1 statistical analysis - graphical techniques dr. jerrell t. stracener, sae fellow leadership in...

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Statistical Analysis - Graphical Techniques

Dr. Jerrell T. Stracener, SAE Fellow

Leadership in Engineering

EMIS 7370/5370 STAT 5340 :

PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS

Systems Engineering Program

Department of Engineering Management, Information and Systems

Stracener_EMIS 7370/STAT 5340_Sum 08_07.03.08

1.psd

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Time Series Graph or Run Chart

Box Plot

Histogram and Relative Frequency Histogram

Frequency Distribution

Probability Plotting

*

A plot of the data set x1, x2, , xn in the order

in which the data were obtained

Used to detect trends or patterns in the data

over time

Time Series Graph or Run Chart

*

A pictorial summary used to describe the

most prominent statistical features of the data

set, x1, x2, , xn, including its:

- Center or location

- Spread or variability

- Extent and nature of any deviation from symmetry

- Identification of outliers

Box Plot

*

Shows only certain statistics rather than all the

data, namely

- median

- quartiles

- smallest and greatest values in the sample

Immediate visuals of a box plot are the center,

the spread, and the overall range of the data

Box Plot

*

Given the following random sample of size 25:

38, 10, 60, 90, 88, 96, 1, 41, 86, 14, 25, 5, 16,

22, 29, 34, 55, 36, 37, 36, 91, 47, 43, 30, 98

Arranged in order from least to greatest:

1, 5, 10, 14, 16, 22, 25, 29, 30, 34, 36, 36, 37,

38, 41, 43, 47, 55, 60, 86, 88, 90, 91, 96, 98

Box Plot

*

First, find the median, the value exactly in the

middle of an ordered set of numbers.

The median is 37

Next, we consider only the values to the left of

the median:

1, 5, 10, 14, 16, 22, 25, 29, 30, 34, 36, 36

We now find the median of this set of numbers.

The median for this group is (22 + 25)/2 = 23.5,

which is the lower quartile.

Box Plot

*

Now consider the values to the right of the

median.

38, 41, 43, 47, 55, 60, 86, 88, 90, 91, 96, 98

The median for this set is (60 + 86)/2 = 73, which

is the upper quartile.

We are now ready to find the interquartile range

(IQR), which is the difference between the upper

and lower quartiles, 73 - 23.5 = 49.5

49.5 is the interquartile range

Box Plot

*

The lower quartile 23.5

The median is 37

The upper quartile 73

The interquartile range is 49.5

The mean is 45.1

upper

quartile

Box Plot

0

10

20

30

40

50

60

70

80

90

100

lower

extreme

upper

extreme

lower

quartile

median

mean

*

A graph of the observed frequencies in the data

set, x1, x2, , xn versus data magnitude to

visually indicate its statistical properties, including

- shape

- location or central tendency

- scatter or variability

Histogram

Guidelines for Constructing Histograms Discrete Data

*

If the data x1, x2, , xn are from a discrete

random variable with possible values y1, y2, , yk

count the number of occurrences of each value

of y and associate the frequency fi with yi,

for i = 1, , k,

Note that


*

If the data x1, x2, , xn are from a continuous

random variable

- select the number of intervals or cells, r,

to be a number between 3 and 20, as an

initial value use r = (n)1/2, where n is the

number of observations

- establish r intervals of equal width, starting

just below the smallest value of x

- count the number of values of x within

each interval to obtain the frequency

associated with each interval

- construct graph by plotting (fi, i) for

i = 1, 2, , k


*

To illustrate the construction of a relative frequency distribution,

consider the following data which represent the lives of 40 carbatteries of a given type recorded to the nearest tenth of a year.The batteries were guaranteed to last 3 years.

Histogram and Relative Frequency Example

Sheet1

Car Battery Lives

2.24.13.54.53.23.732.6

3.41.63.13.33.83.14.73.7

2.54.33.43.62.93.33.93.1

3.33.13.74.43.24.11.93.4

4.73.83.22.63.934.23.5

Sheet2

Sheet3

*

For this example, using the guidelines for constructing a histogram,

the number of classes selected is 7 with a class width of 0.5. The

frequency and relative frequency distribution for the data are shown

in the following table.

Histogram and Relative Frequency Example

Sheet1

Car Battery LivesRelative Frequency Distribution of

2.24.13.54.53.23.732.6Battery Lives

3.41.63.13.33.83.14.73.7ClassClassFrequencyRelative

2.54.33.43.62.93.33.93.1intervalmidpointffrequency

3.33.13.74.43.24.11.93.41.5-1.91.720.050

4.73.83.22.63.934.23.52.0-2.42.210.025

2.5-2.92.740.100

3.0-3.43.2150.375

3.5-3.93.7100.250

4.0-4.44.250.125

4.5-4.94.730.075

Total401.000

Sheet2

Sheet3

*

The following diagram is a relative frequency histogram of the battery

lives with an approximate estimate of the probability density function

superimposed.

