Control Chart

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Introduction

• used to determine if a manufacturing or business process is in a state of statistical control

• used to detect/identify assignable causes. always has a central line for the average, an upper line.

• for the upper control limit and a lower line for the

lower control limit.

• also known as Shewhart charts or process-behavior charts

Control Chart

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Attributes Control Charts

• charts applied to data that follow a discrete distribution.

Variable Control Charts • charts applied to data that follow a continuous distribution.

• Are typically used used in pairs:• monitors process average • monitors the variation in the process

• A quality characteristic that is measured on a numerical scale is called a variable.• dimension• length, width• weight• temperature• volume

Control Chart

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• X-bar and R chart (also called averages and range chart)

• X-bar and s chart

• Moving average–moving range chart (also called MA–MR chart)

• Target charts (also called difference charts, deviation charts and nominal charts)

• CUSUM (cumulative sum chart)

• EWMA (exponentially weighted moving average chart)

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• is a type of control chart used to monitor variables data when samples are collected atis a type of control chart used to monitor variables data when samples are collected at regular intervals from a business or industrial process.

• is advantageous in the following situations:• The sample size is relatively small (n ≤

10)• The sample size is constant• Humans must perform the calculations for

the chart

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X bar Charts

• is used to monitor the average value, or mean, of a process over time.

• Mean chart or average chart

R Chart

• is a control chart that is used to monitor process variation when the variable of interest is a quantitative measure.

• Range Chart

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Variable Control ChartX

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• The R chart is examined first before the X bar chart

• If the R chart indicates the sample variability is in statistical control, the X bar chart is examined to determine if the sample mean is also in statistical control.

• If the sample variability is not in statistical control, then the entire process is judged to be not in statistical control regardless of what the X bar chart indicates.

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Variable Control ChartX

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s Reading Control ChartsControl chart is out of statistical control if:

Variable Control ChartX

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• in order to construct x bar and R charts, we must first find the upper- and lower-control limits:

R D= LCL

R D= UCL

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Limits Control ChartR

RA - x = LCL

RA + x = UCL

2

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Limits Control Chart x

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Variable Control ChartX

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Variable Control ChartX

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1. Select k successive subgroups where k is at least 20, in which there are n measurements in each subgroup. Typically n is between 1 and 9. 3, 4, or 5 measurements per subgroup is quite common.

2. Find the range of each subgroup R(i) where R(i)=biggest value - smallest value for each subgroup i.

3. Find the centerline for the R chart, denoted by RBAR=summation of R(i)/ k

4. Find the UCL and LCL

5. Plot the subgroup data and determine if the process is in statistical control.

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Variable Control ChartX

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rt EXAMPLE:

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Sample Observation (xi) Average Range (R)

1 11.90 11.92 12.09 11.91 12.01

2 12.03 12.03 11.92 11.97 12.07

3 11.92 12.02 11.93 12.01 12.07

4 11.96 12.06 12.00 11.91 11.98

5 11.95 12.10 12.03 12.07 12.00

6 11.99 11.98 11.94 12.06 12.06

7 12.00 12.04 11.92 12.00 12.07

8 12.02 12.06 11.94 12.07 12.00

9 12.01 12.06 11.94 11.91 11.94

10 11.92 12.05 11.92 12.09 12.07

Variable Control ChartX

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Sample Observation (xi) Average Range (R)

1 11.90 11.92 12.09 11.91 12.01 11.97 0.19

2 12.03 12.03 11.92 11.97 12.07 12.00 0.15

3 11.92 12.02 11.93 12.01 12.07 11.99 0.15

4 11.96 12.06 12.00 11.91 11.98 11.98 0.15

5 11.95 12.10 12.03 12.07 12.00 12.03 0.15

6 11.99 11.98 11.94 12.06 12.06 12.01 0.12

7 12.00 12.04 11.92 12.00 12.07 12.01 0.15

8 12.02 12.06 11.94 12.07 12.00 12.02 0.13

9 12.01 12.06 11.94 11.91 11.94 11.97 0.15

10 11.92 12.05 11.92 12.09 12.07 12.01 0.1715.0R

Variable Control ChartX

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0

0.32

)15.0)(0(RD = LCL

)15.0)(11.2(RD = UCL

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Variable Control ChartX

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R-CHART

LCL = 0.00

R = 0.15

UCL = 0.32

Variable Control ChartX

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t Steps in Constructing the X Bar Chart

1.Find the mean of each subgroup and the grand mean of all subgroups.2. Find the UCL and LCL 3. Plot the LCL, UCL, centerline, and subgroup means 4. Interpret the data using the following guidelines to determine if the process is in control:

Variable Control ChartX

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t Calculating the sample means and the grand meanSample Observation (xi) Average Range

(R)

