Control charts

CONTROL CHARTS

ARAVIND BABU RSAHUL HAMEED.H

M.E-INDUSTRIAL ENGG

1

AGENDA

• What is control chart?

• History of control chart

• Types of data

• Defect and defective

• Types of control charts

• Control limits vs specification limits

• Variable control charts

• Attribute control charts

CONTROL CHART

A statistical tool to study the variation in the process over time.

A control chart 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.

• These lines are determined from historical data.

CL = Mean x

UCL = x + 3σ

Y-axis

X-axis

LCL = x - 3σ

CONTROL CHART

Purpose:

• Analyze the past data and determine the performance of the process

• Measure control of the process against standards

HISTORY

• Invented by Walter Andrew Shewhart, father of statistical quality control in 1920.

• The company's engineers had been seeking to improve the reliability of their telephony

transmission systems. • There was a stronger business need to reduce the frequency of failures and repairs

• Shewhart framed the problem in terms of Common- and special-causes of variation and, on May 16, 1924, wrote an internal memo introducing the control chart as a tool for distinguishing between the two.

• He understood data from physical processes typically produce a "normal distribution curve“.

TYPES OF DATA

DISCRETE DATA CONTINUOUS DATAInfinite number of values between whole numbers

Data that can be counted

DEFECTS AND DEFECTIVES

Defects :

• A defect is any item or service that exhibits a departure from specifications.

• A defect does not necessarily mean that the product or service cannot be used.

• A defect indicates only that the product result is not entirely as intended.

Example: crack, bend in a shaft

Defectives:

• A defective is an item or service that is considered

completely unacceptable for use.

• Each item or service experience is either considered

defective or not—there are only two choices.

LOTS AND SAMPLES

CONTROL LIMITS• Voice of the process

• Calculated from Data

• Appear on control charts

• Appear to subgroups

• Guide for process actions

• What the process is doing

SPECIFICATION LIMITS• Voice of the customer

• Defined from the customer

• Appear on histograms

• Apply to items

• Separate good items from bad items

• What we want the process do

TYPES OF CONTROL CHARTS

CONTROL CHARTSGeneral Procedure of constructing a chart:A control chart consists of:• A graph such that part number is plotted along X-axis and attribute measure along Y-

axis.

• The mean of this statistic using all the samples is calculated (e.g., the mean of the means, mean of the ranges, mean of the proportions)

• A centre line (CL) is drawn at the value of the mean.

CL = Mean x

Y-axis

X-axis

CONTROL CHARTS

• The standard deviation σ of the statistic is also calculated using all the samples.

• Upper and lower control limits indicate the threshold at which the process output is considered undesired and are drawn at 3 standard deviation from the centre line.

CL = Mean x

UCL = x + 3σ

LCL = x - 3σ

Y-axis

X-axis

CONTROL CHARTS• Plot the attribute measure representing a statistic (e.g., a mean, range, proportion) of

measurements of a quality characteristic in samples taken from the process at different times (i.e., the da for each part number and join all the points to form a curve).

• If anyone of the points in the curve exceeds the upper and lower control limits, then the process is said to be out of control

CL = Mean x

UCL = x + 3σ

LCL = x - 3σA

B

C

D

EF

GH

I

J

K

L M

Y-axis

X-axis

CONTROL CHARTS FOR VARIABLES

CONTROL CHART FOR VARIABLES

A single measurable quality characteristic ,such as dimension, weight, or volume, is called

variable.

Our objectives for this section are to learn how to use control charts to monitor continuous

data.

We want to learn the assumptions behind the charts, their application, and their interpretation.

Since statistical control for continuous data depends on both the mean and the variability,

variables control charts are constructed to monitor each.

The most commonly used chart to monitor the mean is called the X-BAR chart.

There are two commonly used charts used to monitor the variability: the R chart and the s

chart.15

The X-BAR Chart:

This chart is called the X-BAR chart because the statistic being plotted is the sample mean. The

reason for taking a sample is because we are not always sure of the process distribution. By using

the sample mean we can "invoke" the central limit theorem to assume normality.

The R chart

I. The R chart is used to monitor process variability when sample sizes are small (n<10), or to simplify the

calculations made by process operators.

II. This chart is called the R chart because the statistic being plotted is the sample range.III. Using the R chart, the estimate of the process standard deviation,σ ,is R/d2.

