chee320

CHEE320 - Fall 2001

J. McLellan

CHEE320

Module 1: Graphical Methods for Analyzing Data, and Descriptive Statistics

CHEE320 - Fall 2001

J. McLellan 2

Graphical Methods for Analyzing Data

What is the pattern of variability?

Techniques

• histograms• dot plots• stem and leaf plots• box plots• quantile plots

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Histogram

• summary of frequency with which certain ranges of values occur

• ranges - “bins”

• choosing bin size - influences ability to recognize pattern

» too large - data clustered in a few bins - no indication of spread of data

» too small - data distributed with a few points in each bin - no indication of concentration of data

» there are quantitative rules for choosing the number of bins - typically automated in statistical software

• not automated in Excel!

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Histogram (lco90.STA 1v*768c)

LCO90

No

of

ob

s

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

<= 630(630,640]

(640,650](650,660]

(660,670](670,680]

(680,690](690,700]

(700,710](710,720]

(720,730]> 730

Histogram - Important Features

number of peaks

centre of gravity

spread in the data

tails? - extreme data points

max, min data values- range of values

symmetry?

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Dot Plots

• similar to histogram » plot data by value on horizontal axis» stack repeated values vertically» look for similar shape features as for histogram» e.g., data set for solder thickness» {0.1, 0.12, 0.09, 0.07, 0.09, 0.11, 0.1, 0.13, 0.1}

0.06 0.09 0.1 0.11 0.120.07 0.08 0.13

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Stem and Leaf Plots

• illustrate variability pattern using the numerical data itself• choose base division - “stem” • build “leaves” by taking digit next to base division

Decimal point is 1 place to the right of the colon

1 : 0124 1 : 58888 2 : 001112 2 : 557 3 : 3 : 6

12.0010.0014.0020.0018.0018.0025.0021.0036.0044.00

11.0015.0022.0021.0027.0025.0018.0021.0018.0020.00

Data

ToothDiscolorationby Fluoride

10-14

15-19

20-24

25-29

Stems

Leaves

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Stem and Leaf Plots

Solder example» numbers viewed as 0.070, 0.080, 0.090, 0.100, 0.110,…» decision - what is the stem?

• considerations similar to histogram - size of bins

Decimal point is 2 places to the left of the colon

7 : 0 8 : 9 : 00 10 : 000 11 : 0 12 : 0 13 : 0

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Box Plots

• graphical representation of “quartile” information» quartiles - describe how data occurs - ordering» 1st quartile - separates bottom 25% of data » 2nd quartile (median) - separates bottom 50% of data» 3rd quartile - separates bottom 75% of data

and extreme data values» add “whiskers” - extend from box to largest data point within

• upper quartile + 1.5 * interquartile range• lower quartile - 1.5 * interquartile range

» interquartile range = Q3 - Q1» plot outliers - data points outside Q3 + 1.5*IQR,

Q1-1.5*IQR

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Box Plot - for solder data

Non-Outlier MaxNon-Outlier Min

75%25%

Median

Box Plot (jsolder.STA 10v*10c)

TH

ICK

NE

S

THICKNES: 0

0.065

0.075

0.085

0.095

0.105

0.115

0.125

0.135

THICKNES

Interpretation• no outliers• relatively symmetric

distribution• longer tails on both sides• fairly tightly clustered

about centre

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Non-Outlier MaxNon-Outlier Min

75%25%

Median

Box Plot (teethdisc.STA 10v*20c)

DIS

CO

LO

R

VAR2: 1

8

12

16

20

24

28

32

DISCOLOR

Box Plot - for teeth discoloration

Interpretation• no outliers• asymmetric distribution -

long lower tail• some tails on both sides• fairly tightly clustered at

higher range of discoloration

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Quantile Plots

• plot cumulative progression of data» values vs. cumulative fraction of data » comparison to standard distribution shapes

• e.g., normal distribution, lognormal distribution, …

» can be plotted on special axes • analogous to semi-log graphs

to provide visual test for closeness to given distribution• e.g., test to see if data are normally distributed

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Quantile Plot - teeth discolorationQuantile-Quantile Plot of DISCOLOR (teethdisc.STA 10v*20c)

Distribution: Normal

y=20.798+7.916*x+eps

Theoretical Quantile

Ob

serv

ed

Va

lue

.01 .05 .1 .25 .5 .75 .9 .95 .99

5

10

15

20

25

30

35

40

45

50

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

Interpretation• data don’t follow linear

progression– underlying distribution

not normal?Note the irregular spacing - similar to “semi-log” paper - cumulative points shouldfollow linear progression on this scale if distribution is normal.

