What is a control chart?

The control chart is a graph used to study how a process changes over time. Data is plotted in time order. It does not show cause for a variation.

WHY THESE EVOLVED?

No 2 things are alikeVariation exists - even if variation is small, precision instruments show differences.To keep track of these variations we use control charts.

Source of variationRandom and Non Random variations. Equipment- tool wear, electrical fluctuations for weldingMaterial- tensile strength, moisture content (e.g. raw material) Environment- temperature, light, humidityOperator- method, motivation level, training

Inspection- inspector, inspection equipment, environment

Statistical stabilityA process is statistically stable over time (with respect to characteristic X) if the distribution of X does not change over time

Process CapabilityProcess Capabilityis a measure of the ability of the process to meet specifications.

TYPES OF CONTROL CHARTS:Control chart for variables X R chart X S chart

Control chart for attributesn- chartnp chartc- chartu- chart

Variable A single quality characteristic that can be measured on a numerical scale. E.g. height, weight , age etc

Attribute- A quality characteristic that can not be measured but is a discrete one . Attribute data arise when you are determining only the presence or absence of something: success or failure

CHOICE OF SAMPLE SIZE:

Large sample size is easy to take but the control chart will be smooth which means that average will normalize the small defects.

So small sample sizes should be taken such that smaller problems can be found out.

Average Run LengthThe average run length (ARL) is a very important way of determining the appropriate sample size and sampling frequency.Let p = probability that any point exceeds the control limits. Then,

The average run length gives us the length of time (or number of samples) that should plot in control before a point plots outside the control limits.

( X R CHART )

A control chart mainly consists of control limits which set up a quality characteristic.

In X R chart, two charts are drawn. These are X chart and R chart.

X bar chart monitors the between sample variabilityR chart monitors the within sample variability.

Select quality characteristic.Select sample number and sample size.Find out the corresponding mean and range.Find x and RFind the control limits.Draw the chart by plotting sample number on abscissa/ordinate and sample size on ordinate/abscissa.Steps to draw chart :

Control limits for R chart:

D4= 1+(3D3/D2) and D3 is max (0, 1-(3D3/D2)

Control limits for X Chart :

Where A2 is 3 (D2n)

X BAR AND R CHART EXAMPLE

Chart1

Chart2

10.731810.856245333310.6005813333

10.754610.856245333310.6005813333

10.758610.856245333310.6005813333

10.72710.856245333310.6005813333

10.72410.856245333310.6005813333

10.705210.856245333310.6005813333

10.734610.856245333310.6005813333

10.62410.856245333310.6005813333

10.710410.856245333310.6005813333

10.731810.856245333310.6005813333

10.747610.856245333310.6005813333

10.768210.856245333310.6005813333

10.733210.856245333310.6005813333

10.783210.856245333310.6005813333

10.69210.856245333310.6005813333

Sample

Means

Chart3

0.1160.4650440

0.2590.4650440

0.1710.4650440

0.2210.4650440

0.1190.4650440

0.1430.4650440

0.2740.4650440

0.6690.4650440

0.1320.4650440

0.1790.4650440

0.1630.4650440

0.250.4650440

0.3490.4650440

0.1580.4650440

0.1030.4650440

Sample

Ranges

Sheet1

x-barRange

SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangeUCLLCLUCLLCLnA2D3D4

110.68210.68910.77610.79810.71410.7320.11610.856245333310.60058133330.465044021.8803.27

210.78710.8610.60110.74610.77910.7550.25910.856245333310.60058133330.465044031.0202.57

310.7810.66710.83810.78510.72310.7590.17110.856245333310.60058133330.465044040.7302.28

410.59110.72710.81210.77510.7310.7270.22110.856245333310.60058133330.465044050.5802.11

510.69310.70810.7910.75810.67110.7240.11910.856245333310.60058133330.465044060.4802

610.74910.71410.73810.71910.60610.7050.14310.856245333310.60058133330.465044070.420.081.92

710.79110.71310.68910.87710.60310.7350.27410.856245333310.60058133330.465044080.370.141.86

810.74410.77910.1110.73710.7510.6240.66910.856245333310.60058133330.465044090.340.181.82

910.76910.77310.64110.64410.72510.7100.13210.856245333310.60058133330.4650440100.310.221.78

1010.71810.67110.70810.8510.71210.7320.17910.856245333310.60058133330.4650440110.290.261.74