Histogram and Relative Frequency

Chart1

1.7

2.2

2.7

3.2

3.7

4.2

4.7

Battery Lives (years)

Relative Frequency

Relative frequency histogram

0.05

0.025

0.1

0.375

0.25

0.125

0.075

Sheet1

Car Battery LivesRelative Frequency Distribution of

2.24.13.54.53.23.732.6Battery Lives

3.41.63.13.33.83.14.73.7ClassClassFrequencyRelative

2.54.33.43.62.93.33.93.1intervalmidpointffrequency

3.33.13.74.43.24.11.93.41.5-1.91.720.050

4.73.83.22.63.934.23.52.0-2.42.210.025

2.5-2.92.740.100

3.0-3.43.2150.375

3.5-3.93.7100.250

4.0-4.44.250.125

4.5-4.94.730.075

Sheet2

Sheet3

*

Data are plotted on special graph paper

designed for a particular distribution

- Normal- Weibull

- Lognormal- Exponential

If the assumed model is adequate, the plotted

points will tend to fall in a straight line

If the model is inadequate, the plot will not

be linear and the type & extent of departures

can be seen

Once a model appears to fit the data

reasonably will, percentiles and parameters can

be estimated from the plot

Probability Plotting

*

Step 1: Obtain special graph paper, known asprobability paper, designed for the distribution under

examination. Weibull, Lognormal and Normal paper

are available at:

http://www.weibull.com/GPaper/index.htm

Step 2: Rank the sample values from smallest

to largest in magnitude i.e., X1 X2 ..., Xn.

Probability Plotting Procedure

*

Step 3:

Plot the Xis on the paper versus or

, depending on whether the marked axis

on the paper refers to the % or the proportion

of observations. The axis of the graph paper on

which the Xis are plotted will be referred to as

the observational scale, and the axis for

as the cumulative scale.

Step 4: If a straight line appears to fit the data,

draw a line on the graph, by eye.

Step 5: Estimate the model parameters from

the graph.

Probability Plotting General Procedure

*

If

the cumulative probability distribution function isWe now need to linearize this function into the form

y = ax +b

Weibull Probability Plotting Paper

*

Then

which is the equation of a straight line of the form

y = ax +b


*

where

and


*

which is a linear equation with a slope of b and an intercept of . Now the x- and y-axes of the Weibull probability plotting paper can be constructed. The x-axis is simply logarithmic, since x = ln(T) and


*

cumulative

probability(in %)

x


*

To illustrate the process let 10, 20, 30, 40, 50, and 80 be a random sample of size n = 6.

Probability Plotting - Example

*

We need value estimates corresponding to each of the sample values in order to plot the data on the Weibull probability paper. These estimates are accomplished with what are called median ranks.


*

Median ranks represent the 50% confidence level (best guess) estimate for the true value of F(t), based on the total sample size and the order number (first, second, etc.) of the data.


*

There is an approximation that can be used to estimate median ranks, called Benards approximation. It has the form:

where n is the sample size and i is the sample order number. Tables of median ranks can be found in may statistics and reliability texts.


*

Based on Benards approximation, we can now calculate F(t) for each observed value of X. These are shown in the following table:

For example, for x2=20,

^

^


Sheet1

i

11010.9%

22026.6%

33042.2%

44057.8%

55073.4%

68089.1%

Sheet2

Sheet3

Sheet1

ixiF(xi)

11010.9%

22026.6%

33042.2%

44057.8%

55073.4%

68089.1%

Sheet2

Sheet3

*

cumulativeprobability

(in %)

x


*

Now that we have y-coordinate values to go with the x-coordinate sample values so we can plot the

points on Weibull probability paper.

F(x)(in %)

x

^


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The line represents the estimated relationship between x and F(x):

x


F(x)(in %)

^

*

In this example, the points on Weibull probability paper fall in a fairly linear fashion, indicating that the Weibull distribution provides a good fit to the data. If the points did not seem to follow a straight line, we might want to consider using another probability distribution to analyze the data.


*


*


*

Probability Paper - Normal

*

Probability Paper - Lognormal

*

Probability Paper - Exponential

*

Given the following random sample of size n=8, which probability distribution provides the best fit?

Example - Probability Plotting

Sheet5

ixi

179.4096765982

288.1209305386

391.0639417067

498.7309365679

5104.1536168283

6105.1019

7106.5036374508

8112.035434338

Sheet4

Sheet1

Sheet2

Sheet3

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40 specimens are cut from a plate for tensile tests. The tensile tests were made, resulting in Tensile Strength, x, as follows:

Perform a statistical analysis of the tensile strength data.

