Statistical Quality Control

By : Vilas Pathak

HistoryControl charting is one of the tools of

Statistical Quality Control(SQC) It is the most technically sophisticated tool of SQC. It was developed in the 1920s by Dr. Walter A. Shewhart of the Bell Telephone Labs. Dr. Shewhart developed the control charts as an statistical approach to the study of manufacturing process variation for the purpose of improving the economic effectiveness of the process. These methods are based on continuous monitoring of process variation.

Background Information A typical control chart is a graphical display of a

quality characteristic that has been measured or computed from a sample versus the sample number or time. The chart contains a centre line that represents the average value of the quality characteristic corresponding to the in-control state. Two other horizontal lines, called the upper control limit(UCL) & the lower control limit(LCL) are also drawn. These control limits are chosen so that if the process is in control, nearly all of the sample points will fall between them. As long as the points plot within the control limits, the process is assumed to be in control, and no action is necessary.

Background Information

However, a point that plots outside of the control limits is interpreted as evidence that the process is out of control, & investigation & corrective action is required to find and eliminate the assignable causes responsible for this behaviour. The control points are connected with straight line segments for easy visualization.

Background Information

Even if all the points plot inside the control limits, if they behave in a systematic or nonrandom manner, then this is an indication that the process is out of control.

Uses of Control charts Control chart is a device for describing

in a precise manner what is meant by statistical control. Its uses are

1. It is a proven technique for improving productivity.

2. It is effective in defect prevention. 3. It prevents unnecessary process

adjustments. 4. It provides diagnostic information. 5. It provides information about

process capability.

Types of control charts

1. Control charts for Attributes. • p chart • c chart• u chart

2. Control charts for Variables. • X bar chart• R chart

Control Charts for Attributes

1. Introduction 2. p chart 3. c chart 4. Example 5. u chart

Introduction

Many quality characteristics cannot be conveniently represented numerically. In such cases, each item inspected is classified as either conforming or nonconforming to the specifications on that quality characteristic. Quality characteristics of this type are called attributes. Examples are non-functional semiconductor chips, warped connecting rods, etc,.

p charts

This chart shows the fraction of nonconforming or defective product produced by a manufacturing process. It is also called the control chart for fraction nonconforming.

c charts

This shows the number of defects or nonconformities produced by a manufacturing process. An Example for construction of c chart is provided.

Construction of c Control charts

• The c chart is drawn with the Plate number on the x axis & the number of conformities on the y axis. The lines showing the Upper Control Limit, the Lower Control Limit & the mean are also drawn. The figure below shows the control chart.

• It can be seen from the figure that the process is out of control as one of the points is out of the control limits. The possible reasons for the out of control behaviour can be understood by looking at the Analysis of patterns on control charts.

Surface defects have been counted on 25 rectangular steel plates, and the data are shown below. The control chart for nonconformities is set up using this data.

No NC No NC No NC No NC No NC

1 1 6 2 11 1 16 1 21 3

2 0 7 5 12 0 17 3 22 1

3 4 8 0 13 8 18 5 23 0

4 3 9 2 14 0 19 4 24 2

5 1 10 1 15 2 20 6 25 4

• The Upper Control Limit(UCL) and the Lower Control Limit(LCL) are calculated as follows:

• LCL = mean -3 * sqrt(mean) = 0 (because less than 0)

• UCL = mean +3 * sqrt(mean) = 6.968

• The c chart is drawn with the Plate number on the x axis & the number of conformities on the y axis. The lines showing the Upper Control Limit, the Lower Control Limit & the mean are also drawn. The figure below shows the control chart.

u charts

This chart shows the nonconformities per unit produced by a manufacturing process.

Control Charts for Variables

• Introduction • X bar and R charts • Example of X bar and R charts

IntroductionMany quality characteristics can be expressed in terms

of a numerical measurement. A single measurable quality characteristic, such as a dimension, weight, or volume, is called a Variable. Control charts for variables are used extensively. They usually lead to more efficient control procedures & provide more information about process performance than attributes control charts. When dealing with a quality characteristic that is a variable, it is a standard practice to control both the mean value of the quality characteristic and its variability. Control of the process average or mean quality level is usually with the control chart for means, or the x bar chart. The control of the process range is done by using the control chart for range, or the R chart.

X bar and R charts

The X bar chart is developed from the average of each subgroup data. The R chart is developed from the ranges of each subgroup data, which is calculated by subtracting the maximum and the minimum value in each subgroup.

Construction of Control charts

Plastic Keychains are being produced in a company named Etcetra. The plastic material is first moulded & then trimmed to the required shape. The cure times during the moulding process affect the edge quality of the keychains produced. The aim is to achieve statistical control of the cure times using X bar and R charts. Cure time data of twenty-five samples, each of size four, have been taken when the process is assumed to be in control.

