3_measure phase -introduction to statistics _14!1!15 [autosaved] [compatibility mode](1)

IBM-09: Six Sigma – Tools and TechniquesMeasure Phase

Introduction to Basic Statistics

A. Ramesh PhDDepartment of Management StudiesIndian Institute of Technology [email protected]

Measure Phase – Basic Statistics

What is Data?: Refer to facts usually collected as the result of experience, observation or

measurement

Consist of numbers, words, or images

Lowest level of abstraction from which information and knowledge arederived

DATA Information Knowledge

“ I believe in God - Rest bring data!” – A famous quote


WHAT DO THESE WORDS & NUMBERS MEAN TO YOU ?

2.87, 2.85, 2.88, 2.85, 2.86, 2.85, 2.81, 2.82, 2.83, 2.85, 2.84, 2.84, 2.85, 2.86, 2.85, 2.84, 2.85, 2.85, 2.87, 2.81, 2.85, 2.82, 2.83, 2.85, 2.85, 2.86, 2.85, 2.86, 2.89, 2.85, 2.84, 2.84, 2.85, 2.85, 2.83, 2.82, 2.86, 2.83, 2.85, 2.86, 2.85, 2.84, 2.84, 2.87, 2.85, 2.86, 2.85, 2.84, 2.90, 2.88

PROBABLY NOTHING MUCH !

ROOF

GLASS

TREE

SKY

LAMP

FLOWERS

WIRES

WINDOWS

GRASSPOLE

TREE

TILES


Statistics helps in summarizing and understanding the data

How about following pictures?

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Can Statistics Be Trusted?“There are three kinds of lies:

Lies, damned lies, and statistics.”‐‐Mark Twain

“It is easy to lie with statistics. But it is easier to lie without them.” ‐‐Frederick Mosteller

“Figures won’t lie but liars will figure.”‐‐Charles Grosvenor

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7

Statistics..

• Plays an important role in many facets of human endeavour

• Occurs remarkably frequently in our everyday lives

• It is often incorrectly thought of as just a collection of data, graphs and diagrams

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What is Statistics?

• Science of gathering, analyzing, interpreting, and presenting data

• Branch of mathematics• Facts and figures• Statistics is the scientific method that enables us to

make decisions as responsibly as possible.

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Statistics: Science of variability..?

• Virtually everything varies• Variation occurs among individuals• Variation occurs within any one individual as

time passes

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Concept of Variation:

Variation is natural and can not be avoided

Customer experiences the variation not the average.

Lower the variation, better the customer experience

What Customer wants

What customer experiences

(Variation)

What we measure

(Average)


Sources of Variation:

Any changes to the above factors would directly impact the process performance and causes for variation


Variation – Example:

One day he reaches little earlier (6:55)

Another day he reaches little late (7:05)

6.55 7.05

A man wants to reach his work place exactly by 7:00 a.m

Can we identify cause for this variation?

No ! We may not be !It may be affected by factors which

Affects the time he takes to travelHe cannot controlVary randomly

E.g..) The normal traffic he encountersunder normal course of travel

These Variation is called asInherent Variation or

Common Cause Variation orWhite Noise


Variation – Example:

YES. We Can ! It may be because of some specific circumstances which do not occur in the normal scheme of actions.E.g..)

• His watch was running fast• He got a lift• He had a Client Call• He had some important work to be

finished before 7.30

These variations are called asSpecial Cause Variation or

Black Noise

TODAY HE IS VERY EARLY !

Can We find out what is cause for this?

6.00


Reacting to common cause vs. special cause:


Sampling:

Sampling is the process of:

Collecting only a portion of the data that is available or could be available,and drawing conclusions about the total population (statistical inference)

From the sample, we infer that the

average resolutiontime (x) is 1.2 days

Population Sample

xx

x

xxx

x

x

x

xx

x x

x

x

xx

x

xxx

x

N = 567 daysn = 3 days

What is the AverageResolution time?

*Within a certain confidence band or

margin of error

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Population Versus Sample• Population — the whole

– a collection of persons, objects, or items under study– The entire group of individuals in a statistical study we

want information about.

