JF608: Quality Control - Unit 1

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UNIT 1- UNIT 1- BASIC STATISTICS

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LOGOOUTLINEOUTLINE

Introduction

Statistical Process Control (S.P.C.)

Measure of Central Tendency

Measure of Dispersion

Frequency Distribution

The Normal Curve

LOGOINTRODUCTIONINTRODUCTION

Definition of Statistics:

Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting numerical data to assist in making more effective decisions.


Who Uses Statistics?

Statistical techniques are used extensively in marketing, accounting, quality control, consumers, professional sports people, hospital administrators, educators, politicians, physicians, etc...

LOGOINTRODUCTIONINTRODUCTIONTypes of Statistics

Descriptive Statistics: Describes the characteristics of a product or process

using information collected on it. Methods of organizing, summarizing, and presenting

data in an informative way.

Inferential Statistics: Draws conclusions on unknown process parameters

based on information contained in a sample. A decision, estimate, prediction, or generalization

about a population, based on a sample. Uses probability


Type of Variable

A. Qualitative or Attribute variable - The characteristic being studied is nonnumeric. Examples: Gender, religious affiliation, type of automobile owned, state of birth, eye color are examples.

B. Quantitative variable - Information is reported numerically. Examples: Balance in your checking account, minutes remaining in class, or number of children in a family.


Type of Variable

Type of Variable

Qualitative Quantitative

• brand of PC• marital status• hair colours

ContinuousDiscrete

• amount of income tax paid

• weight of a student• yearly rainfall in

Malacca

• children in a family• TV sets owned• strokes on a golf

hole

LOGOSTATISTICAL PROCESS CONTROLSTATISTICAL PROCESS CONTROL

Statistical Process Control (S.P.C.)

This is a control system which uses statistical techniques for knowing, all the time, changes in the process.

It is an effective method in preventing defects and helps continuous quality improvement.


WHAT DOES S.P.C. MEAN?Statistical:Statistics are tools used to make predictions

on performance.There are a number of simple methods for

analysing data and, if applied correctly, can lead to predictions with a high degree of accuracy.


WHAT DOES S.P.C. MEAN?Process:The process involves people, machines,

materials, methods, management and environment working together to produce an output, such as an end product.

People Machines Material

Management Methods Environment

Output


WHAT DOES S.P.C. MEAN?

Control:Controlling a process is guiding

it and comparing actual performance against a target/nominal value.

Then identifying when and what corrective action is necessary to achieve

the target.


S.P.CStatistics aid in making decisions about a

process based on sample data and the results predict the process as a whole.


POPULATION & SAMPLE

X = Sample mean s = Sample standard Deviationµ = Population mean σ= Population Standard deviation


POPULATION & SAMPLE

Population :Includes each element from the set of observations that can be

made.

Sample:Consists only of observations drawn from the population.

Population parameter (µ,σ)The mean of a population is denoted by the symbol μ.

Sample statistic (x , s)The mean of a sample is denoted by the symbol x. A quality

calculated from sample of observation.

LOGO

VARIATIONSVariation exists in all processes.Variation can be categorized as either:

Example: Let us taking a pie and cutting it into pieces, making each pieces the same size as best we can.

This is inherent variability so even very good product.

Common or Random causes of variationOR

Assignable Cause Variation

STATISTICAL PROCESS CONTROLSTATISTICAL PROCESS CONTROL

LOGO

PROCESS VARIATIONNo industrial process or machine is able to

produce consecutive items which are identical in appearance, length, weight, thickness etc.

The differences may be large or very small, but they are always there.

The differences are known as ‘variation’. This is the reason why ‘tolerances’ are used.


LOGO

Process VariationsSTATISTICAL PROCESS CONTROLSTATISTICAL PROCESS CONTROL

Process Element Variable Examples

Machine Speed, operating temperature, feed rate

Tools Shape, wear rate

Fixtures Dimensional accuracy

Materials Composition, dimensions

Operator Choice of set-up, fatigue

Maintenance Lubrication, calibration

Environment Humidity, temperature

LOGO

TYPES OF VARIATION (1)

1. Random variation Random causes that we cannot identify. Unavoidable, e.g. slight differences in process

variables like diameter, weight, service time, temperature, equipment, tooling, employee actions, facility environment, materials, measurement system, etc.

