Statistical Process ControlStatistical Process ControlProcesses that are not in a state

of statistical control show excessive variations or exhibit variations that change with time

Control charts are used to detect whether a process is statistically stable.

Control charts differentiates between variations that is normally expected of the process due chance or common causes that change over time due to assignable or special causes

Common cause variationCommon cause variationVariations due to common causesare inherent to the process because of:

◦ the nature of the system◦ the way the system is managed◦ the way the process is organized and operated

can only be removed by◦ making modifications to the process◦ changing the process

are the responsibility of higher management

Special Cause VariationSpecial Cause VariationVariations due to special causes arelocalized in natureexceptions to the systemconsidered abnormalitiesoften specific to a

◦ certain operator◦ certain machine◦ certain batch of material, etc.

Investigation and removal of variations due to special causes are key to process improvement

Causes of VariationCauses of Variation

Two basic categories of variation in output include common causes and assignable causes.

Common causes are the purely random, unidentifiable sources of variation that are unavoidable with the current process.

◦ If process variability results solely from common causes of variation, a typical assumption is that the distribution is symmetric, with most observations near the center.

Assignable causes of variation are any variation-causing factors that can be identified and eliminated, such as a machine needing repair.

Statistical Process Control Statistical Process Control (SPC) Charts(SPC) Charts

Statistical process control (SPC) charts are used to help us distinguish between common and assignable causes of variation. We will cover 2 types:

Variables Control Charts: Service or product characteristics that is continuous and can be measured, such as weight, length, volume, or time.

Attributes Control Charts: Service or product characteristics that can be counted...pass/fail, good/bad, rating scale. Color, inspection, test results (pass/fail), number of defects, types of defects

SPC charts Explained

Control ChartsControl ChartsControl Charts are run charts with

superimposed normal distributions

Purpose of Control ChartsPurpose of Control ChartsControl charts provide a graphical

means for testing hypotheses about the data being monitored. 

Control and warning limitsControl and warning limits

The probability of a sample having a particular value is given by its location on the chart. Assuming that the plotted statistic is normally distributed, the probability of a value lying beyond the:

warning limits is approximately 0.025 or 2.5% chance (plus or minus 2-sigma from the mean)

control limits is approximately 0.001 or 0.1% chance (plus or minus 3 sigma from the mean), this is rare and indicates that◦ the variation is due to an assignable cause◦ the process is out-of-statistical control

Out of Control ProcessesOut of Control ProcessesRun rules are rules that are used to indicate out-of-

statistical control situations. Typical run rules for Shewhart X-charts with control and warning limits are:

a point lying beyond the control limits2 consecutive points lying beyond the warning limits

(0.025x0.025x100 = 0.06% chance of occurring)7 or more consecutive points lying on one side of the

mean ( 0.57x100 = 0.8% chance of occurring and indicates a shift in the mean of the process)

5 or 6 consecutive points going in the same direction (indicates a trend)

Other run rules can be formulated using similar principles

Nominal

UCL

LCLVar

iati

on

s

Sample number

Control ChartsControl Charts

Nominal

UCL

LCLVar

iati

on

s

Sample number

Control ChartsControl Charts

Nominal

UCL

LCLVar

iati

on

s

Sample number

Control ChartsControl Charts

Nominal

UCL

LCLVar

iati

on

s

Sample number

Control ChartsControl Charts

SamplingSampling

Sampling plan: A plan that specifies a sample size, the time between successive samples, and decision rules that determine when action should be taken.

Sample size: A quantity of randomly selected observations of process outputs.

Why do I need to sample? The case of potato chips

Means and RangesMeans and Ranges

The mean of the sample is the sum of all observations in a sample divided by the number of observations in the sample.

The range of the sample is the difference between the largest observation in a sample and the smallest.

The mean of the process is the sum of all sample means divided by the number of samples

The mean range for the process is the sum of all ranges divided by the number of samples

Statistical ProcessStatistical ProcessControl MethodsControl Methods

Control Charts for variables are used to monitor the mean and variability of the process distribution.