• Census — gathering data from the entire population• Sample — a portion of the whole

– a subset of the population– a part of the population from which we actually collect

information, used to draw conclusions about the whole (statistical inference

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Statistics can be split into two broad categories

1. Descriptive statistics

2. Statistical inference

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Descriptive Statistics

Collect data ex. Survey

Present data ex. Tables and graphs

Characterize data ex. Sample mean = iX

n

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Descriptive statistics..

• Encompasses the following:– Graphical or pictorial display– Condensation of large masses of data into a form

such as tables– Preparation of summary measures to give a

concise description of complex information (e.g. an average figure)

– Exhibition of patterns that may be found in sets of information

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Inferential Statistics

Estimation ex. Estimate the population

mean weight using the sample mean weight

Hypothesis testing ex. Test the claim that the

population mean weight is 120 pounds

Drawing conclusions and/or making decisions concerning a population based on sample results.

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Inferential Statistics..

• Especially relates to:– Determining whether characteristics of a situation

are unusual or if they have happened by chance– Estimating values of numerical quantities and

determining the reliability of those estimates– Using past occurrences to attempt to predict the

future

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Process of Inferential Statistics

Population

(parameter)

)(statisticx

Sample estimate to

x Calculate

Select arandom sample

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Population vs. Sample

Population Sample

Measures used to describe the population are called parameters

Measures computed from sample data are called statistics

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Parameter vs. Statistic

• Parameter — descriptive measure of the population– Usually represented by Greek letters

• Statistic — descriptive measure of a sample– Usually represented by Roman letters

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Symbols for Population Parameters

denotes population parameter2 denotes population variance

denotes population standard deviation

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Symbols for Sample Statistics

x denotes sample mean2S denotes sample variance

S denotes sample standard deviation

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Types of Variables

Categorical (qualitative) variables have values that can only be placed into categories, such as “yes” and “no.”

Numerical (quantitative) variables have values that represent quantities.

Types of Variables

Data

Categorical Numerical

Discrete ContinuousExamples:

Marital Status Political Party Eye Color

(Defined categories)Examples:

Number of Children Defects per hour

(Counted items)

Examples:

Weight Voltage(Measured characteristics)

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Levels of Data Measurement

• Nominal — Lowest level of measurement• Ordinal• Interval• Ratio — Highest level of measurement

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Levels of Measurement

A nominal scale classifies data into distinct categories in which no ranking is implied.

Categorical Variables Categories

Personal Computer Ownership

Type of Stocks Owned

Internet Provider

Yes / No

Microsoft Network / AOL

Growth Value Other


An ordinal scale classifies data into distinct categories in which ranking is implied

Categorical Variable Ordered Categories

Student class designation Freshman, Sophomore, Junior, Senior

Product satisfaction Satisfied, Neutral, Unsatisfied

Faculty rank Professor, Associate Professor, Assistant Professor, Instructor

Standard & Poor’s bond ratings AAA, AA, A, BBB, BB, B, CCC, CC, C, DDD, DD, D

Student Grades A, B, C, D, F


An interval scale is an ordered scale in which the difference between measurements is a meaningful quantity but the measurements do not have a true zero point.

A ratio scale is an ordered scale in which the difference between the measurements is a meaningful quantity and the measurements have a true zero point.

Interval and Ratio Scales

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Usage Potential of VariousLevels of Data

Nominal

OrdinalIntervalRatio

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35

Data Level, Operations, and Statistical Methods

Data Level

Nominal

Ordinal

Interval

Ratio

Meaningful Operations

Classifying and Counting

All of the above plus Ranking

All of the above plus Addition, Subtraction

All of the above plus multiplication and division

StatisticalMethods

Nonparametric

Nonparametric

Parametric

Parametric

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36

Data preparation rules

• Data presented must be– factual– relevant

Before presentation always check:• the source of the data• that the data has been accurately transcribed• the figures are relevant to the problem

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37

Methods of visual presentation of data

• Table

1st Qtr 2nd Qtr 3rd Qtr 4th QtrEast 20.4 27.4 90 20.4West 30.6 38.6 34.6 31.6North 45.9 46.9 45 43.9