Also called common/natural cause variation To reduce random variation, we must reduce

variation in the inputs and the process. As long as the distribution remains in specified

limits, the process said be ‘in control’.


LOGO

TYPES OF VARIATION (2)

2. Non-random variation Also called special cause variation or

assignable cause variation Caused by equipment out of adjustment,

worn tooling, operator errors, poor training, defective materials, measurement errors, new batch of raw materials etc.

The process is not behaving as it usually does.

The cause should be identified and corrected.


LOGO

Mean = the calculated average of all the values in a given data set

Median = the central value of a data set arranged in order

Mode = the value which occurs with most frequency in a given data set

MEASURE OF CENTRAL TENDENCYMEASURE OF CENTRAL TENDENCY

LOGOMEASURE OF CENTRAL TENDENCYMEASURE OF CENTRAL TENDENCY

Ungrouped Data Grouped DataMean for population data:

Mean for sample data:

where: = the sum af all values N = the population size n = the sample size, = the population mean = the sample mean

Mean for population data:

Mean for sample data:

where: = midpoint = frequency of a class

Mean

Nxu

nxx

x

x

u

Nfxu

nfxx

xf

LOGO

Example (Ungrouped Data) The following data give the prices (rounded to

thousand RM) of five homes sold recently in NEC.

158 189 265 127 191 Find the mean sale price for these homes.


LOGO

Solution

Thus, these five homes were sold for an average price of RM186 thousand @ RM186 000.

The mean has the advantage that its calculation includes each value of the data set.


1865

9305

191127265189158

nxx

LOGO

Example (Grouped Data)

The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Calculate the mean.


Number of order f

10 – 12 4 13 – 15 12

16 – 18 20

19 – 21 14

n = 50

LOGO

Solution Because the data set includes only 50 days, it represents

a sample. The value of is calculated in the following table:


fx

Number of order f x fx

10 – 12 4 11 44

13 – 15 12 14 168

16 – 18 20 17 340

19 – 21 14 20 280

n = 50 Σfx = 832

LOGO

Solution

The value of mean sample is:

Thus, this mail-order company received an average of 16.64 orders per day during these 50 days.


64.1650

832 nfxx

LOGO

The measures of central tendency such as mean, median and mode do not reveal the whole picture of the distribution of a data set.

Two data sets with the same mean may have a completely different spreads.

The variation among the values of observations for one data set may be much larger or smaller than for the other data set

Relative Dispersion Measurement :i. Rangeii. Standard Deviation

MEASURE OF DIPERSIONMEASURE OF DIPERSION

LOGO

Range (Ungrouped Data)

ExampleFind the range of production for this data set


RANGE = Largest value – Smallest value

StateTotal Area

(square miles)

Arkansas 53,182

Louisiana 49,651

Oklahoma 69,903

Texas 267,277

LOGO

SolutionFind the range of production for this data set

Range = Largest value – Smallest value = 267 277 – 49 651 = 217 626

Disadvantages: being influenced by outliers. Based on two values only. All other values in a data set are ignored.


LOGO

Range (Grouped Data)

ExampleFind the range for this data set


Range = Upper bound of last class – Lower bound of first class

Class Frequency

41 - 50 1

51 - 60 3

61 - 70 7

71 - 80 13

81 – 90 10

91 - 100 6

TOTAL 40

Solution

Upper bound of last class = 100.5Lower bound of first class = 40.5Range = 100.5 – 40.5 = 60

LOGO

Range (Ungrouped Data)

ExampleFind the range of production for this data set


RANGE = Largest value – Smallest value

StateTotal Area

(square miles)