1110.78710.82110.76410.65810.70810.7480.16310.856245333310.60058133330.4650440

1210.62210.80210.81810.87210.72710.7680.25010.856245333310.60058133330.4650440

1310.65710.82210.89310.54410.7510.7330.34910.856245333310.60058133330.4650440

1410.80610.74910.85910.80110.70110.7830.15810.856245333310.60058133330.4650440

1510.6610.68110.64410.74710.72810.6920.10310.856245333310.60058133330.4650440

Averages10.7280.220400

A20.58X-BarUCLLCL

D3010.856245333310.6005813333

D42.11

Mean10.728R-Bar0.4650440

Range0.220

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Sheet1

nA2D3D4

21.8803.27

31.0202.57

40.7302.28

50.5802.11

60.4802.00

70.420.081.92

80.370.141.86

90.340.181.82

100.310.221.78

110.290.261.74

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#REF!

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#REF!

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Chart1

Chart2

10.731810.856245333310.6005813333

10.754610.856245333310.6005813333

10.758610.856245333310.6005813333

10.72710.856245333310.6005813333

10.72410.856245333310.6005813333

10.705210.856245333310.6005813333

10.734610.856245333310.6005813333

10.62410.856245333310.6005813333

10.710410.856245333310.6005813333

10.731810.856245333310.6005813333

10.747610.856245333310.6005813333

10.768210.856245333310.6005813333

10.733210.856245333310.6005813333

10.783210.856245333310.6005813333

10.69210.856245333310.6005813333

Sample

Means

Chart3

0.1160.46504400.2204

0.2590.46504400.2204

0.1710.46504400.2204

0.2210.46504400.2204

0.1190.46504400.2204

0.1430.46504400.2204

0.2740.46504400.2204

0.6690.46504400.2204

0.1320.46504400.2204

0.1790.46504400.2204

0.1630.46504400.2204

0.250.46504400.2204

0.3490.46504400.2204

0.1580.46504400.2204

0.1030.46504400.2204

Range

UCL

LCL

R-bar

Sample

R

Sheet1

x-barRange

SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangeUCLLCLUCLLCLr-barnA2D3D4

110.68210.68910.77610.79810.71410.7320.11610.856245333310.60058133330.46504400.22040021.8803.27

210.78710.8610.60110.74610.77910.7550.25910.856245333310.60058133330.46504400.22040031.0202.57

310.7810.66710.83810.78510.72310.7590.17110.856245333310.60058133330.46504400.22040040.7302.28

410.59110.72710.81210.77510.7310.7270.22110.856245333310.60058133330.46504400.22040050.5802.11

510.69310.70810.7910.75810.67110.7240.11910.856245333310.60058133330.46504400.22040060.4802

610.74910.71410.73810.71910.60610.7050.14310.856245333310.60058133330.46504400.22040070.420.081.92

710.79110.71310.68910.87710.60310.7350.27410.856245333310.60058133330.46504400.22040080.370.141.86

810.74410.77910.1110.73710.7510.6240.66910.856245333310.60058133330.46504400.22040090.340.181.82

910.76910.77310.64110.64410.72510.7100.13210.856245333310.60058133330.46504400.220400100.310.221.78

1010.71810.67110.70810.8510.71210.7320.17910.856245333310.60058133330.46504400.220400110.290.261.74

1110.78710.82110.76410.65810.70810.7480.16310.856245333310.60058133330.46504400.220400

1210.62210.80210.81810.87210.72710.7680.25010.856245333310.60058133330.46504400.220400

1310.65710.82210.89310.54410.7510.7330.34910.856245333310.60058133330.46504400.220400

1410.80610.74910.85910.80110.70110.7830.15810.856245333310.60058133330.46504400.220400

1510.6610.68110.64410.74710.72810.6920.10310.856245333310.60058133330.46504400.220400