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Graphical Methods for Quality Investigations

• primary purpose - help organize information in quality investigation

Examples • Pareto Charts• Fishbone diagrams - Ishikawa diagrams

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Pareto Chart

• used to rank factors • typically present as a bar chart, listing in descending

order of significance• significance can be determined by

» number count - e.g., of defects attributed to specific causes

» by size of effect - e.g., based on coefficients in regression model

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Example - Circuit DefectsNumber of Defects Attributed to:Stamping_Oper_ID1Stamping_Missing 1Sold._Short 1Wire_Incorrect 1Raw_Cd_Damaged 1Comp._Extra_Part 2Comp._Missing 2Comp._Damaged 2TST_Mark_White_Mark 3Tst._Mark_EC_Mark 3Raw_CD_Shroud_Re. 3Sold._Splatter 5Comp._Improper_16Sold._Opens 7Sold._Cold_Joint 20Sold._Insufficient 40

Data fromMontgomery

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Pareto Chart

• for circuit defect data

Pareto Chart & Analysis; NO_DEFCT

40

20

7 6 5 3 3 3 2 2 2 1 1 1 1 10%

20%

40%

60%

80%

100%

0

20

40

60

80

100

Sol

d._I

nsuf

ficie

nt

Sol

d._C

old_

Join

t

Sol

d._O

pens

Com

p._I

mpr

oper

_1

Sol

d._S

plat

ter

TS

T_M

ark_

Whi

te_M

ark

Tst.

_Mar

k_E

C_M

ark

Raw

_CD

_Shr

oud_

Re.

Com

p._D

amag

ed

Com

p._E

xtra

_Par

t

Com

p._M

issi

ng

Sol

d._S

hort

Wir

e_In

corr

ect

Raw

_Cd_

Dam

aged

Sta

mpi

ng_O

per_

ID

Sta

mpi

ng_M

issi

ng

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Fishbone Diagrams

• organize causes in analysis» have spine, with cause types branching from spine, and

sub-groups branching further

Example - factors influencing poor conversion in reactive extrusion

poor conversion

initiator type

half-life

barrel temperature

temperaturedistribution along barreltemperature

control

polymer grade

catalyst used- metallocene/Ziegler-Natta

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Looking for time trends in data...

• Time sequence plot– look for

» jumps» ramps to new values» meandering - indicates time correlation in data» large amount of variation about general trend - indication

of large variance

} indicate shift in mean operation

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Time Sequence Plot - Naphtha 90% Point

90

% p

oin

t (d

eg

ree

s F

)

390

400

410

420

430

440

450

460

470

480

0 30 60 90 120 150 180 210 240 270

Time Sequence Plot

- for naphtha 90% point - indicates amount of heavy hydrocarbons present in gasoline range material

excursion - suddenshift in operation

meandering about average operating point- time correlation in data

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Monitoring process operation• Quality Control Charts

– time sequence plots with added indications of variation» account for fluctuations in values associated with natural

process noise» look for significant jumps - shifts - that exceed normal

range of variation of values» if significant shift occurs, stop and look for “assignable

causes”» essentially graphical “hypothesis tests”» can plot - measurements, sample averages, ranges,

standard deviations, ...

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Example - Monitoring Process Mean

• is the average process operation constant?• collect samples at time intervals, compute average, and

plot in time sequence plot• indication of process variation - standard deviation

estimated from prior data» propagates through sample average calculation» if “s” is sample standard deviation, calculated averages will

lie between of the historical average 99% of the time if the mean operation has NOT shifted

» values outside this range suggest that a shift in the mean operation has occurred - alarm - “something has happened”

ns /3

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• time sequence plot with these alarm limits is referred to as a “Shewhart X-bar Chart”

» X-bar - sample mean of XXX-BAR Mean:74.0012 (74.0012) Proc. sigma:.009785 (.009785) n:5

Samples

73.9880

74.0012

74.0143

1 5 10 15 20 25

upper andlower controllimits

centre-lineor targetline - indicatesmean when process is operating properly

no points exceed limits in a stateof statistical control

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X-BAR Mean:74.0022 (74.0022) Proc. sigma:.011832 (.011832) n:5

Samples

73.9864

74.0022

74.0181

1 5 10 15 20 25

• X-bar chartPoint exceedsregion of natural variation- significant shift has occurred

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Visualizing relationships between variables

Techniques

• scatterplots• scatterplot matrices

» also referred to as “casement plots”

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Scatterplots

,,, are also referred to as “x-y diagrams”• plot values of one variable against another • look for systematic trend in data

» nature of trend• linear?

• exponential?

• quadratic?

» degree of scatter - does spread increase/decrease over range?

• indication that variance isn’t constant over range of data

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Scatterplots - Example

Scatterplot (teeth 4v*20c)

FLUORIDE

DIS

CO

LO

R

5

10

15

20

25

30

35

40

45

50

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

trend - possiblynonlinear?

