Engineering Statistics Handbook Episode 3 Part 1 ppt

Analysis Probability plots are a graphical test for assessing if a particular distribution provides an adequate fit to a data set.. A quantitative enhancement to the probability plot is

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9 0.0 0.0000 0.0065 -0.01

10 0.0 0.0000 0.0020 0.00

STATISTIC = NUMBER OF RUNS UP

OF LENGTH I OR MORE

I STAT EXP(STAT) SD(STAT) Z

1 58.0 64.8333 4.1439 -1.65

2 23.0 24.1667 2.7729 -0.42

3 15.0 6.4083 2.1363 4.02

4 3.0 1.3278 1.1043 1.51

5 0.0 0.2264 0.4716 -0.48

6 0.0 0.0328 0.1809 -0.18

7 0.0 0.0041 0.0644 -0.06

8 0.0 0.0005 0.0215 -0.02

9 0.0 0.0000 0.0068 -0.01

10 0.0 0.0000 0.0021 0.00

RUNS DOWN STATISTIC = NUMBER OF RUNS DOWN

OF LENGTH EXACTLY I

1 33.0 40.6667 6.4079 -1.20

2 18.0 17.7583 3.3021 0.07

3 3.0 5.0806 2.0096 -1.04

4 3.0 1.1014 1.0154 1.87

5 1.0 0.1936 0.4367 1.85

6 0.0 0.0287 0.1692 -0.17

7 0.0 0.0037 0.0607 -0.06

8 0.0 0.0004 0.0204 -0.02

9 0.0 0.0000 0.0065 -0.01

10 0.0 0.0000 0.0020 0.00

STATISTIC = NUMBER OF RUNS DOWN

OF LENGTH I OR MORE

1 58.0 64.8333 4.1439 -1.65

2 25.0 24.1667 2.7729 0.30

3 7.0 6.4083 2.1363 0.28

4 4.0 1.3278 1.1043 2.42

5 1.0 0.2264 0.4716 1.64

6 0.0 0.0328 0.1809 -0.18

7 0.0 0.0041 0.0644 -0.06

8 0.0 0.0005 0.0215 -0.02

9 0.0 0.0000 0.0068 -0.01

10 0.0 0.0000 0.0021 0.00

RUNS TOTAL = RUNS UP + RUNS DOWN STATISTIC = NUMBER OF RUNS TOTAL

1.4.2.8.3 Quantitative Output and Interpretation

http://www.itl.nist.gov/div898/handbook/eda/section4/eda4283.htm (4 of 8) [5/1/2006 9:58:59 AM]

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OF LENGTH EXACTLY I

1 68.0 81.3333 9.0621 -1.47

2 26.0 35.5167 4.6698 -2.04

3 15.0 10.1611 2.8420 1.70

4 6.0 2.2028 1.4360 2.64

5 1.0 0.3871 0.6176 0.99

6 0.0 0.0574 0.2392 -0.24

7 0.0 0.0074 0.0858 -0.09

8 0.0 0.0008 0.0289 -0.03

9 0.0 0.0001 0.0092 -0.01

10 0.0 0.0000 0.0028 0.00

STATISTIC = NUMBER OF RUNS TOTAL

OF LENGTH I OR MORE

1 116.0 129.6667 5.8604 -2.33

2 48.0 48.3333 3.9215 -0.09

3 22.0 12.8167 3.0213 3.04

4 7.0 2.6556 1.5617 2.78

5 1.0 0.4528 0.6669 0.82

6 0.0 0.0657 0.2559 -0.26

7 0.0 0.0083 0.0911 -0.09

8 0.0 0.0009 0.0305 -0.03

9 0.0 0.0001 0.0097 -0.01

10 0.0 0.0000 0.0029 0.00

LENGTH OF THE LONGEST RUN UP = 4 LENGTH OF THE LONGEST RUN DOWN = 5 LENGTH OF THE LONGEST RUN UP OR DOWN = 5 NUMBER OF POSITIVE DIFFERENCES = 98

NUMBER OF NEGATIVE DIFFERENCES = 95 NUMBER OF ZERO DIFFERENCES = 1

Values in the column labeled "Z" greater than 1.96 or less than -1.96 are statistically significant at the 5% level The runs test does indicate some non-randomness.

Although the autocorrelation plot and the runs test indicate some mild non-randomness, the violation of the randomness assumption is not serious enough to warrant developing a more sophisticated model It is common in practice that some of the assumptions are mildly violated and it is a judgement call as to whether or not the violations are serious enough to warrant developing a more sophisticated model for the data.

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Analysis

Probability plots are a graphical test for assessing if a particular distribution provides an adequate fit to a data set.

A quantitative enhancement to the probability plot is the correlation coefficient of the points on the probability plot For this data set the correlation coefficient is 0.996 Since this is greater than the critical value of 0.987 (this is a tabulated value ), the normality assumption is not rejected.