Averages10.7280.220400

A20.58X-BarUCLLCL

D3010.856245333310.6005813333

D42.11

Mean10.728R-Bar0.4650440

Range0.220

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Chart1

Chart2

10.731810.856245333310.600581333310.7284133333

10.754610.856245333310.600581333310.7284133333

10.758610.856245333310.600581333310.7284133333

10.72710.856245333310.600581333310.7284133333

10.72410.856245333310.600581333310.7284133333

10.705210.856245333310.600581333310.7284133333

10.734610.856245333310.600581333310.7284133333

10.62410.856245333310.600581333310.7284133333

10.710410.856245333310.600581333310.7284133333

10.731810.856245333310.600581333310.7284133333

10.747610.856245333310.600581333310.7284133333

10.768210.856245333310.600581333310.7284133333

10.733210.856245333310.600581333310.7284133333

10.783210.856245333310.600581333310.7284133333

10.69210.856245333310.600581333310.7284133333

Sample mean

UCL

LCL

grand mean of x

Sample

Means

Chart3

0.1160.4650440

0.2590.4650440

0.1710.4650440

0.2210.4650440

0.1190.4650440

0.1430.4650440

0.2740.4650440

0.6690.4650440

0.1320.4650440

0.1790.4650440

0.1630.4650440

0.250.4650440

0.3490.4650440

0.1580.4650440

0.1030.4650440

Sample

Ranges

Sheet1

x-barRange

SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangeUCLLCLUCLLCLx-bar-barnA2D3D4

110.68210.68910.77610.79810.71410.7320.11610.856245333310.60058133330.465044010.72821.8803.27

210.78710.8610.60110.74610.77910.7550.25910.856245333310.60058133330.465044010.72831.0202.57

310.7810.66710.83810.78510.72310.7590.17110.856245333310.60058133330.465044010.72840.7302.28

410.59110.72710.81210.77510.7310.7270.22110.856245333310.60058133330.465044010.72850.5802.11

510.69310.70810.7910.75810.67110.7240.11910.856245333310.60058133330.465044010.72860.4802

610.74910.71410.73810.71910.60610.7050.14310.856245333310.60058133330.465044010.72870.420.081.92

710.79110.71310.68910.87710.60310.7350.27410.856245333310.60058133330.465044010.72880.370.141.86

810.74410.77910.1110.73710.7510.6240.66910.856245333310.60058133330.465044010.72890.340.181.82

910.76910.77310.64110.64410.72510.7100.13210.856245333310.60058133330.465044010.728100.310.221.78

1010.71810.67110.70810.8510.71210.7320.17910.856245333310.60058133330.465044010.728110.290.261.74

1110.78710.82110.76410.65810.70810.7480.16310.856245333310.60058133330.465044010.728

1210.62210.80210.81810.87210.72710.7680.25010.856245333310.60058133330.465044010.728

1310.65710.82210.89310.54410.7510.7330.34910.856245333310.60058133330.465044010.728

1410.80610.74910.85910.80110.70110.7830.15810.856245333310.60058133330.465044010.728

1510.6610.68110.64410.74710.72810.6920.10310.856245333310.60058133330.465044010.728

Averages10.7280.220400

A20.58X-BarUCLLCL

D3010.856245333310.6005813333

D42.11

Mean10.728R-Bar0.4650440

Range0.220

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Trial Control Limits

The control limits obtained from equations of range 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.

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 controlTrial control limits and the out-of-control process

( x - s control chart )It consists of two charts : X bar ChartS Chart

In this chart, we are using standard deviation as a measure of variation.

Both R and s measure dispersion of data When n

R chart- simple, only use XH (highest) and XL(lowest)S chart- more calculation - use ALL xis more accurate, need calculate sub-group sample standard deviation

STANDARD DEVIATION (S) :

The upper and lower control limits for the chart are given as

where A3 is found in the Appendix

X BAR S CHART EXAMPLE

Process Out Of Control

1. A point falls outside control limits

assignable cause presentprocess producing unstable process must be investigated, corrected

2. Unnatural runs of variation even within 3 limits :7 or more points above or below center line (in a row)10 out of 11 points on one side12 out of 14 points on one side6 points increasing/decreasing2 out of 3 in Zone A (WL)4 out of 5 in Zone B.

3. For two zones 1.5 each 2 or more points beyond 1.5

ANALYSIS FOR OUT-OF-CONTROL Patterns1. Change/Jump in levelshift in meanCauses - process parameters change, diff / new operator, change in raw material

2. Trend or steady change in leveldrifting mean common, upward or downward directiontool wear, gradual change in temp. viscosity of chemical used

Recurring cyclesWavy, periodic high & low pointsRecurring effects of temp., humidity (morning vs. evening)

4.Two populations (mixture) many points near or outside limits due tolarge difference in material quality2 or more machinesdifferent test method

Thank YouMake Presentation much more fun

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