lost time and cost, then narrower control limits, perhaps at 2.5 sigma are

    appropriate.

    x

    x2

    2s

    x

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    Changing sample size on the and R chart

    There are two situations in which the sample size n is not constant.

    Each sample may consists of a different number of observations.

    A permanent change in sample size because of cost or because the process has exhibited

    good stability and fewer resources are being allocated for process monitoring.

    In the second situation it is easy to recompute the new control limits directly from the old

    ones without collecting additional samples based on the new sample size.

    x

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    For the chart the new control limits arex

    Where the center line is unchanged and the factor is selected for the new sample size.x 2A

    For the R chart the new control limits are

    Where and are selected for the new sample size.3

    D4

    D

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    Example: Consider the previous example where the limits were based on the sample size

    of five. Since the process exhibits good control, the process engineering personnel want to

    reduce the sample size to three and to calculate the new control limits.

    For the chart the new control limits arex

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    For the R chart the new control limits are

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    Interpretation of and R chart

    In the interpreting patterns on the chart, we must determine whether or not theR chart

    is in control.

    If the both and R charts exhibit a nonrandom pattern, the best strategy is to eliminate

    the R chart assignable causes first. In many cases, this will automatically eliminate the

    nonrandom pattern on the chart.

    Never attempt to interpret the chart when the R chart indicates an out of control

    condition.

    x

    x

    x

    x

    x

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    Example:

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    Control limits of the mean and range chart:

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    Range chart:

    0 5 10 15 20-4

    -2

    0

    2

    4

    6

    8

    10

    12

    14

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    Mean chart:

    0 5 10 15 20

    28

    30

    32

    34

    36

    38

    40

    42

    X: 12

    Y: 38.6

    X: 15

    Y: 37.1

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    Control limits of the mean and range chart for 22 samples:

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    0 2 4 6 8 10 12 14 16 18 20 22-4

    -2

    0

    2

    4

    6

    8

    10

    12

    Range chart:

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    0 2 4 6 8 10 12 14 16 18 20 22

    28

    30

    32

    34

    36

    38

    40

    Mean chart:

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    0 5 10 15 2026

    28

    30

    32

    34

    36

    38

    40

    42

    0 5 10 15 2026

    28

    30

    32

    34

    36

    38

    40

    42

    0 5 10 15 2026

    28

    30

    32

    34

    36

    38

    40

    42