40 Specimens

Sheet1

ixixixix

148.51155.02153.13154.6

254.71255.72249.13249.9

347.81349.92355.63344.5

456.91454.82446.23452.9

554.81549.72552.03554.4

657.91658.92656.63660.2

744.91752.72752.93750.2

853.01857.82852.23857.4

954.71946.82954.13954.8

1046.72049.23042.34061.2

Sheet2

Sheet3

*

Time Series plot:

By visual inspection of the scatter plot, there seems to be no trend.

40 Specimens

Chart1

52

53.9

50.9

50.1

54.1

52.8

49.2

54

53.2

53.6

54.2

53.4

52.6

51.6

50.2

52.8

52.2

52.8

53.1

52.2

54

52

52.2

51.8

50.2

51.6

51.8

51.7

51.9

52.1

51.6

51.5

52.4

53.4

50.5

51.6

51.2

52

50.6

49.6

Chart2

48.4988205647

54.7372948352

47.8297053126

56.8690708354

54.8171825761

57.8673435888

44.9246541797

53.0099142855

54.74485501

46.6513185023

55.0364617487

55.7413428799

49.9336109845

54.8472697977

49.6839092142

58.8838289735

52.7233211363

57.7531451603

46.7783749121

49.2002797222

53.0525013749

49.0763781221

55.6090398225

46.1983901408

52.0162572178

56.5687215788

52.8689767128

52.2033516466

54.1127220862

42.3493517195

54.6492955385

49.9327753863

44.5385736788

52.8546180652

54.4428857159

60.2144424596

50.2023496159

57.3826965996

54.8274996566

61.1844867711

Sheet1

48.5

54.7

47.8

56.9

54.8

57.9

44.9

53.0

54.7

46.7

55.0

55.7

49.9

54.8

49.7

58.9

52.7

57.8

46.8

49.2

53.1

49.1

55.6

46.2

52.0

56.6

52.9

52.2

54.1

42.3

54.6

49.9

44.5

52.9

54.4

60.2

50.2

57.4

54.8

61.2

Sheet2

Sheet3

*

40 Specimens

Using the descriptive statistics function in Excel, the following were calculated:

Sheet4

Descriptive Statistics

Count40

Minimum42.35

Maximum61.18

Range18.84

Sum2104.82

Mean52.62

Median53.03

Sample Variance19.83

Standard Deviation4.45

Kurtosis2.51

Skewness-0.34

Sheet1

48.5-4.116.987626856876-70.016285897250288.579466228453

54.72.14.4811359804809.48597955368920.080579675551

47.852.6-4.822.950999465460-109.951815385863526.748376463549

56.94.218.05098758349176.692245040325325.838152739336

54.82.24.82574185689910.60098279729523.287784469425

57.95.227.530149920915144.448442893810757.909154668077

44.9-7.759.224895495179-455.7812494322673507.588246414870

53.00.40.1517015504840.0590860597560.023013360419

54.72.14.5132009414489.58797747540920.368982737883

46.7-6.035.630237791792-212.6807000667251269.513845099640

55.02.45.83723138103714.10296288165134.073270195759

55.73.19.74012558436330.39812767461394.870046399155

49.9-2.77.218971989155-19.39603838952052.113556580202

54.82.24.95883573453011.04255531063124.590051842056

49.7-2.98.623127622266-25.32195429427474.358329989881

58.96.339.230223657845245.7147209598801539.010448244510

52.70.10.0105875215980.0010894102670.000112095614

57.85.126.344811590985135.220533631175694.049097764518

46.8-5.834.129554524825-199.3865824794261164.826492063000

49.2-3.411.697396821757-40.006801804565136.829092405648

53.10.40.1866896021200.0806640819090.034853007540

49.1-3.512.560271485509-44.514196636877157.760419789698

55.63.08.93181584847526.69375318882479.777334351062

46.2-6.441.242537554041-264.8610325460901700.946903896490

52.0-0.60.365019256716-0.2205330402320.133239057773

56.63.915.58904229542761.550155296847243.018239688613

52.90.20.0617777286360.0153549322660.003816487755

52.2-0.40.173950560154-0.0725502222840.030258797378

54.11.52.2269492406613.3232688535494.959302920481

42.3-10.3105.494955644077-1083.54646795874011129.185666345800

54.62.04.1163138228998.35146597685816.944039488588

49.9-2.77.223462882354-19.41414049934252.178416012753

44.5-8.165.316328104624-527.8768821946994266.222717070830

52.90.20.0548461795010.0128445700340.003008103406

54.41.83.3213614091266.05304918620211.031441610032

60.27.657.669094292588437.9400827981623325.724436527380

50.2-2.45.847090755130-14.13870887371634.188470298723

57.44.822.679226471064108.004625843982514.347313325788

54.82.24.87117653930310.75104808009023.728360877053

61.28.673.343146256475628.1152017869815379.217102798720

19.335314995007-27.723493035941937.352335997336

skew-0.326077679

kurt2.5072657191

Sheet2

Sheet3

*

40 Specimens

From looking at the Histogram and the Normal Probability Plot, we see that the tensile strength can be estimated by a normal distribution.