1 11.90 11.92 12.09 11.91 12.01 11.97 0.19

2 12.03 12.03 11.92 11.97 12.07 12.00 0.15

3 11.92 12.02 11.93 12.01 12.07 11.99 0.15

4 11.96 12.06 12.00 11.91 11.98 11.98 0.15

5 11.95 12.10 12.03 12.07 12.00 12.03 0.15

6 11.99 11.98 11.94 12.06 12.06 12.01 0.12

7 12.00 12.04 11.92 12.00 12.07 12.01 0.15

8 12.02 12.06 11.94 12.07 12.00 12.02 0.13

9 12.01 12.06 11.94 11.91 11.94 11.97 0.15

10 11.92 12.05 11.92 12.09 12.07 12.01 0.17

00.12X

Variable Control ChartX

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91.1115058012

09.1215058012

2

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)=.(.-R - AxLCL =

)=.(.R + AxUCL =

Variable Control ChartX

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LCL = 11.90

UCL = 12.10

X = 12.00

Variable Control ChartX

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tX-bar and S Chart

1. The sample size n is moderately large, n > 10 or 12

2. The sample size n is variable

The Construction of X-bar and S Chart

Setting up and operating control charts for X-bar and S requires about the same sequence of step as those for the X-bar and R charts, except that for each sample we must calculate the average X-bar and sample standard deviation S.

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tStandard Deviation

1

)( 2

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Variance

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Form

ula

1 For σ not given

Form

ula

2 For σ given

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Variable Control ChartX

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tExample

Piston for automotive engine are produced by a forging process. We wish to establish statistical control of inside diameter of the ring manufactured by this process using X-bar and S charts.

Twenty-five (25) samples, each of size five (5), have been taken when we think the process is in control. The inside diameter measurement data from these samples are shown in table.

TABLE

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996.7300940565.0001.74

014.7400940435.1001.74

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)=.(-S - BxLCL =

)=.(S + BxUCL =

Calculating the UCL and LCL (X bar Chart)

0053.000940565.0

0135.000940435.1

3

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)=.(SLCL = B

)=.(SUCL = B

Calculating the UCL and LCL (S Chart)

ANSWER-1 ANSWER-2 Table of Constant

Variable Control Chart

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MA-MR charts

• In situations where data are collected slowly over a period of time, or where data are expensive to collect, moving average charts are beneficial.

• Moving Average / Range Charts are a set of control charts for variables data (data that is both quantitative and continuous in measurement, such as a measured dimension or time). The Moving Average chart monitors the process location over time, based on the average of the current subgroup and one or more prior subgroups. The Moving Range chart monitors the variation between the subgroups over time.

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Variable Control Chart

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t Image for MA- MR chart

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Moving range (MR)

• n= number of measuremens in moving average

• MR= l current measurement – previous measurement I

• R = total of MRs/ total numbers of MRs• X = total of measurements/ total numbers

of measurements

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Variable Control Chart

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tFormula MR

UCL= 3.267 x RLCL= 0

Formula MA

UCL= X +3LCL = X - 3Where R/ 1.128

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Example

1) 100 10) 96.5 19) 100.92) 101.7 11) 105.2 20) 98.63) 104.5 12) 95.1 21) 105.94) 105.2 13) 93.25) 99.6 14) 93.66) 101.4 15) 103.37) 94.5 16) 100.18) 1010.6 17) 98.39) 99.1 18) 98.5

Observations (X)

n=2where;

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Variable Control Chart

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101.7 100 100.9 1.7104.5 101.7 103.1 2.8105.2 104.5 104.9 0.799.6 105.2 102.4 5.6

101.4 99.6 100.5 1.894.5 101.4 98 6.9

101.6 94.5 98.1 7.199.1 101.6 101.6 2.596.5 99.1 97.8 2.6

105.2 96.5 100.9 8.795.1 105.2 100.2 10.193.2 95.1 93.4 1.993.6 93.2 98.5 0.4

103.3 93.6 101.7 9.7100.1 103.3 99.2 3.298.3 100.1 98.4 1.898.5 98.3 99.7 0.2

100.9 98.5 99.7 2.498.6 100.9 99.8 2.3

105.9 98.6 102.3 7.3

Tabl

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Variable Control Chart

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For m

A n=2R=79.7/20=3.985X= 2096.8/21= 99.85

For MR

UCL= 3.267 x R LCL= 0 = 3.267 X 3.985 = 13.02

For m

R

For MA σ= R/ 1.128= 3985/1.128= 3.53

UCL= X + 3 LCL= X - 3

UCL= 107.35 LCL= 92.35

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Variable Control Chart

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s Difference Chart• Is a type of Short Run SPC (Statistical Process Control)

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s Difference Chart

• Red Line – Our production rate for the past 6 months.

• Green Line - The competitor’s production rate for the past 6 months.

