Lecture 4 Spr 2011 - Revised

8/13/2019 Lecture 4 Spr 2011 - Revised

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MENG 0237

Probability and Statistics forManufacturing

Lecture 4

08/31/2011

Dr. M. Calhoun

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http://www.tuskegee.edu/global/category.asp?C=34069http://www.tuskegee.edu/global/category.asp?C=34069


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Upcoming Events

Practice problem set #1to be completed by

due Wednesday 7 September 2011

2.2, 2.6, 2.12, 2.13, 2.33, 2.39, 2.66, 2.67

Problems posted on Black Board

Career Fair Thursday 29 September 2011

Test #1 16 September 2011

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http://www.tuskegee.edu/global/category.asp?C=34069


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Chapter 2: Treatment of Data

Chapter Outline

I. Pareto Charts

II. Dot DiagramsIII. Frequency Distributions

IV. Histograms

V. Stem and Leaf Displays

VI. Descriptive MeasuresVII. Quantiles and Quartiles

VIII. Box Plots

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Pareto Charts

Pareto Chart is a series of bars whose heights reflect

the frequency or impact of problems

The categories represented by the taller bars on the

left are relatively more significant then those on the

right

These charts are based on the Paretos Principle

which states that 80 percent of the problems comefrom 20 percent of the causes

Pareto charts are used to identify those factors that

have the greatest cumulative effect on the system4


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Pareto Charts

Defects Recorded for Computer Chips

Category Frequency Percentage

Cumulative

Percentage

Holes not open 182 65.00 65.00

holes too large 55 19.64 84.64

poor connections 31 11.07 95.71

incorrect size chip 5 1.79 97.50

other 7 2.50 100.00

280 100.00 100.00

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0

20

4060

80

100

120

140

160

180

200

Data Recorded for Computer Chips

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Example

A computer controlled lathe has a below par

performance. Workers recorded the following causes

and their frequencies:

Cause Frequency

Power fluctuations 6

Unstable controller 22

Operator error 13

Worn tool not replaced 2

other 5

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Example

Using the data in the table shown previously, do

the following:

Prioritize each cause by frequency

Determine the total number of defects

Determine the percentage that each cause

contributes to the problem Develop a cumulative percentage for all of the

data

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Example

Performance Issues for Computer Controlled Lathe

Category Frequency PercentageCumulative

Percentage

unstable controller 22 45.83 45.83

operator error 13 27.08 72.92

power fluctuations 6 12.50 85.42

worn toolunreplaced 2 4.17 89.58

other 5 10.42 100.00

48 100.00 100.00

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-4 -2 0 2 4 6 8

Dot Diagrams

What observations can be made about thisdata?

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Examples of Dot Diagrams

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Dot Diagrams

Can be useful in order to expose outliers Example: In 1987, physicists observed neutrinos for the first time from

a supernova that occurred outside of our solar system. At a site in

Kamiokande, Japan, the following times between neutrinos were

recorded:

0.107 0.196 0.021 0.283 0.179 0.854 0.58 0.19 7.3 1.18 2.0

0 1 2 3 4 5 6 7 8

The corresponding dot diagram:

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Frequency Distributions

Temperatures Recorded for 30 consecutive days

50 45 49 50 43

49 50 49 45 49

47 47 44 51 51

44 47 46 50 44

51 49 43 43 49

45 46 45 51 46

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Temperatures Recorded for 30 consecutive days

51 51 51 51 50

50 50 50 49 49

49 49 49 49 47

47 47 46 46 46

45 45 45 45 44

44 44 43 43 43

First the data must be ordered

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Then the frequency with which each discrete observationcan be observed

Temperature Frequency

51 4

50 4

49 6

48 0

47 3

46 345 4

44 3

43 3

30 18


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Bar Chart

0

1

2

3

4

5

6

7

51 50 49 48 47 46 45 44 43

Temperatures Recorded Over 30 Days

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Bar Charts

Bar charts are used to describe categorical data.

Example: The manager of an auto dealership ispreparing a year-end summary of sales data. He

wants to use bar charts to display:

Number of vehicles sold by each associate

Total profit attained by each associate

A comparison between current year andprevious year profits for each associate

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Bar Charts

The sales data from the dealership for sales perperson is:

Bill Mike Nancy Sarah

# of sales 24 37 15 24

$ of sales 142,980 138,195 107,164 69,993

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Bar Charts

NancySarahBillMike

40

30

20

10

0

Sales Associate

Count

15

2424

37

Chart of Sales Associate

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Bar Charts

SarahNancyMikeBill

$160,000

$140,000

$120,000

$100,000

$80,000

$60,000

$40,000

$20,000

$0

Sales Associate

Sum

ofProfit

$69,993

$107,164

$138,195$142,980

Chart of Sum( Profit )

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Grouped Frequency Distributions

In some cases it is necessary to group the values of thedata to summarize the data properly.