• tooth discoloration data - discoloration vs. fluoride

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Scatterplot - Example


BRUSHING

DIS

CO

LO

R

5

10

15

20

25

30

35

40

45

50

4 5 6 7 8 9 10 11 12 13

• tooth discoloration data -discoloration vs. brushing

signficant trend?- doesn’t appear tobe present

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Scatterplot - Example


BRUSHING

DIS

CO

LO

R

5

10

15

20

25

30

35

40

45

50

4 5 6 7 8 9 10 11 12 13

Variance appearsto decrease as # of brushings increases

• tooth discoloration data -discoloration vs. brushing

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Scatterplot matrices

… are a table of scatterplots for a set of variables

Look for - » systematic trend between “independent” variable and

dependent variables - to be described by estimated model

» systematic trend between supposedly independent variables - indicates that these quantities are correlated

• correlation can negatively ifluence model estimation results

• not independent information

• scatterplot matrices can be generated automatically with statistical software, manually using Excel

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Scatterplot Matrices - tooth data

Matrix Plot (teeth 4v*20c)

FLUORIDE

AGE

BRUSHING

DISCOLOR

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Describing Data Quantitatively

Approach - describe the pattern of variability using a few parameters

» efficient means of summarizing

Techniques• average - (sample “mean”)• sample standard deviation and variance• median• quartiles• interquartile range• ...

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Sample Mean - “Average”

Given “n” observations xi :

Notes - » sensitive to extreme data values - outliers - value can be

artificially raised or lowered

n

iixn

x1

1

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Sample Variance

• sum of squared deviations about the average» squaring - notion of distance (squared)» average - is the centre of gravity

• sample variance provides a measure of dispersion - spread - about the centre of gravity

n

ii xx

ns

1

22 )(1

1

Note that we divideby “n-1”, and NOT“n” - degrees of freedom argument

Note - there is an alternativeform of this equation whichis more convenient forcomputation.

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Sample Standard Deviation

… is simply

• sample standard deviation provides a more direct link to dispersion

» e.g., for Normal distribution• 95% of values lie within 2 standard devn’s of the mean

• 99% of values like within 3 standard devn’s of the mean

2ss

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Range

• provides a measure of spread in the data• defined as

maximum data value - minimum data value

• can be sensitive to extreme data points• is often monitored in quality control charts to see if

process variance is changing

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“Order” Statistics

… summarize the progression of observations in the data set

Quartiles » divide the data in quarters

Deciles» divide the data in tenths ...

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Quartiles

• order data - N data points {yi}, i=1,…N

• if N is odd, » median is observation

• if N is even,» median is

• i.e., midpoint between two middle points

2/)1( Ny

2

122

NN yy

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Quartiles - Q1 and Q3

• Q1: Compute (N+1)/4 = A.B

• Q3: Compute 3(N+1)/4 = A.B

» i.e., interpolate between adjacent points» Note - there are other conventions as well - e.g., for Q1,

take bottom half of data set, and take midpoint between middle two points if there are an even number of points...

)(*1 1 AAA yyByQ

)(*3 1 AAA yyByQ

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Quartiles - Example

• solder data set» observations» 0.1, 0.12, 0.09, 0.07, 0.09, 0.11, 0.1, 0.13, 0.1» ordered: 0.07, 0.09, 0.09, 0.1, 0.1, 0.1, 0.11, 0.12, 0.13» 9 points --> median is 5th observation: 0.1» Q1: (N+1)/4 = 2.5

• Q1 = 0.09+0.5*(0.09-0.09) = 0.9

» Q3: 3(N+1)/4 = 7.5• Q3 = 0.11 + 0.5*(0.12-0.11) = 0.115

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Robustness

… refers to whether a given descriptive statistic is sensitive to extreme data points

Examples

• sample mean» is sensitive to extreme points - extreme value pulls

average toward the extreme

• sample variance » sensitive to extreme points - large deviation from the

sample mean leads to inflated variance

• median, quartiles » relatively insensitive to extreme data points

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Robustness -Solder Data Example

• replace 0.13 by 0.5 - output from Excel

thickness

Mean 0.142222Median 0.1Mode 0.1Standard Deviation 0.134887Sample Variance 0.018194Range 0.43Minimum 0.07Maximum 0.5

thickness

Mean 0.101111Median 0.1Mode 0.1Standard Deviation 0.017638Sample Variance 0.000311Range 0.06Minimum 0.07Maximum 0.13

With 0.13 With 0.5

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Robustness

• Other robust statistics» “m-estimator” - involves iterative filtering out of extreme

data values, based on data distribution» trimmed mean - other bases for eliminating extreme data

point effect» median absolute deviation

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