THE S CHART

i. The s chart is used to monitor process variability when sample sizes are large (n*10), or when a computer is available to automate the calculations.

ii. This chart is called the s chart because the statistic being plotted is the sample standard deviation.

iii. Using the s chart, the estimate of the process standard deviation, σ, is

PROCEDURE FOR USING VARIABLES CONTROL CHARTS:

I. Determine the variable to monitor.

II. At predetermined, even intervals, take samples of size n (usually n=4 or 5).

III. Compute X BAR and R (or s) for each sample, and plot them on their respective control

charts. Use the following relationships:

IV. After collecting a sufficient number of samples, k (k>20), compute the control limits for the

charts. The following additional calculations will be necessary:

V. If any points fall outside of the control limits, conclude that the process is out of control, and

begin a search for an assignable or special cause. When the special cause is identified,

remove that point and return to step 4 to re-evaluate the remaining points.

VI. If all the points are within limits, conclude that the process is in control, and use the calculated

limits for future monitoring of the process.

EXAMPLE PROBLEM

A large hotel in a resort area has a housekeeping staff that cleans and prepares all of the hotel's

guestrooms daily. In an effort to improve service through reducing variation in the time required to

clean and prepare a room, a series of measurements is taken of the times to service rooms in one

section of the hotel. Cleaning times for five rooms selected each day for 25 consecutive days

appear below:

Day Room 1 Room 2 Room 3 Room 4 Room 5

1 15.6 14.3 17.7 14.3 15

2 15 14.8 16.8 16.9 17.4

3 16.4 15.1 15.7 17.3 16.6

4 14.2 14.8 17.3 15 16.4

5 16.4 16.3 17.6 17.9 14.9

6 14.9 17.2 17.2 15.3 14.1

7 17.9 17.9 14.7 17 14.5

8 14 17.7 16.9 14 14.9

9 17.6 16.5 15.3 14.5 15.1

10 14.6 14 14.7 16.9 14.2

11 14.6 15.5 15.9 14.8 14.2

12 15.3 15.3 15.9 15 17.8

13 17.4 14.9 17.7 16.6 14.7

14 15.3 16.9 17.9 17.2 17.5

15 14.8 15.1 16.6 16.3 14.5

16 16.1 14.6 17.5 16.9 17.7

17 14.2 14.7 15.3 15.7 14.3

18 14.6 17.2 16 16.7 16.3

19 15.9 16.5 16.1 15 17.8

20 16.2 14.8 14.8 15 15.3

21 16.3 15.3 14 17.4 14.5

22 15 17.6 14.5 17.5 17.8

23 16.4 15.9 16.7 15.7 16.9

24 16.6 15.1 14.1 17.4 17.8

25 17 17.5 17.4 16.2 17.9

CALCULATE THE MEAN, RANGE, STANDARD DEVIATION

Day Average Range St. Dev

1 15.4 3.4 1.41

2 16.2 2.6 1.19

3 16.2 2.2 0.85

4 15.5 3.1 1.27

5 16.6 3 1.19

6 15.7 3.1 1.4

7 16.4 3.4 1.69

8 15.5 3.7 1.71

9 15.8 3.1 1.24

10 14.9 2.9 1.16

11 15 1.7 0.69

12 15.9 2.8 1.13

13 16.3 3 1.39

14 17 2.6 1

15 15.5 2.1 0.93

16 16.6 3.1 1.26

17 14.8 1.5 0.65

18 16.2 2.6 0.98

19 16.3 2.8 1.02

20 15.2 1.4 0.58

21 15.5 3.4 1.37

22 16.5 3.3 1.59

23 16.3 1.2 0.51

24 16.2 3.7 1.56

25 17.2 1.7 0.64

CALCULATE THE CONTROL LIMITS

15.94 2.7 1.14

X R s

x Chart Control Limits

UCL = x + A R

LCL = x - A R

2

2

R Chart Control Limits

UCL = D R

LCL = D R

4

3

X-BAR , R CHART X- BAR ,S CHART

TABLE FOR CONSTANTS

SQC A MODERN INTRODUCTION 6th Edition,D.C. Montgomery

CONSTANTS FOR X-BAR ,R & X-BAR, S CHARTS

X-BAR , R CHART X-BAR , S CHART

A2 D4 D3

0.577 2.004 0

A3 B4 B3

1.427 2.089 0

X-BAR CHART

CL =15.94UCL = 15.94+ 0.577(2.7) = 17.49LCL = 15.94 – 0.577(2.7) = 14.38

R CHART

CL = 2.7UCL = 2.004(2.7) = 5.41LCL = 0 (2.7) = 0

X-BAR CHART

CL =15.94UCL = 15.94+ 1.427(1.14) = 17.56LCL = 15.94 – 1.427(1.14) = 14.31

S CHART

CL = 1.14UCL = 2.089(1.14) = 2.38LCL = 0 (1.14) = 0

X-BAR , R CONTROL CHART

X-BAR,S CONTROL CHART

CONTROL CHARTS FOR ATTRIBUTES

CONTROL CHARTS

CLASSIFICATION OF ATTRIBUTES CONTROL CHARTS


CLASSIFICATION CHARTS

P-chart

NP-chart

COUNT CHARTS

C-chart

U-chart


Classification charts:Classification charts deal with either the fraction of items or the

number of items in a series of subgroups.