Arkansas 53,182

Louisiana 49,651

Oklahoma 69,903

Texas 267,277

LOGO

Variance and Standard Deviation Standard deviation is the most used measure of dispersion. A Standard Deviation value tells how closely the values of a data set clustered

around the mean. Lower value of standard deviation indicates that the data set value are spread

over relatively smaller range around the mean. Larger value of data set indicates that the data set value are spread over

relatively larger around the mean (far from mean). Standard deviation is obtained the positive root of the variance


LOGO

Variance and Standard DeviationUngrouped Data


VarianceStandard Deviation

Population

Sample

NNxx

22

2

1

22

2

nnxx

s

2

2ss

LOGO

ExampleLet x denote the total production (in unit) of company

Find the variance and standard deviation


Company ProductionA 62B 93

C 126

D 75

E 34

LOGO

Solution


Company Production (x) x2

A 62 3844B 93 8649

C 126 15876

D 75 5625

E 34 1156

Σx=390 Σx2=35150

5.118215

5

2390

35150

1

22

2

nnxx

s

3875.3450.1182

,;50.11822

sTherefore

sSince

LOGO

Variance and Standard DeviationGrouped Data


VarianceStandard Deviation

Population

Sample

NNfx

fx

2

22

1

22

2

nnfx

fxs

2

2ss

LOGO

ExampleFind the variance and standard deviation for the following data:


No. of order f10 – 12 413 – 15 1216 – 18 20

19 – 21 14

TOTAL n = 50

LOGOMEASURE OF DIPERSIONMEASURE OF DIPERSION

No. of order f x x2 fx fx2

10 – 12 4 11 121 44 48413 – 15 12 14 196 168 235216 – 18 20 17 289 340 5780

19 – 21 14 20 400 280 5600

TOTAL n = 50 832 14216

5820.715050

283214216

1

22

2

nnfx

fxs

Solution

Variance

75.2

5820.7

2

ss

Standard Deviation

112

1012int

pomidx

LOGO

A frequency distribution for quantitative data lists all the classes and the number of values that belong to each class.

Data presented in form of frequency distribution are called grouped data.

FREQUENCY DISTRIBUTIONFREQUENCY DISTRIBUTION

LOGO

The class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class. Also called real class limit.

To find the midpoint of the upper limit of the first class and the lower limit of the second class, we divide the sum of these two limits by 2.

Example :


5.4002

401400

boundaryclassLower

LOGO

Class Width (class size)

Example

Class Midpoint or Mark

Example :


boundaryLowerboundaryUpperwidthClass

2005.4005.600 classfirsttheofWidth

2

limlimint

itUpperitLowermidpoClass

5.5002

6004011

classsttheofWidth

LOGOFREQUENCY DISTRIBUTIONFREQUENCY DISTRIBUTION

LOGO

1. To decide the number of classes, we used Sturge’s formula, which is

Where; c = the no. of classes n = the no. of observations in the data set

2. Class width,

This class width is rounded to a convenient number3. Lower Limit of the First Class or the Starting Point Use the smallest value in the data set

CONSTRUCTING FREQUENCY DISTRIBUTION CONSTRUCTING FREQUENCY DISTRIBUTION TABLESTABLES

nc log3.31

c

rangei

classesofnumber

valuesmallestvalueestli

arg

LOGO

ExampleThe following data give the total home runs hit by all players of each of the 30 Major League

Baseball teams during 2004 season


LOGO

Solution1.

2.

3. Starting point = 135


class

ncclassesofNumber

689.8

48.13.31

log3.31,

18

8.176

135242,

iwidthClass

LOGO

Frequency Distribution for Data


Total Home Runs

Class Boundaries

Tally f

135-152 134.5 - 152.5 IIII IIII 10

153-170 152.5 - 170.5 II 2

171-188 170.5 - 188.5 IIII 5

189-206 188.5 - 206.5 IIII I 6

207-224 206.5 - 224.5 III 3

225-242 224.5 - 242.5 IIII 4

Σf=30

LOGO

Histograms


134.5 152.5 170.5 188.5 206.5 224.5 242.5

LOGOTHE NORMAL CURVETHE NORMAL CURVE

The Histogram and the Normal Curve


The Theoretical Normal Curve


Properties of the Normal Curve: Theoretical construction Also called Bell Curve or Gaussian Curve Perfectly symmetrical normal distribution The mean of a distribution is the midpoint of the

curve The tails of the curve are infinite Mean of the curve = median = mode The “area under the curve” is measured in standard

deviations from the mean


Properties of the Normal Curve: Has a mean = 0 and standard deviation = 1. General relationships:±1 s = about 68.26%

