Statistical Process Control

A process can be described as a transformation of set of inputs into desired outputs.

Inputs PROCESS

Outputs

What is a process?

Data Collection

• We need information to arrive at a conclusion about a problem. Information is the outcome of the data collected. Statistical process control relies on the data and its analysis.

• There are many type of data

= Measures where the metric is composed of a classification in one of two (or more) categories is called Attribute data.

-Good/Bad

-Yes/No

=Measures where the metric consists of a number which indicates a precise value is called Variable data.

-Time

-Miles/Hr

4

PopulationPopulation Vs. Vs. Sample Sample (Certainty Vs. Uncertainty) (Certainty Vs. Uncertainty) A sample is just a subset of all possible values

A whole set of data is called a population

populationsample

Since the sample does not contain all the possible values, there is some uncertainty about the population. Hence Hence any statistics, such as mean and any statistics, such as mean and standard deviation, are just standard deviation, are just estimatesestimates of the true population parameters.of the true population parameters.

5

WHY STATISTICS?THE ROLE OF STATISTICS ………

USLT

LSL

Statistics is the art of collecting, classifying, presenting, interpreting and analyzing numerical data, as well as making

conclusions about the system from which the data was obtained.

6

Descriptive Statistics

Descriptive Statistics is the branch of statistics which most people are familiar.

It characterizes and summarizes the most prominent features of a given set of data (means, medians, standard deviations, percentiles, graphs, tables and charts.

7

Inferential Statistics

Inferential Statistics is the branch of statistics that deals with drawing conclusions about a population based on information obtained from a sample drawn from that population.

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WHAT IS THE MEAN?

ORDERED DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 4

The mean is simply the average value of the data.

n=12

xi 2

mean xxn

i 212

17.

Mean

9

WHAT IS THE MEDIAN?

ORDERED DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 4

If we rank order (descending or ascending) the data set ,we find the value half way (50%) through the data points and is called the median value. Median is the middle data point of a serious.

Median Value

Median

50% of data

points

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WHAT IS THE MODE?

ORDERED DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 4

If we rank order (descending or ascending) the data set We find that a single value occurs more often than any other. This is called the mode.

.

Mode Mode

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WHAT IS THE RANGE?ORDERED DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 4

The range is a very common metric .

To calculate the range simply subtract the minimum value in the sample from the maximum value.

Range

RangeMaxMin

Range x xMAX MIN 4 5 9( )

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WHAT IS THE VARIANCE/STANDARD DEVIATION?

The variance (s2) is a very robust metric .

The standard deviation(s) is the square root of the variance and is the most commonly used measure of dispersion.

s

X X

ni2

2

1

61 67

12 15 6

.

.

DATA SET-5-3-1-10000013

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 4

XX

n

i

2

12-.17

X Xi -5-(-.17)=-4.83

-3-(-.17)=-2.83

-1-(-.17)=-.83

-1-(-.17)=-.83

0-(-.17)=.17

0-(-.17)=.17

0-(-.17)=.17

0-(-.17)=.17

0-(-.17)=.17

1-(-.17)=1.17

3-(-.17)=3.17

4-(-.17)=4.17

X Xi 2

(-4.83)2=23.32

(-2.83)2=8.01

(-.83)2=.69

(-.83)2=.69

(.17)2=.03

(.17)2=.03

(.17)2=.03

(.17)2=.03

(.17)2=.03

(1.17)2=1.37

(3.17)2=10.05

(4.17)2=17.39

61.67

• Measures performance of a process• Uses mathematics (i.e., statistics)• Involves collecting, organizing, & interpreting

data • Objective: Regulate product quality• Used to

– Control the process as products are produced– Inspect samples of finished products

Statistical Process Control (SPC)

Steps of a Process Control System are

Define Measure Comparing with a standard Evaluation of the sample for its acceptability Corrective action Evaluate corrective action

CONTROL CHART

Control Charts

• Control Charts is a graph used to assess and maintain the stability of the production process.

• These charts describe where the process is in terms of current performance and helps the resources to work with the process to make decisions to enhance the future quality of products and services.

• Control Charts serve two basic functions:

1. Decision-making: These charts help the supervisor in deciding the course of action to be initiated for the information revealed from the process.

ex. investigate for potential problems, adjust the process, allow the process to run as it is.

2.Problem–Solving: By observing the patterns on the chart, the supervisor can determine what adjustments need to be made to bring the process ‘in control’.