Using the histogram feature of excel the following data was calculated:

and the graph:

Sheet4


Count40

Sum2104.82

Mean52.62

Standard Error0.70

Median53.03



Kurtosis-0.39

Skewness-0.34

Range18.84

Minimum42.35

Maximum61.18

Sheet5

BinFrequency

42.34935171951

45.48854089482

48.627730075

51.76691924536

54.906108420615

58.04529759588

More3

Sheet6

BinFrequency

400

453

5010

5516

609

More2

Sheet1

48.540

54.745

47.850

56.955

54.860

57.9

44.9

53.0

54.7

46.7

55.0

55.7

49.9

54.8

49.7

58.9

52.7

57.8

46.8

49.2

53.1

49.1

55.6

46.2

52.0

56.6

52.9

52.2

54.1

42.3

54.6

49.9

44.5

52.9

54.4

60.2

50.2

57.4

54.8

61.2

Sheet2

Sheet3

Chart1

40

45

50

55

60

More

Histogram of Tensile Strengths

0

3

10

16

9

2

Sheet4


Count40

Sum2104.82

Mean52.62

Standard Error0.70

Median53.03



Kurtosis-0.39

Skewness-0.34

Range18.84

Minimum42.35

Maximum61.18

Sheet5

BinFrequency

42.34935171951

45.48854089482

48.627730075

51.76691924536

54.906108420615

58.04529759588

More3

Sheet6

BinFrequency

400

453

5010

5516

609

More2

Sheet6


Sheet1

48.540

54.745

47.850

56.955

54.860

57.9

44.9

53.0

54.7

46.7

55.0

55.7

49.9

54.8

49.7

58.9

52.7

57.8

46.8

49.2

53.1

49.1

55.6

46.2

52.0

56.6

52.9

52.2

54.1

42.3

54.6

49.9

44.5

52.9

54.4

60.2

50.2

57.4

54.8

61.2

Sheet2

Sheet3

*

40 Specimens

Box Plot

The lower quartile 49.45

The median is 53.03

The mean 52.6

The upper quartile 55.3

The interquartile range is 5.86

40

45

50

55

60

65

lower

extreme

upper

extreme

lower

quartile

upper

quartile

median

mean

*

40 Specimens

Chart1

4042.349351719542.3493517195

4044.538573678861.1844863892

4044.9246541797

4046.1983901408

4046.6513185023

4046.7783749121

4047.8297053126

4048.4988205647

4049.0763781221

4049.2002797222

4049.6839092142

4049.9327753863

4049.9336109845

4050.2023496159

4052.0162572178

52.2033516466

52.7233211363

52.8546180652

52.8689767128

53.0099142855

53.0525013749

54.1127220862

54.4428857159

54.6492955385

54.7372948352

54.74485501

54.8171825761

54.8274996566

54.8472697977

55.0364617487

55.6090398225

55.7413428799

56.5687215788

56.8690708354

57.3826965996

57.7531451603

57.8673435888

58.8838289735

60.2144424596

61.1844867711

Normal Probability Plot

0.10%

1%

5%

10%

20%

30%

40%

50%

60%

70%

80%

90%

95%

99%

99.90%

12.9170633852

23.103258526

22.6964484019

22.0838968642

28.6344415043

72.7658067418

30.2617639943

31.5905357536

34.6213904675

33.7242648937

39.900543369

35.4399356991

43.7071879685

36.9005478104

46.9598411553

38.1891835685

50

39.3542462814

53.0401588447

40.426676959

56.2928120315

41.4275940682

60.099456631

42.372113411

65.3786095325

43.2715581963

69.7382360057

44.1346868582

77.9161031358

44.9684570311

87.0829366148

45.7785848892

46.5699134918

47.3465764924

48.1122709869

48.8703530107

49.6239466782

50.3760533218

51.1296469893

51.8877290131

52.6534235076

53.4300865082

54.2214151108

55.0315429689

55.8653131418

56.7284418037

57.627886589

58.5724059318

59.573323041

60.6457537186

61.8108164315

63.0994521896

64.5600643009

66.2757351063

68.4094642464

71.3655584957

76.896741474

Sheet1

48.5Standard Deviation4.6282284525

54.7Mean52.620426178

47.8

56.9

54.8

57.9

44.9

53.0

54.7

46.7

55.0

55.7

49.9

54.8

49.7

58.9

52.7

57.8

46.8

49.2

53.1

49.1

55.6

46.2

52.0

56.6

52.9

52.2

54.1

42.3

54.6

49.9

44.5

52.9

54.4

60.2

50.2

57.4

54.8

61.2

Sheet1

4042.349351719542.3493517195

4044.538573678861.1844863892

4044.9246541797

4046.1983901408

4046.6513185023

4046.7783749121

4047.8297053126

4048.4988205647

4049.0763781221

4049.2002797222

4049.6839092142