R-chart (Range Chart) is used to monitor process variability.

- x-chart is used to see whether the process is generating output, on average, consistent with a target value set by management for the process or whether its current performance, with respect to the average of the performance measure, is consistent with past performance.

◦ If the standard deviation of the process is known, we can place UCL and LCL at “z” standard deviations from the mean at the desired confidence level.

Another way to develop Another way to develop control charts so for small control charts so for small sample sizessample sizes

The control limits for the x-chart are: UCLx = x + A2R and LCLx = x - A2R

Where

X = central line of the chart, which can be either the average of past sample means or a target value set for the process.

A2 = constant to provide three-sigma limits for the sample mean.

The control limits for the R-chart are UCLR = D4R and LCLR = D3R

where

R = average of several past R values and the central line of the chart.

D3,D4 = constants that provide 3 standard deviations (three-sigma) limits for a given sample size.

– =– =

=

Control Chart FactorsControl Chart FactorsTABLE 5.1 | FACTORS FOR CALCULATING THREE-SIGMA LIMITS FOR

| THE x-CHART AND R-CHART

Size of Sample (n)

Factor for UCL and LCL for x-Chart (A2)

Factor for LCL for R-Chart (D3)

Factor for UCL for R-Chart (D4)

2 1.880 0 3.267

3 1.023 0 2.575

4 0.729 0 2.282

5 0.577 0 2.115

6 0.483 0 2.004

7 0.419 0.076 1.924

8 0.373 0.136 1.864

9 0.337 0.184 1.816

10 0.308 0.223 1.777

Year Quarter Exam 1 Exam 2 Exam 3

2009 Quarter 1 85 82 90

2009 Quarter 2 73 67 77

2009 Quarter 3 85 85 85

2009 Quarter 4 90 73 83

2010 Quarter 1 70 83 98

2010 Quarter 2 89 81 83

2010 Quarter 3 95 93 91

2010 Quarter 4 72 83 95

Over the past 2 years, Professor Matta has Over the past 2 years, Professor Matta has been asked to teach one section of Process been asked to teach one section of Process Analytics in each quarter (4 Quarters/ Year). Analytics in each quarter (4 Quarters/ Year). Each time he taught, he would give 3 exams. Each time he taught, he would give 3 exams. The class average grade on these exams over The class average grade on these exams over the last 8 quarters have been as follows: the last 8 quarters have been as follows:

Exam Grades in Prof Matta’s PA Exam Grades in Prof Matta’s PA class during the past 2 years class during the past 2 years with Averages:with Averages:

Year Quarter Exam 1 Exam 2 Exam 3 Average

2009 Quarter 1 85 82 90 85.67

2009 Quarter 2 73 67 77 72.33

2009 Quarter 3 85 85 85 85.00

2009 Quarter 4 90 73 83 82.00

2010 Quarter 1 70 83 98 83.67

2010 Quarter 2 89 81 83 84.33

2010 Quarter 3 95 93 91 93.00

2010 Quarter 4 72 83 95 83.33

Overall Average 83.67

Exam Grades in Prof Matta’s PA Exam Grades in Prof Matta’s PA class during the past 2 years with class during the past 2 years with Averages and Ranges:Averages and Ranges:

Year Quarter Exam 1 Exam 2 Exam 3 Average Range

2009 Quarter 1 85 82 90 85.67 8.00

2009 Quarter 2 73 67 77 72.33 10.00

2009 Quarter 3 85 85 85 85.00 0.00

2009 Quarter 4 90 73 83 82.00 17.00

2010 Quarter 1 70 83 98 83.67 28.00

2010 Quarter 2 89 81 83 84.33 8.00

2010 Quarter 3 95 93 91 93.00 4.00

2010 Quarter 4 72 83 95 83.33 23.00

Overall Average 83.67 12.25

For Prof. Matta’s Case:For Prof. Matta’s Case:

The sample size = 3 (3 exams/quarter)