±2 s = about 95.44%±3 s = about 99.72%

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

68.26%

95.44%

99.72%


Standard Scores

One use of the normal curve is to explore Standard Scores. Standard Scores are expressed in standard deviation units, making it much easier to compare variables measured on different scales.

There are many kinds of Standard Scores. The most common standard score is the ‘z’ scores.

A ‘z’ score states the number of standard deviations by which the original score lies above or below the mean of a normal curve.


The Z Score

The normal curve is not a single curve but a family of curves, each of which is determined by its mean and standard deviation.

In order to work with a variety of normal curves, we cannot have a table for every possible combination of means and standard deviations.


The Z Score

What we need is a standardized normal curve which can be used for any normally distributed variable. Such a curve is called the Standard Normal Curve.

sxxZ i


The Standard Normal Curve

The Standard Normal Curve (z distribution) is the distribution of normally distributed standard scores with mean equal to zero and a standard deviation of one.

A z score is nothing more than a figure, which represents how many standard deviation units a raw score is away from the mean.


Example 1:

A normal curve has an average of 55.38 and a standard deviation of 1.95. What percentage of the area under the curve will fall between the limits of 52.5 and 56.5


Solution:

Given data,

5.565.52,95.138.55

21

XandXLimits

x

]57.0[7157.0,

57.095.1

38.555.56]48.1[0694.0,

48.195.1

38.555.52

22

22

11

11

ATableAppendixZForAArea

xxZ


xxZ

LOGOTHE NORMAL CURVETHE NORMAL CURVESolution:The area under the normal distribution curve is

Therefore, the area under the curve between limits 52.5 and 56.5 = A2 – A1

= 0.7157 – 0.0694 = 0.6463 = 64.63%

µ

52.38

x2

56.5

Area, A2

Area, A1

x1

52.5


Example 2:

The life of an equipment in hours is a random variable following normal distribution having a mean life of 5600 hours with standard deviation of 840 hours.i.What % of equipment will fail between 5000 and 6200 hours.ii.What % will survive more than 6000 hours.iii.What % will fail below 3500 hours.

LOGOTHE NORMAL CURVETHE NORMAL CURVESolution:Given data,

i.Percentage of equipment that will fail between 5000 and 6200 hours.Let x1 = 5000 hours, x2 = 6200 hours

hourshoursx

8405600

]71.0[7611.0,

71.0840

56006200]71.0[2389.0,

71.0840

56005000

22

22

11

11


xxZ


xxZ

LOGOTHE NORMAL CURVETHE NORMAL CURVESolution:

Area under the curve between 5000 hours and 6200 hours

= A2 – A1

= 0.7611 – 0.2389 = 0.5222 = 52.22%

µ

5600

x2

6200

Area, A2

Area, A1

x1

5000

LOGOTHE NORMAL CURVETHE NORMAL CURVESolution:ii. Percentage of equipment that will survive more

than 6000 hours.

Hence, the percentage is= 1 – A1 = 1 – 0.6844 = 0.3156 = 31.56%

]48.0[6844.0,

48.0840

56006000

1 ATableAppendixZForAArea

xxZ

Total area = 1

Area, A

μ5600

x6000

Area, A1

LOGOTHE NORMAL CURVETHE NORMAL CURVESolution:iii. Percentage of equipment that will fail below 3500

hours.

Hence, the percentage is= 0.062%

]5.2[0062.0,

5.2840

56003500

1 ATableAppendixZForAArea

xxZ

μ5600

x3000

Area, A1

LOGO

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JF608: Quality Control - Unit 1