4049.9327753863

4049.9336109845

4050.2023496159

4052.0162572178

52.2033516466

52.7233211363

52.8546180652

52.8689767128

53.0099142855

53.0525013749

54.1127220862

54.4428857159

54.6492955385

54.7372948352

54.74485501

54.8171825761

54.8274996566

54.8472697977

55.0364617487

55.6090398225

55.7413428799

56.5687215788

56.8690708354

57.3826965996

57.7531451603

57.8673435888

58.8838289735

60.2144424596

61.1844867711


99.90%

99%

95%

90%

80%

70%

60%

50%

40%

30%

20%

10%

5%

1%

0.10%

12.9170633852

23.103258526

22.6964484019

22.0838968642

28.6344415043

72.7658067418

30.2617639943

31.5905357536

34.6213904675

33.7242648937

39.900543369

35.4399356991

43.7071879685

36.9005478104

46.9598411553

38.1891835685

50

39.3542462814

53.0401588447

40.426676959

56.2928120315

41.4275940682

60.099456631

42.372113411

65.3786095325

43.2715581963

69.7382360057

44.1346868582

77.9161031358

44.9684570311

87.0829366148

45.7785848892

46.5699134918

47.3465764924

48.1122709869

48.8703530107

49.6239466782

50.3760533218

51.1296469893

51.8877290131

52.6534235076

53.4300865082

54.2214151108

55.0315429689

55.8653131418

56.7284418037

57.627886589

58.5724059318

59.573323041

60.6457537186

61.8108164315

63.0994521896

64.5600643009

66.2757351063

68.4094642464

71.3655584957

76.896741474

Sheet2

4012.917063385242.323.10325852622.696448401942.3493517195

4022.083896864244.528.634441504372.765806741861.1844863892

4030.261763994344.931.5905357536

4034.621390467546.233.7242648937

4039.90054336946.735.4399356991

4043.707187968546.836.9005478104

4046.959841155347.838.1891835685

405048.539.3542462814

4053.040158844749.140.426676959

4056.292812031549.241.4275940682

4060.09945663149.742.372113411

4065.378609532549.943.2715581963

4069.738236005749.944.1346868582

4077.916103135850.244.9684570311

4087.082936614852.045.7785848892

52.246.5699134918

52.747.3465764924

52.948.1122709869

52.948.8703530107

53.049.6239466782

53.150.3760533218

54.151.1296469893

54.451.8877290131

54.652.6534235076

54.753.4300865082

54.754.2214151108

54.855.0315429689

54.855.8653131418

54.856.7284418037

55.057.627886589

55.658.5724059318

55.759.573323041

56.660.6457537186

56.961.8108164315

57.463.0994521896

57.864.5600643009

57.966.2757351063

58.968.4094642464

60.271.3655584957

61.276.896741474

182.324100554160.5347589852

182.916556771561.4499744086

183.528555756362.4371945276

184.087656255663.516691979

184.206856904964.7183436638

184.813774088166.0890522238

185.097331798867.709498934

185.890979612969.7382360057

186.780670067572.5694748224

196.20799192477.9161031358

Sheet3

*

40 Specimens

Chart1

1042.3493517195

1044.5385736788

1044.9246541797

1046.1983901408

1046.6513185023

1046.7783749121

1047.8297053126

1048.4988205647

1049.0763781221

1049.2002797222

1049.6839092142

1049.9327753863

1049.9336109845

1050.2023496159

1052.0162572178

52.2033516466

52.7233211363

52.8546180652

52.8689767128

53.0099142855

53.0525013749

54.1127220862

54.4428857159

54.6492955385

54.7372948352

54.74485501

54.8171825761

54.8274996566

54.8472697977

55.0364617487

55.6090398225

55.7413428799

56.5687215788

56.8690708354

57.3826965996

57.7531451603

57.8673435888

58.8838289735

60.2144424596

61.1844867711

LogNormal Probability Plot

0.10%

1%

5%

10%

20%

30%

40%

50%

60%

70%

80%

90%

95%

99%

99.90%

12.9170633852

23.103258526

22.0838968642

28.6344415043

30.2617639943

31.5905357536

34.6213904675

33.7242648937

39.900543369

35.4399356991

43.7071879685

36.9005478104

46.9598411553

38.1891835685

50

39.3542462814

53.0401588447

40.426676959

56.2928120315

41.4275940682

60.099456631

42.372113411

65.3786095325

43.2715581963