Chi-square and Kolmogorov-Smirnov goodness-of-fit tests are alternative methods for assessing distributional adequacy The Wilk-Shapiro and Anderson-Darling tests can be used to test for normality Dataplot generates the following output for the

Anderson-Darling normality test.

ANDERSON-DARLING 1-SAMPLE TEST THAT THE DATA CAME FROM A NORMAL DISTRIBUTION

1 STATISTICS:

NUMBER OF OBSERVATIONS = 195 MEAN = 9.261460 STANDARD DEVIATION = 0.2278881E-01

ANDERSON-DARLING TEST STATISTIC VALUE = 0.1264954 ADJUSTED TEST STATISTIC VALUE = 0.1290070

2 CRITICAL VALUES:

90 % POINT = 0.6560000

95 % POINT = 0.7870000 97.5 % POINT = 0.9180000

99 % POINT = 1.092000

3 CONCLUSION (AT THE 5% LEVEL):

THE DATA DO COME FROM A NORMAL DISTRIBUTION

The Anderson-Darling test also does not reject the normality assumption because the test statistic, 0.129, is less than the critical value at the 5% significance level of 0.918.

Outlier

Analysis A test for outliers is the Grubbs' test Dataplot generated the following output for Grubbs'

test.

GRUBBS TEST FOR OUTLIERS (ASSUMPTION: NORMALITY)

1 STATISTICS:

NUMBER OF OBSERVATIONS = 195 MINIMUM = 9.196848 MEAN = 9.261460 MAXIMUM = 9.327973 STANDARD DEVIATION = 0.2278881E-01

GRUBBS TEST STATISTIC = 2.918673

2 PERCENT POINTS OF THE REFERENCE DISTRIBUTION

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FOR GRUBBS TEST STATISTIC

0 % POINT = 0.000000

50 % POINT = 2.984294

75 % POINT = 3.181226

90 % POINT = 3.424672

95 % POINT = 3.597898 97.5 % POINT = 3.763061

99 % POINT = 3.970215

100 % POINT = 13.89263

3 CONCLUSION (AT THE 5% LEVEL):

THERE ARE NO OUTLIERS

For this data set, Grubbs' test does not detect any outliers at the 25%, 10%, 5%, and 1% significance levels.

Model Since the underlying assumptions were validated both graphically and analytically, with a

mild violation of the randomness assumption, we conclude that a reasonable model for the data is:

We can express the uncertainty for C, here estimated by 9.26146, as the 95% confidence interval (9.258242,9.26479).

Univariate

Report It is sometimes useful and convenient to summarize the above results in a report The report for the heat flow meter data follows.

Analysis for heat flow meter data

1: Sample Size = 195

2: Location Mean = 9.26146 Standard Deviation of Mean = 0.001632 95% Confidence Interval for Mean = (9.258242,9.264679) Drift with respect to location? = NO

3: Variation Standard Deviation = 0.022789 95% Confidence Interval for SD = (0.02073,0.025307) Drift with respect to variation?

(based on Bartlett's test on quarters

of the data) = NO

4: Randomness Autocorrelation = 0.280579 Data are Random?

(as measured by autocorrelation) = NO

5: Distribution Normal PPCC = 0.998965 Data are Normal?

(as measured by Normal PPCC) = YES

6: Statistical Control (i.e., no drift in location or scale,

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data are random, distribution is fixed, here we are testing only for fixed normal)

Data Set is in Statistical Control? = YES

7: Outliers?

(as determined by Grubbs' test) = NO

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4 Generate a normal probability

plot.

4 The normal probability plot verifies that the normal distribution is a reasonable distribution for these data.

4 Generate summary statistics, quantitative

analysis, and print a univariate report.

1 Generate a table of summary

statistics.

2 Generate the mean, a confidence

interval for the mean, and compute

a linear fit to detect drift in

location.

3 Generate the standard deviation, a

confidence interval for the standard

deviation, and detect drift in variation

by dividing the data into quarters and

computing Bartlett's test for equal

standard deviations.

4 Check for randomness by generating an

autocorrelation plot and a runs test.

5 Check for normality by computing the

normal probability plot correlation

coefficient.

6 Check for outliers using Grubbs' test.

7 Print a univariate report (this assumes

steps 2 thru 6 have already been run).

1 The summary statistics table displays 25+ statistics.

2 The mean is 9.261 and a 95%

confidence interval is (9.258,9.265).

The linear fit indicates no drift in location since the slope parameter estimate is essentially zero.

3 The standard deviation is 0.023 with

a 95% confidence interval of (0.0207,0.0253) Bartlett's test indicates no significant change in variation.

4 The lag 1 autocorrelation is 0.28.

From the autocorrelation plot, this is statistically significant at the 95%

level.

5 The normal probability plot correlation coefficient is 0.999 At the 5% level,

we cannot reject the normality assumption.