A2 1.02

D3 0

D4 2.57

X-Bar 83.67

R-Bar 12.25

UCLx = X-Bar + A2 * R-Bar 96.16

LCLx = X-Bar - A2 * R-Bar 71.17

UCLr = D4* R-Bar 31.48

LCLr = D3 * R-Bar 0.00

Control Charts for Prof Matta’s Control Charts for Prof Matta’s Exams:Exams:

Control Charts Example:Control Charts Example:At Quikie Car Wash, the wash

process is advertised to take less than 7 minutes. Consequently, management has set a target average of 390 seconds for the wash process. Suppose that the average range for a sample of 9 cars is 10 seconds. Establish the means and ranges control limits using this data.

Solved ExampleSolved Example

X = 390 sec, n = 9, R= 10 secFrom Table 5.1 in your book,A2 = 0.337, D3 = 0.184, D4 = 1.816UCLR= D4 R= 1.816(10 sec) = 18.16 secLCLR= D3 R= 0.184(10 sec) = 1.84 secUCLx = x + A2 R= 390 sec + 0.337(10 sec) =

393.37 secLCLx = x - A2 R = 390 sec – 0.337(10 sec) =

386.63 sec

Marlin Bottling Company Problem 5 in the book chapter 2in the book chapter 2Marlin Company

     

Sample 1 2 3 4  X-BAR R

1 0.60 0.61 0.59 0.60 0.60 0.02

2 0.60 0.60 0.61 0.60 0.60 0.01

3 0.58 0.57 0.59 0.59 0.58 0.02

4 0.62 0.61 0.60 0.59 0.60 0.03

5 0.59 0.61 0.61 0.60 0.60 0.02

6 0.59 0.58 0.62 0.58 0.59 0.04

Marlin Co. Bottling – In class Calculations

Factor for UCL Factor forFactor

Size of and LCL for LCL for UCL for

Sample x-Charts R-Charts R-Charts

(n) (A2) (D3) (D4)

2 1.880 03.267

3 1.023 02.575

4 0.729 02.282

5 0.577 02.115

6 0.483 0 2.00

Control Charts Control Charts for Attributesfor Attributes

p-chart: A chart used for controlling the proportion of defective services or products generated by the process.

pp = = pp(1 – (1 – pp))//nnWheren = sample sizep = central line on the chart, which can be either the historical average population proportion defective or a target value.

z = normal deviate (number of standard deviations from the average)

Control limits are: UCLp = p+zpp and LCLp = p−zp– –

Hometown BankHometown BankExampleExample

The operations manager of the booking services department of Hometown Bank is concerned about the number of wrong customer account numbers recorded by Hometown personnel.

Each week a random sample of 2,500 deposits is taken, and the number of incorrect account numbers is recorded. The results for the past 12 weeks are shown in the following table.

Is the booking process out of statistical control? Use three-sigma control limits.

Sample Wrong ProportionNumber Account # Defective

1 15 0.006 2 12 0.0048 3 19 0.0076 4 2 0.0008 5 19 0.0076 6 4 0.0016 7 24 0.0096 8 7 0.0028 9 10 0.00410 17 0.006811 15 0.00612 3 0.0012

Total 147

Hometown BankHometown BankUsing a p-Chart to monitor a processUsing a p-Chart to monitor a process

n = 2500

p =147

12(2500)= 0.0049

pp = = pp(1 – (1 – pp))//nn

pp = = 0.00490.0049(1 – (1 – 0.00490.0049)/)/25002500

pp = 0.0014 = 0.0014

UCLp = 0.0049 + 3(0.0014)

= 0.0091 LCLp = 0.0049 – 3(0.0014) = 0.0007

Hometown BankHometown BankUsing a p-Chart to monitor a processUsing a p-Chart to monitor a process

Example

In class ProblemIn class Problem

In Class ProblemIn Class Problem

Control ChartsControl ChartsTwo types of error are possible with

control chartsA type I error occurs when a

process is thought to be out of control when in fact it is not

A type II error occurs when a process is thought to be in control when it is actually out of statistical control

These errors can be controlled by the choice of control limits