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• Graphs

0102030405060708090

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

EastWestNorth

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• Pie chart

1st Qtr2nd Qtr3rd Qtr4th Qtr

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• Multiple bar chart

0 20 40 60 80 100

1st Qtr

2nd Qtr

3rd Qtr

4th Qtr

NorthWestEast

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• Simple pictogram

020406080

100

1st Qtr 2nd Qtr 3rd Qtr 4th QtrEast

North

West

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42

Frequency distributions

• Frequency tables

Class Interval Frequency Cumulative Frequency< 20 13 13<40 18 31<60 25 56<80 15 71<100 9 80

Observation Table

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Frequency

0

5

10

15

20

25

30

< 20 <40 <60 <80 <100

Frequency

Frequency diagramsFrequency

0

5

10

15

20

25

30

< 20 <40 <60 <80 <100

Frequency

Cumulative Frequency

0102030405060708090

< 20 <40 <60 <80 <100

Cumulative Frequency

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44

Ungrouped Versus Grouped Data

• Ungrouped data• have not been summarized in any way• are also called raw data

• Grouped data• have been organized into a frequency

distribution

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45

Example of Ungrouped Data

42

30

53

50

52

30

55

49

61

74

26

58

40

40

28

36

30

33

31

37

32

37

30

32

23

32

58

43

30

29

34

50

47

31

35

26

64

46

40

43

57

30

49

40

25

50

52

32

60

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Ages of a Sample of Managers from

XYZ

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46

Frequency Distribution of Ages

Class Interval Frequency20-under 30 630-under 40 1840-under 50 1150-under 60 1160-under 70 370-under 80 1

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Data Range

42

30

53

50

52

30

55

49

61

74

26

58

40

40

28

36

30

33

31

37

32

37

30

32

23

32

58

43

30

29

34

50

47

31

35

26

64

46

40

43

57

30

49

40

25

50

52

32

60

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Smallest

Largest

Range = Largest - Smallest = 74 - 23= 51

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Number of Classes and Class Width• The number of classes should be between 5 and 15.

• Fewer than 5 classes cause excessive summarization.• More than 15 classes leave too much detail.

• Class Width• Divide the range by the number of classes for an approximate

class width

• Round up to a convenient number

10=Width Class

8.5 =651 = Width Class eApproximat

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Class Midpoint

Class Midpoint = beginning class endpoint + ending class endpoint

2

= 30 + 40

2= 35

Class Midpoint = class beginning point + 12

class width

= 30 + 12

10

= 35

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50

Relative FrequencyRelative

Class Interval Frequency Frequency20-under 30 6 .1230-under 40 18 .3640-under 50 11 .2250-under 60 11 .2260-under 70 3 .0670-under 80 1 .02

Total 50 1.00

650

1850

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Cumulative FrequencyCumulative

Class Interval Frequency Frequency20-under 30 6 630-under 40 18 2440-under 50 11 3550-under 60 11 4660-under 70 3 4970-under 80 1 50

Total 50

18 + 611 + 24

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Class Midpoints, Relative Frequencies, and Cumulative Frequencies

Relative CumulativeClass Interval Frequency Midpoint Frequency Frequency20-under 30 6 25 .12 630-under 40 18 35 .36 2440-under 50 11 45 .22 3550-under 60 11 55 .22 4660-under 70 3 65 .06 4970-under 80 1 75 .02 50

Total 50 1.00

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Cumulative Relative Frequencies

Relative Cumulative Cumulative Relative

Class Interval Frequency Frequency Frequency Frequency20-under 30 6 .12 6 .1230-under 40 18 .36 24 .4840-under 50 11 .22 35 .7050-under 60 11 .22 46 .9260-under 70 3 .06 49 .9870-under 80 1 .02 50 1.00

Total 50 1.00

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54

Common Statistical Graphs

• Histogram -- vertical bar chart of frequencies• Frequency Polygon -- line graph of frequencies• Ogive -- line graph of cumulative frequencies• Pie Chart -- proportional representation for

categories of a whole• Stem and Leaf Plot• Pareto Chart• Scatter Plot

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55

Histogram

Class Interval Frequency20-under 30 630-under 40 1840-under 50 1150-under 60 1160-under 70 370-under 80 1 0