69.7382360057

44.1346868582

77.9161031358

44.9684570311

87.0829366148

45.7785848892

46.5699134918

47.3465764924

48.1122709869

48.8703530107

49.6239466782

50.3760533218

51.1296469893

51.8877290131

52.6534235076

53.4300865082

54.2214151108

55.0315429689

55.8653131418

56.7284418037

57.627886589

58.5724059318

59.573323041

60.6457537186

61.8108164315

63.0994521896

64.5600643009

66.2757351063

68.4094642464

71.3655584957

76.896741474

Sheet1

48.5Standard Deviation4.7453095531

54.7Mean52.4394683838

47.8

56.9

54.8

57.9

44.9

53.0

54.7

46.7

55.0

55.7

49.9

54.8

49.7

58.9

52.7

57.8

46.8

49.2

53.1

49.1

55.6

46.2

52.0

56.6

52.9

52.2

54.1

42.3

54.6

49.9

44.5

52.9

54.4

60.2

50.2

57.4

54.8

61.2

Sheet1

1042.3493517195

1044.5385736788

1044.9246541797

1046.1983901408

1046.6513185023

1046.7783749121

1047.8297053126

1048.4988205647

1049.0763781221

1049.2002797222

1049.6839092142

1049.9327753863

1049.9336109845

1050.2023496159

1052.0162572178

52.2033516466

52.7233211363

52.8546180652

52.8689767128

53.0099142855

53.0525013749

54.1127220862

54.4428857159

54.6492955385

54.7372948352

54.74485501

54.8171825761

54.8274996566

54.8472697977

55.0364617487

55.6090398225

55.7413428799

56.5687215788

56.8690708354

57.3826965996

57.7531451603

57.8673435888

58.8838289735

60.2144424596

61.1844867711


99.90%

99%

95%

90%

80%

70%

60%

50%

40%

30%

20%

10%

5%

1%

0.10%

12.9170633852

23.103258526

22.0838968642

28.6344415043

30.2617639943

31.5905357536

34.6213904675

33.7242648937

39.900543369

35.4399356991

43.7071879685

36.9005478104

46.9598411553

38.1891835685

50

39.3542462814

53.0401588447

40.426676959

56.2928120315

41.4275940682

60.099456631

42.372113411

65.3786095325

43.2715581963

69.7382360057

44.1346868582

77.9161031358

44.9684570311

87.0829366148

45.7785848892

46.5699134918

47.3465764924

48.1122709869

48.8703530107

49.6239466782

50.3760533218

51.1296469893

51.8877290131

52.6534235076

53.4300865082

54.2214151108

55.0315429689

55.8653131418

56.7284418037

57.627886589

58.5724059318

59.573323041

60.6457537186

61.8108164315

63.0994521896

64.5600643009

66.2757351063

68.4094642464

71.3655584957

76.896741474

Sheet2

1012.917063385242.323.10325852643.819745992823.9743649032

1022.083896864244.528.634441504388.74773325594789.8973559713

1030.261763994344.931.5905357536

1034.621390467546.233.7242648937

1039.90054336946.735.4399356991

1043.707187968546.836.9005478104

1046.959841155347.838.1891835685

105048.539.3542462814

1053.040158844749.140.426676959

1056.292812031549.241.4275940682

1060.09945663149.742.372113411

1065.378609532549.943.2715581963

1069.738236005749.944.1346868582

1077.916103135850.244.9684570311

1087.082936614852.045.7785848892

52.246.5699134918

52.747.3465764924

52.948.1122709869

52.948.8703530107

53.049.6239466782

53.150.3760533218

54.151.1296469893

54.451.8877290131

54.652.6534235076

54.753.4300865082

54.754.2214151108

54.855.0315429689

54.855.8653131418

54.856.7284418037

55.057.627886589

55.658.5724059318

55.759.573323041

56.660.6457537186

56.961.8108164315

57.463.0994521896

57.864.5600643009

57.966.2757351063

58.968.4094642464

60.271.3655584957

61.276.896741474

Sheet3

*

40 Specimens

Chart1

3.70849358223.74595311333.7084935822

3.70849358223.79635563793.7895736004

3.70849358223.80498673493.8706536186

3.70849358223.8329449523.9517336368

3.70849358223.84270119064.032813655

3.70849358223.84542102164.1138936733

3.70849358223.8676468965

3.70849358223.8815394794

3.70849358223.8933778217

3.70849358223.8958993089

3.70849358223.9056811224

3.70849358223.9106776085

3.70849358223.9106943428

3.70849358223.9160618307