• Shaded Region – Is the difference between the 2 production rate

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s Deviation Column Chart

A B C D E F

1 Budget and Actual Revenues

2 Budget Actual Dev Pos Dev Neg Dev

3 AB 1200 1250 4.2% 4.2% 0.0%

4 CD 1000 900 -10.0% 0.0% -10.0%

5 EF 900 950 5.6% 5.6% 0.0%

6 GH 1150 1100 -4.3% 0.0% -4.3%

Variable Control Chart

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s Deviation Column Chart

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s • ComputationDeviation = (Actual – Budget) / Budget

Variable Control Chart

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• Nominal Short Run SPC• Target Short Run SPC

Variable Control Chart

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s Short Run Nominal X Bar and R Chart

• The nominal x bar and R chart is used to monitor the behavior of a process running different part numbers and still retain the ability to assess control.

• This is done by coding the actual measured readings in a subgroup as a variation from a common reference point, in this case the nominal print specification.

Variable Control Chart

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s Nominal and Target X Bar and R Chart

UCL = + LCL = + UCLR = LCLR =

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s Historical Average or Nominal?

• Use the nominal unless the absolute value of the difference between the nominal and historical value is greater than the critical value of f1 times the sample standard deviation.

• If (Nominal – Historical Average) ≤ Then use Nominal

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Suppose the historical average based on 10 measurements taken from the last time a given part number was run is 20.4 and the sample standard deviation was 1.07. Determine if the nominal of 20.0 or the historical average of 20.4 should be used.

SOLUTION

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s • Calculate the difference• Multiply the f1 value times the

standard deviation• Compare the difference and

the product

CUSUM

EWMA

CON

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contin

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Cumulative Sum Chart Tab

CUSUM charts show cumulative sums of subgroup or individual measurements from a target value. CUSUM charts can help you decide whether a process is in a state of statistical control by detecting small, sustained shifts in the process mean.

EWMA

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CUSUM

• CUSUM works as follows: Let us collect m samples, each of size n, and compute the mean of each sample. Then the cumulative sum (CUSUM) control chart is formed. In either case, as long as the process remains in control centered at , the CUSUM plot will show variation in a random pattern centered about zero. If the process mean shifts upward, the charted CUSUM points will eventually drift upwards, and vice versa if the process mean decreases.

• A visual procedure proposed by Barnard in 1959, known as the V-Mask, is sometimes used to determine whether a process is out of control. A V-Mask is an overlay shape in the form of a V on its side that is superimposed on the graph of the cumulative sums.

EWMA

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CUSUM

EWMA

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CUSUM

The origin point of the V-Mask (see diagram below) is placed on top of the latest cumulative sum point and past points are examined to see if any fall above or below the sides of the V. As long as all the previous points lie between the sides of the V, the process is in control. Otherwise (even if one point lies outside) the process is suspected of being out of control.

EWMA

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CUSUM• In practice, designing and manually constructing a V-Mask is a

complicated procedure. A CUSUM spreadsheet style procedure will be shown below is more practical, unless you have statistical software that automates the V-Mask methodology. Before describing the spreadsheet approach, we will look briefly at an example of a V-Mask in graph form.

• An example will be used to illustrate the construction and application of a V-Mask. The 20 data points 324.925, 324.675, 324.725, 324.350, 325.350, 325.225, 324.125, 324.525, 325.225, 324.600, 324.625, 325.150, 328.325, 327.250, 327.825, 328.500, 326.675, 327.775, 326.875, 328.350

• Are each the average of samples of size 4 taken from a process that has an estimated mean of 325. Based on process data, the process standard deviation is 1.27 and therefore the sample means have a standard deviation of 1.27/(41/2) = 0.635.

• We can design a V-Mask using h and k or we can use an alpha and beta design approach. For the latter approach we must specify.

In our example we choose α = 0.0027, and β= 0.01. Finally, we decide we want to quickly detect a shift as large as 1 sigma, which sets δ = 1.When the V-Mask is placed over the last data point, the mask clearly indicates an out of control situation.

EWMA

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CUSUM

We next move the V-Mask and back to the first point that indicated the process was out of control. This is point number 14, as shown below.

We next move the V-Mask and back to the first point that indicated the process was out of control. This is point number 14, as shown below. Most users of CUSUM procedures prefer tabular charts over the V-Mask. The V-Mask is actually a carry-over of the pre-computer era. The tabular method can be quickly implemented by standard spreadsheet software. To generate the tabular form we use the h and k parameters expressed in the original data units.