P – chart:The P-Chart monitors the percent of samples having the condition, relative to

either a fixed or varying sample size.

EXAMPLE SCENARIO:Consider a shop purchasing bundles of readymade shirts from same

manufacturer. The number of shirts in each bundle and the defective item are as follows. Plot a suitable control chart.

Keywords: Changing sample size (No. of shirts in the bundle), defectives.

No. of shirts in the bundle

No. of defective shirts

20 5

15 3

20 2

25 1

20 5

CONTROL CHARTS FOR ATTRIBUTES(P chart)A team in an accounting group has been working on improving the processing of invoices. The

team is trying to reduce the cost of processing invoices by decreasing the fraction of invoices with errors. The team developed the following operational definition for a defective invoice: an invoice is defective if it has incorrect price, incorrect quantity, incorrect coding, incorrect address, or incorrect name. The team decided to pull a random sample of 50 invoices per day. If the invoice had one or more errors it was defective. The data from the last 20 days are given in the table. Draw the p chart.

Sample no No of defectives Fraction defective

1 4 0.082 3 0.063 8 0.164 12 0.245 7 0.146 15 0.307 20 0.408 13 0.269 9 0.1810 8 0.1611 5 0.10

Sample no No of defectives Fraction defective

12 14 0.28

13 9 0.18

14 11 0.22

15 2 0.04

16 9 0.18

17 3 0.06

18 13 0.26

19 6 0.12

20 9 0.18

180

CONTROL CHARTS FOR ATTRIBUTES (P chart)

CALCULATION:

P =Σpi/n

= 18/(20 x 50)

= 0.18

UCL = p + 3 √(p x (1-p)/n)

= 0.18 + 3 √(0.18 x (1 - 0.18)/50)

= 0.343

LCL = p - 3 √(p x (1-p)/n)

= 0.18 - 3 √(0.18 x (1- 0.18)/50)

= 0.017

CONTROL CHARTS FOR ATTRIBUTES (P chart)

The process is out of control because lot number 7 exceeds the upper control limit.

191715131197531

0.4

0.3

0.2

0.1

0.0

Sample

Pro

port

ion

_P=0.18

UCL=0.3430

LCL=0.0170

1

P Chart of C2


NP-chart:The NP-Chart monitors the number of times a condition occurs, relative to a

constant sample size.

Example scenario:Consider a ball manufacturing company. It produces 1000 balls daily. The

number of defective balls produced each day for 10 days are 10, 13, 9, 3, 7, 16, 21,19, 12, 10. Plot the control chart.

Keywords: Sample size doesn’t change (1000 balls), defectives

CONTROL CHARTS FOR ATTRIBUTES (NP chart)

The data representing the results of inspecting l00 units of personal computer produced for the past 10 days. Does the process appear to be in control.

Sample lot number Sample size No. of defectives

1 100 8

2 100 7

3 100 12

4 100 5

5 100 18

6 100 2

7 100 10

8 100 16

9 100 14

10 100 6

Total 1000 98


CALCULATION:

Average number of defective = np =98/10 = 9.8

Average fraction defective = p =98/1000 = 0.098

UCL = np + 3 √(np x (1-p))

= 9. 8 + 3 √(9.8 x (1 - 0.098))

= 18.72

LCL = np - 3 √(np x (1-p))

= 9.8 - 3 √(9.8 x (1- 0.098))

= 0.88


The process is in control

10987654321

20

15

10

5

0

Sample

Sam

ple

Count

__NP=9.8

UCL=18.72

LCL=0.88

NP Chart of C3


Count charts:Count charts deal with the number of times a particular characteristic

appear in some given area of opportunity.C-chart: The c-chart monitors the number of times a condition occurs, relative to a constant sample size. In this case, a given sample can have more than one instance of the condition, in which case we count all the times it occurs in the sample.Example scenario:

Consider an engineer buys a set of 10 ropes for his construction project. All the ropes of same length of 100 meters. Each rope is inspected and the number of defects in each rope are found to be 1,1,3,1,2,1,5,3,4 and 2 respectively. Plot the control chart.

Keywords: Sample size doesn’t change (100 m), defects

CONTROL CHARTS FOR ATTRIBUTES (C chart)

The data represents no. of non-conformities per 1000 metres in a telephone cable. From the analysis of these data, you conclude that the process is in control or not?