6 Grubbs' test detects no outliers at the 5% level.

7 The results are summarized in a convenient report.

1.4.2.8.4 Work This Example Yourself

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1 Exploratory Data Analysis

1.4 EDA Case Studies

1.4.2 Case Studies

1.4.2.9 Airplane Polished Window Strength

1.4.2.9.1 Background and Data

Generation This data set was provided by Ed Fuller of the NIST Ceramics Division

in December, 1993 It contains polished window strength data that was used with two other sets of data (constant stress-rate data and strength of indented glass data) A paper by Fuller, et al describes the use of all three data sets to predict lifetime and confidence intervals for a glass airplane window A paper by Pepi describes the all-glass airplane window design.

For this case study, we restrict ourselves to the problem of finding a good distributional model of the polished window strength data.

Purpose of

Analysis

The goal of this case study is to find a good distributional model for the polished window strength data Once a good distributional model has been determined, various percent points for the polished widow strength will be computed.

Since the data were used in a study to predict failure times, this case study is a form of reliability analysis The assessing product reliability

chapter contains a more complete discussion of reliabilty methods This case study is meant to complement that chapter by showing the use of graphical techniques in one aspect of reliability modeling.

Data in reliability analysis do not typically follow a normal distribution; non-parametric methods (techniques that do not rely on a specific

distribution) are frequently recommended for developing confidence intervals for failure data One problem with this approach is that sample sizes are often small due to the expense involved in collecting the data, and non-parametric methods do not work well for small sample sizes For this reason, a parametric method based on a specific distributional model of the data is preferred if the data can be shown to follow a specific distribution Parametric models typically have greater efficiency

at the cost of more specific assumptions about the data, but, it is important to verify that the distributional assumption is indeed valid If the distributional assumption is not justified, then the conclusions drawn 1.4.2.9.1 Background and Data

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from the model may not be valid.

This file can be read by Dataplot with the following commands:

SKIP 25 READ FULLER2.DAT Y

Resulting

Data

The following are the data used for this case study The data are in ksi (= 1,000 psi).

18.830 20.800 21.657 23.030 23.230 24.050 24.321 25.500 25.520 25.800 26.690 26.770 26.780 27.050 27.670 29.900 31.110 33.200 33.730 33.760 33.890 34.760 35.750 35.910 36.980 37.080 37.090 39.580 44.045 45.290 45.381 1.4.2.9.1 Background and Data

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The normal probability plot has a correlation coefficient of 0.980 We can use this number as a reference baseline when comparing the performance of other distributional fits.

Other Potential

Distributions

There is a large number of distributions that would be distributional model candidates for the data However, we will restrict ourselves to consideration of the following distributional models because these have proven to be useful in reliability studies.

Normal distribution

1

Exponential distribution

2

Weibull distribution

3

Lognormal distribution

4

Gamma distribution

5

Power normal distribution

6

Fatigue life distribution

7

1.4.2.9.2 Graphical Output and Interpretation

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Approach There are two basic questions that need to be addressed.

Does a given distributional model provide an adequate fit to the data?

1

Of the candidate distributional models, is there one distribution that fits the data better than the other candidate distributional models?

2

The use of probability plots and probability plot correlation coefficient (PPCC) plots provide answers to both of these questions.

If the distribution does not have a shape parameter, we simply generate a probability plot.

If we fit a straight line to the points on the probability plot, the intercept and slope of that line provide estimates of the location and scale

parameters, respectively.

1

Our critierion for the "best fit" distribution is the one with the most linear probability plot The correlation coefficient of the fitted line of the points

on the probability plot, referred to as the PPCC value, provides a measure

of the linearity of the probability plot, and thus a measure of how well the distribution fits the data The PPCC values for multiple distributions can

be compared to address the second question above.

2

If the distribution does have a shape parameter, then we are actually addressing

a family of distributions rather than a single distribution We first need to find the optimal value of the shape parameter The PPCC plot can be used to determine the optimal parameter We will use the PPCC plots in two stages The first stage will be over a broad range of parameter values while the second stage will be in the neighborhood of the largest values Although we could go further than two stages, for practical purposes two stages is sufficient After

determining an optimal value for the shape parameter, we use the probability plot as above to obtain estimates of the location and scale parameters and to determine the PPCC value This PPCC value can be compared to the PPCC values obtained from other distributional models.

Analyses for

Specific

Distributions

We analyzed the data using the approach described above for the following distributional models:

Normal distribution - from the 4-plot above, the PPCC value was 0.980.

1

Exponential distribution - the exponential distribution is a special case of the Weibull with shape parameter equal to 1 If the Weibull analysis yields a shape parameter close to 1, then we would consider using the simpler exponential model.

2

Weibull distribution

3

Lognormal distribution

4

Gamma distribution

5

Power normal distribution

6

Power lognormal distribution

7

1.4.2.9.2 Graphical Output and Interpretation

Tiêu đề	Quantitative Output and Interpretation
Trường học	National Institute of Standards and Technology
Chuyên ngành	Engineering Statistics
Thể loại	Bài báo
Năm xuất bản	2006
Thành phố	Gaithersburg

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Số trang	18
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