Sample

Observation

Means Range

1 28.25103 2.59745

2 29.12317 5.18266

3 30.90497 4.20284

4 32.90343 8.54130

5 31.55877 5.44605

Sample

Observation

Means Range

6 30.12132 6.77340

7 31.79940 9.85168

8 28.02750 6.09553

9 30.52838 3.99227

10 30.39390 5.67126

Sample

Observation

Means Range

11 28.92971 4.55051

12 29.22609 2.84574

13 31.30221 11.09669

14 30.51698 3.39341

15 29.50873 6.79941

Sample

Observation

Means Range

16 29.85708 11.17004

17 31.51833 6.42911

18 32.16161 7.97918

19 31.57718 3.33398

20 31.10110 7.01093

Sample

Observation

Means Range

21 31.80966 7.31707

22 30.96172 2.79630

23 30.27140 4.59213

24 26.99992 8.03034

25 30.71869 2.60460

Means 30.40289 5.932155

• The means and the ranges in each sample are calculated. The mean of the sample means and the sample ranges are also calculated. The mean for the ranges gives the centre line for the R chart. Using n=4 from the table for calculating the control limits, we get that D3 = 0 and D4=2.282. Therefore the control limits for the R chart are

• LCL = R mean x D3 = 0 UCL = R mean x D4 = 13.525

• The R chart is drawn with the 25 sample ranges plotted on the chart. The Control limits and the canter line are also drawn. This is shown in the figure below. All the points are within the control limits and no particular pattern can be observed. Therefore the process variability is in control.

• Since the R chart indicates the process variablity is in control, the X bar chart is now constructed. The center line is the mean of the sample means. Using the same table, and taking n=4 the control limits calculated are

• UCL = x mean of means + A2 x R mean = 34.733 LCL = x mean of means - A2 x R mean = 26.072

• The X bar chart is drawn with the 25 sample means plotted on the chart. The Control limits and the centre line are drawn too. This is shown in the figure below. No indication of out of control condition is observed from this figure.

Since both the X bar and the R chart exhibit control, the process can be taken to be in control at the stated levels and the control limits can be adopted for use in on-line statistical process control.

Continuation of the X bar and R charts

Twelve additional samples of cure time data from the moulding process were collected from an actual production run. The data from these new samples are shown

Sample

Observation

Means Range

1 31.80830 4.12920

2 34.15686 14.02663

3 35.75480 11.78558

4 25.97470 5.46960

5 30.53310 9.38450

6 41.75284 11.53828

Sample

Observation

Means Range

7 34.75097 7.19799

8 36.47964 8.45185

9 39.72693 6.07178

10 38.41135 6.56774

11 35.76589 6.84682

12 32.52314 16.16116

The X bar and the R charts are drawn with the new data with the same control limits established

before. They are shown below

X bar chart

• Six points fall above UCL. • One point falls below LCL. • One point falls on UCL.

R chart

Two points fall above UCL. • Eight consecutive points are

above the canter line. • Both the charts show that the

process is out of control.

Analysis of Patterns on Control Charts

A control chart may indicate an out-of-control condition either when one or more points fall beyond the control limits, or when the plotted points exhibit some nonrandom pattern of behaviour.

The process is out of control if any one or more of the criteria is met. One or more points outside of the control

limits. This pattern may indicate: • A special cause of variance from a

material, equipment, method, or measurement system change.

• Mis-measurement of a part or parts. • Miscalculated or mis-plotted data points. • Miscalculated or mis-plotted control

limits.

The process is out of control if any one or more of the criteria is met. A run of eight points on one side

of the canter line. This pattern indicates a shift in the process output from changes in the equipment, methods, or materials or a shift in the measurement system

The process is out of control if any one or more of the criteria is met. Two of three consecutive points

outside the 2-sigma warning limits but still inside the control limits. This may be the result of a large shift in the process in the equipment, methods, materials, or operator or a shift in the measurement system.

The process is out of control if any one or more of the criteria is met. Four of five consecutive points

beyond the 1-sigma limits.

The process is out of control if any one or more of the criteria is met. An unusual or nonrandom pattern in the

data. • A trend of seven points in a row upward or

downward. This may show • Gradual deterioration or wear in equipment. • Improvement or deterioration in technique.

• Cycling of data can indicate • Temperature or other recurring changes in the

environment. • Differences between operators or operator

techniques. • Regular rotation of machines. • Differences in measuring or testing devices that

are being used in order.

The process is out of control if any one or more of the criteria is met. Several points near a warning or

control limit.