3.70849358223.9515563085

3.70849358223.9551467006

3.70849358223.965057884

3.70849358223.9675450891

3.70849358223.9678167153

3.70849358223.970478958

3.9712820152

3.9910693169

3.9971521837

4.0009363241

4.002545284

4.0026833919

4.0040036955

4.0041918866

4.0045524097

4.0079959065

4.0183457748

4.0207221136

4.0354562102

4.0407516228

4.0497428049

4.0561778096

4.0581532115

4.0755665024

4.0979122315

4.1138936733

Weibull Probability Plot

0.10%

0.20%

0.30%

0.50%

1%

2%

3%

5%

10%

20%

30%

40%

50%

60%

70%

80%

90%

95%

99%

99.90%

41

44

48

52

56

61

-6.9072550036

-4.3757438213

-7.2526177538

-6.2136072637

-3.2643645676

-7.2526177538

-5.8076411106

-2.7404930065

-7.2526177538

-5.2958121984

-2.3906822386

-7.2526177538

-4.6001492222

-2.1257221216

-7.2526177538

-3.9019386489

-1.9110827892

-7.2526177538

-3.4913669807

-1.7297202729

-2.9701952395

-1.5719525273

-2.2503673296

-1.4317440643

-1.4999400026

-1.3050729026

-1.0309303623

-1.1891197438

-0.6717269721

-1.0818278631

-0.3665129178

-0.9816470555

-0.087421476

-0.8873759959

0.1856267689

-0.7980610834

0.475885014

-0.7129289212

0.8340323556

-0.6313392162

1.0971886321

-0.5527521431

1.5271798468

-0.4767035757

1.9326466116

-0.4027868111

-0.3306383372

-0.2599258342

-0.1903393255

-0.1215813478

-0.0533591521

0.0146233279

0.0826760122

0.1511325382

0.2203655649

0.2908053716

0.3629664347

0.4374890912

0.5152018941

0.5972292379

0.685181261

0.7815251843

0.890405279

1.0197814405

1.1888836465

1.477511537

Sheet1

48.5

54.7

47.8

56.9

54.8

57.9

44.9

53.0

54.7

46.7

55.0

55.7

49.9

54.8

49.7

58.9

52.7

57.8

46.8

49.2

53.1

49.1

55.6

46.2

52.0

56.6

52.9

52.2

54.1

42.3

54.6

49.9

44.5

52.9

54.4

60.2

50.2

57.4

54.8

61.2

Sheet1

3.70849358223.74595311333.7084935822

3.70849358223.79635563793.7895736004

3.70849358223.80498673493.8706536186

3.70849358223.8329449523.9517336368

3.70849358223.84270119064.032813655

3.70849358223.84542102164.1138936733

3.70849358223.8676468965

3.70849358223.8815394794

3.70849358223.8933778217

3.70849358223.8958993089

3.70849358223.9056811224

3.70849358223.9106776085

3.70849358223.9106943428

3.70849358223.9160618307

3.70849358223.9515563085

3.70849358223.9551467006

3.70849358223.965057884

3.70849358223.9675450891

3.70849358223.9678167153

3.70849358223.970478958

3.9712820152

3.9910693169

3.9971521837

4.0009363241

4.002545284

4.0026833919

4.0040036955

4.0041918866

4.0045524097

4.0079959065

4.0183457748

4.0207221136

4.0354562102

4.0407516228

4.0497428049

4.0561778096

4.0581532115

4.0755665024

4.0979122315

4.1138936733


99.90%

99%

95%

90%

80%

70%

60%

50%

40%

30%

20%

10%

5%

3%

2%

1%

0.50%

0.30%

0.20%

0.10%

61

56

52

48

44

41

-6.9072550036

-4.3757438213

-7.2526177538

-6.2136072637

-3.2643645676

-7.2526177538

-5.8076411106

-2.7404930065

-7.2526177538

-5.2958121984

-2.3906822386

-7.2526177538

-4.6001492222

-2.1257221216

-7.2526177538

-3.9019386489

-1.9110827892

-7.2526177538

-3.4913669807

-1.7297202729

-2.9701952395

-1.5719525273

-2.2503673296

-1.4317440643

-1.4999400026

-1.3050729026

-1.0309303623

-1.1891197438

-0.6717269721

-1.0818278631

-0.3665129178

-0.9816470555

-0.087421476

-0.8873759959

0.1856267689

-0.7980610834

0.475885014

-0.7129289212

0.8340323556

-0.6313392162

1.0971886321

-0.5527521431

1.5271798468

-0.4767035757

1.9326466116

-0.4027868111

-0.3306383372

-0.2599258342

-0.1903393255

-0.1215813478

-0.0533591521

0.0146233279

0.0826760122

0.1511325382

0.2203655649

0.2908053716

0.3629664347

0.4374890912

0.5152018941

0.5972292379

0.685181261

0.7815251843

0.890405279

1.0197814405

1.1888836465

1.477511537