Essential features of a control chart

Time

Var

iab

le V

alu

es Upper Control Limit

Central Line

Lower Control Limit

CONTROL CHART

• Show changes in data pattern– e.g., trends

• Make corrections before process is out of control

• Show causes of changes in data– Assignable causes

• Data outside control limits or trend in data

– Natural causes• Random variations around average

Control Chart Purposes

1. Characteristics for which you focus on defects

2. Classify products as either ‘good’ or ‘bad’, or count # defects– e.g., radio works or not

3. Categorical or discrete random variables

AttributesAttributesVariablesVariables

Quality Characteristics

1. Characteristics that you measure, e.g., weight, length

2. May be in whole or in fractional numbers

3. Continuous random variables

Types of Control Charts for Attribute Data

Description Type Sample Size

Control Chart for proportion non conforming units

p Chart May change

Control Chart for no. of non conforming units in a sample

np Chart Must be constant

Control Chart for no. of non conformities in a sample

c Chart Must be constant

Control Chart for no. of non conformities per unit

u Chart May Change

CONTROL CHART

ControlCharts

RChart

VariablesCharts

AttributesCharts

`XChart

PChart

CChart

Control Chart Types

• Type of variables control chart– Interval or ratio scaled numerical data

• Shows sample means over time• Monitors process average and tells whether

changes have occurred. These changes may due to 1. Tool wear

2. Increase in temperature 3. Different method used in the

second shift 4. New stronger material

• Example: Weigh samples of coffee & compute means of samples; Plot

X Chart

• Type of variables control chart– Interval or ratio scaled numerical data

• Shows sample ranges over time– Difference between smallest & largest values in inspection

sample

• Monitors variability in process, it tells us the loss or gain in dispersion. This change may be due to:

1. Worn bearing

2. A loose tool

3. An erratic flow of lubricant to machine

4. Sloppiness of machine operator• Example: Weigh samples of coffee & compute ranges of samples;

Plot

R Chart

Construction of X and R Charts

• Step 1: Select the Characteristics for applying a control chart.

• Step 2: Select the appropriate type of control chart.• Step 3: Collect the data.• Step 4: Choose the rational sub-group i.e Sample• Step 5: Calculate the average ( X) and range R for each

sample.• Step 6: Cal Average of averages of X and average of

range(R)

Construction of X and R Charts

• Steps 7:Cal the limits for X and R Charts.• Steps 8: Plot Centre line (CL) UCL and

LCL on the chart • Steps 9: Plot individual X and R values on

the chart.• Steps 10: Check whether the process is in

control (or) not.• Steps 11: Revise the control limits if the

points are outside.

From Tables

X Chart Control Limits

RAxxLCL

RAxxUCL

2

2

Sub group average X = x1 + x2 +x3 +x4 +x5 / 5

Sub group range R = Max Value – Min value

From Tables

R Chart Control Limits

RD LCL

RD UCL

3R

4R

Problem8.1 from TQM by V.Jayakumar Page No 8.5

• Type of attributes control chart– Nominally scaled categorical data

• e.g., good-bad• Shows % of nonconforming items• Example: Count # defective chairs &

divide by total chairs inspected; Plot– Chair is either defective or not

defective

p Chart for Attributes

p Chart

• p = np / n where p = Fraction of Defective

np = no of Defectives

n = No of items inspected in

sub group

p= Avg Fraction Defective = ∑np/ ∑n = CL

z = 3 for 99.7% limits

p Chart Control Limits

n

ppzpLCL

n

ppzpUCL

p

p

)1(

)1(

Purpose of the p Chart

Identify and correct causes of bad quality

The average proportion of defective articles submitted for inspection,over a period.

To suggest where X and R charts to be used.

Determine average Quality Level.

Problem

• Problem 9.1 Page no 9.3 TQM by V.Jayakumar

np CHART

P and np are quiet same

Whenever subgroup size is variable,p chart is used. If sub group size is constant, then np is used.

FORMULA: Central Line CLnp = n p

Upper Control Limit, UCLnp = n p +3√ n p (1- p )

Lower Control Limit, LCLnp = n p -3 √ n p (1- p )

Where p = ∑ np/∑n =Average Fraction Defective

n = Number of items inspected in subgroup.

Problem

• Problem No 9.11 page No 9.11 in TQM by V.Jayakumar

• Type of attributes control chart– Discrete quantitative data

• Shows number of nonconformities (defects) in a unit – Unit may be chair, steel sheet, car etc.– Size of unit must be constant

• Example: Count no of defects (scratches, chips etc.) in each chair of a sample of 100 chairs; Plot

c Chart

Use 3 for 99.7% limits

c Chart Control Limits

ccLCL

ccUCL

c

c

3

3