1020

0 10 20 30 40 50 60 70 80

Years

Freq

uenc

y

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Histogram Construction

Class Interval Frequency20-under 30 630-under 40 1840-under 50 1150-under 60 1160-under 70 370-under 80 1

010

20

0 10 20 30 40 50 60 70 80

Years

Freq

uenc

y

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57

Frequency Polygon

Class Interval Frequency20-under 30 630-under 40 1840-under 50 1150-under 60 1160-under 70 370-under 80 1 0

1020

0 10 20 30 40 50 60 70 80

Years

Freq

uenc

y

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58

Ogive

CumulativeClass Interval Frequency20-under 30 630-under 40 2440-under 50 3550-under 60 4660-under 70 4970-under 80 50

020

4060

0 10 20 30 40 50 60 70 80

Years

Freq

uenc

y

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59

Relative Frequency Ogive

CumulativeRelative

Class Interval Frequency20-under 30 .1230-under 40 .4840-under 50 .7050-under 60 .9260-under 70 .9870-under 80 1.00

0.000.100.200.300.400.500.600.700.800.901.00

0 10 20 30 40 50 60 70 80

Years

Cum

ulat

ive

Rel

ativ

e Fr

eque

ncy

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60

Complaints by Passengers

COMPLAINT NUMBER PROPORTION DEGREES

Stations, etc. 28,000 .40 144.0

TrainPerformance

14,700 .21 75.6

Equipment 10,500 .15 50.4

Personnel 9,800 .14 50.6

Schedules,etc.

7,000 .10 36.0

Total 70,000 1.00 360.0

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61

Complaints by Passengers

Stations, Etc.40%Train

Performance21%

Equipment15%

Personnel14%

Schedules, Etc.10%

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62

Second Quarter Truck

Production

2d QuarterTruck

ProductionCompany

A

B

C

D

ETotals

357,411

354,936

160,997

34,099

12,747920,190

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63

39%39%

17%4%

1%

A B C D E

Second Quarter Truck Production

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Pie Chart Calculations for Company A

2d QuarterTruck

Production Proportion DegreesCompany

A

B

C

D

ETotals

357,411

354,936

160,997

34,099

12,747920,190

.388

.386

.175

.037

.0141.000

140

139

63

13

5360

357,411920,190

=

.388 360 =

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65

Pareto Chart

0102030405060708090

100

PoorWiring

Short inCoil

DefectivePlug

Other

Freq

uenc

y

0%10%20%30%40%50%60%70%80%90%100%

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66

Scatter Plot

Registered Vehicles (1000's)

Gasoline Sales (1000's of

Gallons)

5 60

15 120

9 90

15 140

7 60

0

100

200

0 5 10 15 20Registered Vehicles

Gas

olin

e Sa

les

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Principles of Excellent Graphs

The graph should not distort the data. The graph should not contain unnecessary

adornments (sometimes referred to as chart junk). The scale on the vertical axis should begin at zero. All axes should be properly labeled. The graph should contain a title. The simplest possible graph should be used for a

given set of data.

Graphical Errors: Chart Junk

1960: $1.00

1970: $1.60

1980: $3.10

1990: $3.80

Minimum Wage

Bad Presentation

Minimum Wage

0

2

4

1960 1970 1980 1990

$

Good Presentation

Graphical Errors: Compressing the Vertical Axis

Good Presentation

Quarterly Sales Quarterly Sales

Bad Presentation

0

25

50

Q1 Q2 Q3 Q4

$

0

100

200

Q1 Q2 Q3 Q4

$

Graphical Errors: No Zero Point on the Vertical Axis

Monthly Sales

36

39

42

45

J F M A M J

$

Graphing the first six months of sales

Monthly Sales

0

394245

J F M A M J

$

36

Good PresentationsBad Presentation

71

Thank You

• http://www.stats.gla.ac.uk/steps/glossary/presenting_data.html

• http://www.ilir.uiuc.edu/courses/lir593/

3_measure phase -introduction to statistics _14!1!15 [autosaved] [compatibility mode](1)

Documents

individuals variation

collection of data

figures statistics

words numbers

normal traffic

important work

normal scheme of actions

important role