Plot Data

-6.90725500363.70849358223.7-4.37574382133.7084935822-7.2526177538

-6.21360726373.70849358223.8-3.26436456763.7895736004-7.2526177538

-5.80764111063.70849358223.8-2.74049300653.8706536186-7.2526177538

-5.29581219843.70849358223.8-2.39068223863.9517336368-7.2526177538

-4.60014922223.70849358223.8-2.12572212164.032813655-7.2526177538

-3.90193864893.70849358223.8-1.91108278924.1138936733-7.2526177538

-3.49136698073.70849358223.9-1.72972027294.1138936733-7.2526177538

-2.97019523953.70849358223.9-1.5719525273

-2.25036732963.70849358223.9-1.4317440643

-1.49994000263.70849358223.9-1.3050729026

-1.03093036233.70849358223.9-1.1891197438

-0.67172697213.70849358223.9-1.0818278631

-0.36651291783.70849358223.9-0.9816470555

-0.0874214763.70849358223.9-0.8873759959

0.18562676893.70849358224.0-0.7980610834

0.4758850143.70849358224.0-0.7129289212

0.83403235563.70849358224.0-0.6313392162

1.09718863213.70849358224.0-0.5527521431

1.52717984683.70849358224.0-0.4767035757

1.93264661163.70849358224.0-0.4027868111

4.0-0.3306383372

4.0-0.2599258342

4.0-0.1903393255

4.0-0.1215813478

4.0-0.0533591521

4.00.0146233279

4.00.0826760122

4.00.1511325382

4.00.2203655649

4.00.2908053716

4.00.3629664347

4.00.4374890912

4.00.5152018941

4.00.5972292379

4.00.685181261

4.10.7815251843

4.10.890405279

4.11.0197814405

4.11.1888836465

4.11.477511537

*

The tensile strength distribution can be estimated by

40 Specimens

f(x)

F(x)

^

^

=

=

k

i

i

n

f

1

2.24.13.54.53.23.732.6

3.41.63.13.33.83.14.73.7

2.54.33.43.62.93.33.93.1

3.33.13.74.43.24.11.93.4

4.73.83.22.63.934.23.5

Car Battery Lives

ClassClassFrequencyRelative

intervalmidpointffrequency

1.5-1.91.720.050

2.0-2.42.210.025

2.5-2.92.740.100

3.0-3.43.2150.375

3.5-3.93.7100.250

4.0-4.44.250.125

4.5-4.94.730.075

Total401.000

Relative Frequency Distribution of

Battery Lives

Relative frequency histogram

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

1.72.22.73.23.74.24.7

Battery Lives (years)

Relative Frequency

+

-

=

4

.

0

3

.

0

100

)

(

n

i

x

F

i

4

.

0

n

3

.

0

i

)

x

(

F

^

i

+

-

=

+

-

=

4

.

0

n

3

.

0

i

100

)

x

(

F

^

i

b

q

-

-

=

t

e

t

1

)

(

F

(

)

,

W

~

T

(

)

(

)

(

)

[

]

(

)

(

)

q

b

b

q

b

q

b

q

b

ln

ln

)

T

(

F

1

1

ln

ln

ln

)

T

(

F

1

ln

ln

)

T

(

F

1

ln

ln

)

T

(

F

1

ln

-

=

-

=

-

-

-

=

-

=

-

-

x

x

x

e

x

-

-

=

)

t

(

F

1

1

ln

ln

y

b

=

a

(

)

t

x

ln

=

(

)

i.e.,

,

ln

q

b

-

=

b

(

)

q

b

b

ln

-

=

x

y

(

)

q

b

ln

-

-

=

)

t

(

F

1

1

ln

ln

y

(

)

%)

100

(

0.4

n

0.3

i

MR

x

F

i

i

+

-

@

=

(

)

%

6

.

26

%

100

*

0.4

6

0.3

2

20

F

=

+

-

=

i

x

i

F(x

i

)

11010.9%

22026.6%

33042.2%

44057.8%

55073.4%

68089.1%

(

)

(

)

x

F

,

x

i x

i

179.40968

288.12093

391.06394

498.73094

5104.1536

6105.1019

7106.5036

8112.0354

148.51155.02153.13154.6

254.71255.72249.13249.9

347.81349.92355.63344.5

456.91454.82446.23452.9

554.81549.72552.03554.4

657.91658.92656.63660.2

744.91752.72752.93750.2

853.01857.82852.23857.4

954.71946.82954.13954.8

1046.72049.23042.34061.2

30.0

35.0

40.0

45.0

50.0

55.0

60.0

65.0

0510152025303540


Count40

Minimum42.35

Maximum61.18

Range18.84

Sum2104.82

Mean52.62

Median53.03



Kurtosis2.51

Skewness-0.34

BinFrequency

400

453

5010

5516

609

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0

2

4

6

8

10

12

14

16

18

4045505560More


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1 statistical analysis - graphical techniques dr. jerrell t. stracener, sae fellow leadership in...

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