EWMA

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CUSUM

We will construct a CUSUM tabular chart for the example described above. For this example, the parameter are h = 4.1959 and k = 0.3175. Using these design values, the tabular form of the example is

Increase in mean

Decrease in mean

Group x x-325 x-325-k Shi 325-k-x Slo CUSUM

1 324.93 -0.07 -0.39 0.00 -0.24 0.00 -0.007

2 324.68 -0.32 -0.64 0.00 0.01 0.01 -0.40

3 324.73 -0.27 -0.59 0.00 -0.04 0.00 -0.67

4 324.35 -0.65 -0.97 0.00 0.33 0.33 -1.32

5 325.35 0.35 0.03 0.03 -0.67 0.00 -0.97

6 325.23 0.23 -0.09 0.00 -0.54 0.00 -0.75

7 324.13 -0.88 -1.19 0.00 0.56 0.56 -1.62

8 324.53 -0.48 -0.79 0.00 0.16 0.72 -2.10

9 325.23 0.23 -0.09 0.00 0.54 0.17 -1.87

10 324.60 -0.40 -0.72 0.00 0.08 0.25 -2.27

11 324.63 -0.38 -0.69 0.00 0.06 0.31 -2.65

12 325.15 0.15 -0.17 0.00 0.47 0.00 -2.50

13 328.33 3.32 3.01 3.01 -3.64 0.00 0.83

14 327.25 2.25 1.93 4.94* -0.57 0.00 3.08

15 327.83 2.82 2.51 7.45* -3.14 0.00 5.90

16 328.50 3.50 3.18 10.63* -3.82 0.00 9.40

17 326.68 1.68 1.36 11.99* -1.99 0.00 11.08

18 327.78 2.77 2.46 14.44* -3.09 0.00 13.85

19 326.88 1.88 1.56 16.00* -2.19 0.00 15.73

20 328.35 3.35 3.03 19.04* -3.67 0.00 19.08

EWMA

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CUSUM

h k

325 4.1959 0.3175

CUSUM

EWMA

EWMA Control Charts

Definition

• The Exponentially Weighted Moving Average (EWMA) is a statistics for monitoring the process that averages the data in a way that gives less and less weight to data as they are further removed in time.

• In statistical quality control, the EWMA chart (or exponentially-weighted moving average chart) is a type of control chart used to monitor either variables or attributes-type data using the monitored business or industrial process's entire history of output. While other control charts treat rational subgroups of samples individually, the EWMA chart tracks the exponentially-weighted moving average of all prior sample means.

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EWMAt = λ Yt + (1 - λ) EWMAt-1 for t = 1, 2, ..., n.

Where:

• EWMA0 is the mean of historical data (target) • Yt is the observation at time t• n is the number of observations to be monitored

including EWMA0• 0 < λ ≤ 1 is a constant that determines the depth of

memory of the EWMA.

The equation is due to Roberts (1959).

The statistic that is calculated is:CO

NSTAN

TTH

ANK

YOU

END

CUSUM

EWMA

Definition of control limits for EWMA

The center line for the control chart is the target value or EWMA0. The control limits are:

UCL = EWMA0 + ksewma LCL = EWMA0 - ksewma

where the factor k is either set equal 3 or chosen using the Lucas and Saccucci (1990) tables. The data are assumed to be independent and these tables also assume a normal population.

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EWMA

Example of calculation of parameters for an EWMA Control chartTo illustrate the construction of an EWMA control chart, consider a process with the following parameters calculated from historical data:

EWMA0 = 50 s = 2.0539

with λ chosen to be 0.3 so that λ / (2-λ) = .3 / 1.7 = 0.1765 and the square root = 0.4201. The control limits are given byUCL = 50 + 3 (0.4201)(2.0539) = 52.5884 LCL = 50 - 3 (0.4201) (2.0539) = 47.4115

Consider the following data consisting of 20 points

52.0 47.0 49.6 51.247.0 51.0 47.6 52.653.0 50.1 49.9 52.4 49.3 51.2 51.3 53.650.1 50.5 47.8 52.1

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EWMA

EWMA statistics for sample data

These data represent control measurements from the process which is to be monitored using the EWMA control chart technique. The corresponding EWMA statistics that are computed from this data set are:

50.60 49.21 50.11 49.92

49.52 49.75 49.36 50.73

50.56 49.85 49.52 51.23

50.18 50.26 50.05 51.94

50.16 50.33 49.38 51.99

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RAW DATA AND EWMA statistics for sample dataCO

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EWMA

The control chart is given belowCO

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EWMA

Interpretation of EWMA Control chart

The red dots are the raw data; the jagged line is the EWMA statistics over time. The chart tells us that the process is in control because all EWMA lie between the control limits. However, there seems to be a trend upwards for the last 5 periods.

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Alina, Jassfer D.Alvarez, Son Robert C.Bautista, Billy JoeCalosa Gilbert Cristobal, Arnel Mark JohnMercado, Kim Nath

THANK YOUGroup 5 QCT

EWMA

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