Sample no No. of Non-conformities1 12 13 34 75 86 107 58 139 010 1911 24

Sample no No. of Non-conformities12 613 914 1115 1516 817 318 619 720 421 922 20


CL = c = 189/22 = 8.59

UCL = c + 3 √c

= 8.59 + 3 √8.59

= 17.38

LCL = c - 3 √c

= 8.59 - 3 √8.59

= 0


The process is out of control

21191715131197531

25

20

15

10

5

0

Sample

Sam

ple

Count

_C=8.59

UCL=17.38

LCL=0

1

1

1

C Chart of C4


U-chart: The u-Chart monitors the percent of samples having the condition, relative to either a fixed or varying sample size. In this case, a given sample can have more than one instance of the condition, in which case we count all the times it occurs in the sample.

Example scenario: 5 lots of cloth produced by a manufacturer are inspected for defects. The sample taken for

each inspection is different. The sample taken and number of defects found are as follows.

Keywords: changing sample size (length of the cloth), defects

Length of the cloth inspected No. of defects

20 m 5

15 m 3

20 m 2

25 m 1

20 m 5

CONTROL CHARTS FOR ATTRIBUTES (U chart)

Lots of cloth produced by a manufacturer are inspected for defects. Because of the nature of inspection process, the size of the inspection sample varies from lot to lot. Calculate the center line and lower control limits for the appropriate control chart

Lot number 100s of square yards No. of defects

1 2 5

2 2.5 7

3 1 3

4 0.9 2

5 1.2 4

6 0.8 1

7 1.4 0

8 1.6 2

Lot number 100s of square yards No. of defects

9 1.9 3

10 1.5 0

11 1.7 2

12 1.7 3

13 2 1

14 1.6 2

15 1.9 4


CALCULATION:

CL = u = Σc/Σa

= 1.646

LCL = u – 3 √(u/a1)

= 1.646 - 3 √(1.646/2)

= 0

UCL= u1 + 3√(u/a1)

= 1.646 + 3 √ (1.646/2)

= 4.36


The process is in control

151413121110987654321

6

5

4

3

2

1

0

Sample

Sam

ple

Count

Per

Unit

_U=1.646

UCL=4.437

LCL=0

U Chart of C6

Tests performed with unequal sample sizes


How to select control chart:

DefectsExample: A crack in the shaft

DefectivesExample: Broken shaft

Constant sample size C chart np chart

Changing sample size U chart p chart

CONTROL CHARTS FOR ATTRIBUTESA large publisher counts the number of keyboard errors that make their way into finished

books. The number of errors and the number of pages in the past 26 publications are

Book no. No. of errors No. of pages

1 49 202

2 63 232

3 57 332

4 33 429

5 54 512

6 37 347

7 38 401

8 45 412

9 65 481

10 62 770

11 40 577

12 21 734

13 35 455

Book no. No. of errors No. of pages

14 48 612

15 50 432

16 41 538

17 45 383

18 51 302

19 49 285

20 38 591

21 70 310

22 55 547

23 63 469

24 33 652

25 14 343

26 44 401


Try yourself…

The following 20 days data represent the findings from a study conducted at a factory that manufactures film canisters. Each day 500 canisters were sampled and inspected. The number of defective film canisters were recorded each day as follows. Is this process in control?

Day No. of nonconforming

1 26

2 25

3 23

4 24

5 26

6 20

7 21

8 27

9 23

10 25

Day No. of nonconforming

11 22

12 26

13 27

14 29

15 20

16 19

17 23

18 19

19 18

20 27

CONTROL CHARTS FOR ATTRIBUTESTry yourself…

Consider the output of a paper mill: the product appears at the end of a web and is rolled onto a spool called reel. Each reel is examined for blemishes. Results of the inspections are as follows. Is this in control?

Reel No. of blemishes

1 4

2 5

3 5

4 10

5 6

6 4

7 5

8 6

9 3

10 6

11 6

12 7

Reel No. of blemishes

13 11

14 9

15 1

16 1

17 6

18 10

19 3

20 7

21 4

22 8

23 7

24 9

25 7


Try yourself…

The following data represent the results of inspecting all units of a personal computer produced for the past 10 days. Is this process in control?

Day Unit Inspected Non-confirming units

1 80 4

2 110 7

3 90 5

4 75 8

5 130 6

6 120 6

7 70 4

8 125 5

9 105 8

10 95 7

ADVANTAGES OF CONTROL CHARTS

A control chart indicate whether the process is in control or out of control.

It determines the process variability and detects unusual variations in a process.

It ensures product quality level.

It warns in time and if process is rectified at that time percentage of rejection can be

reduced.

It provides information about selection of process and setting up of tolerance limits.

THANK YOU

Engineering

Control charts