Example. Student IQ scores of a class of 30 pupils rangefrom 73 to 139.

To include these scores in a frequency distribution youwould need 67 different score values (139 down to 73).This would not summarize the data very much. Insteadwe group scores together and create a grouped

frequency distribution. 24


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If your data has more than 20 score values, you should create agrouped frequency distribution by grouping score valuestogether into class intervals. To create a grouped frequencydistribution:

1. select an interval size so that you have 7-20 class intervals2. create a class interval column and list each of the class

intervals

3. each interval must be the same size, they must not overlap,there may be no gaps within the range of class intervals

4. create a midpoint column for interval midpoints5. create a frequency column

6. enter N = some value at the bottom of the frequency column

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High Temperatures for 50 Days

57 39 52 52 43

50 53 42 58 55

58 50 53 50 49

45 49 51 44 54

49 57 55 59 45

50 45 51 54 58

53 49 52 51 41

52 40 44 49 45

43 47 47 43 51

55 55 46 54 4126


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High Temperatures for 50 Days

57 39 52 52 43

50 53 42 58 55

58 50 53 50 49

45 49 51 44 54

49 57 55 59 45

50 45 51 54 58

53 49 52 51 41

52 40 44 49 45

43 47 47 43 51

55

55

46

54

4127


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Class IntervalInterval Midpoint

(Class Mark)Frequency

57-59 58 6

54-56 55 7

51-53 52 11

48-50 49 9

45-47 46 742-44 43 6

39-41 40 4

N = 50

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Are Temperatures Discrete Measurements?

Class IntervalInterval Midpoint

(Class Mark)Frequency

57-59 58 6

54-56 55 7

51-53 52 11

48-50 49 9

45-47 46 742-44 43 6

39-41 40 4

N = 50

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Convention for Denoting Endpoints for Class Intervals

Class Interval

57-59

54-56

51-53

48-50

45-47

42-4439-41

Class Interval

[57-60)

[54-57)

[51-54)

[48-51)

[45-48)

[42-45)[39-42)

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[54-57)

Included ininterval

NOT included

in interval

The interval shown above wouldinclude the minimum value of 54 upto a maximum value less than 57


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Convention for Denoting Endpoints for Class

Intervals

Class Interval

57-59

54-56

51-53

48-5045-47

42-44

39-41

Class Interval

[57-59)

[54-57)

[51-54)

[48-51)[45-48)

[42-44)

[39-42)

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Cumulative Frequency Distributions

ClassInterval

Frequency

57-59 6

54-56 7

51-53 11

48-50 9

45-47 742-44 6

39-41 4

50

ClassInterval

CumulativeFrequency

> 60 0

> 57 6

> 54 13

> 51 24

> 48 33> 45 40

> 42 46

> 39 50

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0

10

20

30

40

50

60

> 60 > 57 > 54 > 51 > 48 > 45 > 42 > 39

Cumulativefreq

uency

Intervals for emissions

Ogive for Temperatures Observed Over 50 days

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Histograms

Example: Administrators at a health clinic want to knowhow long patients wait to see a physician for annualphysicals. They suspect there might be a differencebetween wait times in the morning versus the

afternoon.

Approximately every two months, administratorsrecord the time that patients spend waiting to beseen for a physical and whether the appointmentoccurs in the morning or afternoon.

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Histograms

Found a method to showcontinuous intervals forhistograms in Minitab! Willcover this in Wednesdaystutorial.

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Histograms

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0

5

10

15

20

25

30

35

Clas

sFrequency

T, Microseconds

Interrequest Times

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Density Histograms

To remedy this, it may be desirable tohave classes of unequal lengths

In order to do this, use

=

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Density HistogramsClass Intervals Frequency Relative Frequency

[0, 2500) 9

9

2500= 0.0036

[2500, 5000) 13

13

2500= 0.0052

[5000, 10000) 1010

5000= 0.0020

[10000, 20000) 8

8

10000= 0.0008

[20000, 40000) 8

8

20000= 0.0004

[40000,60000) 1

1

20000= 0.00005

[60000, 80000) 1

1

20000= 0.00005

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Relative Frequency For Interrequest Times

0

0.001

0.002

0.003

0.004

0.005

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Density

T, microseconds

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Documents

Lecture 4 Spr 2011 - Revised