Reliability Engineering and System Safety 24 (1989) 257-265

Effect of Ranking on Reliability Prediction Using Weibuil Analysis

K. K u m a r

School of Electrical and Electronic Engineering, Nanyang Technological Institute, Nanyang Avenue,

Singapore 2263, Republic of Singapore

(Received 18 March 1988; revised version received 23 June 1988; accepted 29 June 1988)

A BSTRA CT

Reliability prediction heavily relies on the ranking of failure data of a randomly drawn sample from a uniform population. The degree to which the sample represents the population depends on the size of the sample, luck or chance, sampling method and whether the sampling conditions change. However, for this purpose, it is now standard practice to rank (mean or median) the failure data and plot the cumulative distribution function (cdf) against time (normally) on Weibull graph paper. The Weibull parameters obtained from the plot are used to estimate the expected reliability of the component~system under investigation. In the literature no clear direction is given regarding the correct and reliable means of tanking the failure data, i.e. when to use the mean ranking and when to use the median ranking. This paper proposes a systematic use of either the mean or the median ranking depending upon the outcome of the skewness test applied to the data. The paper also highlights the effect of incorrect ranking on the predicted reliability.

I N T R O D U C T I O N

The two or three parameter Weibull distribution is widely used for analysing failure data. A number of reports and papers have reported its usefulness in predicting the reliability of various types of components and systems. Due to economic consideration the sample size for such data is normally small

Present address: School of Electronic and Electrical Engineering, City of Birmingham Polytechnic, Perry Bar, Birmingham B42 2FU, UK.

257 Reliability Engineering and System SaJety 0951-8320/89/$03.50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

258 K. Kumar

which may or may not represent the true picture of failure behaviour of the whole production for the reasons indicated earlier. Therefore, the usual method of estimating population from samples is to construct a frequency histogram to obtain a frequency curve. To overcome the limitations due to small samples, the data are ordered in increasing order of value and a cumulative plot is constructed. This process is known as ranking. The cumulative distribution function (cdf) of a time dependent three parameter Weibull distribution is defined as:

(1)

In this expression t o, q and [~ are referred to as the location or minimum life, characteristic life and shape or slope parameter of the population, respectively, In the situation when to can be zero, (1) reduces to a two parameter distribution given by (2) and the corresponding expression for reliability is given by (3)

Q ( t ) = l - e x p - (2)

R,,, exp{ ~3)

The strength and the usefulness of this distribution lies in the fact that a double logarithmic transformation of(2) results in the equation of a straight line. This forms the basis of Weibull graph paper with l , t as the principal abscissa and 1~1,[1 - Q(t)] -L as the principal ordinate. The slope of the line on this graph paper gives the value ~, and time corresponding to the 63.2% cumulative failure gives the value of q. Having found the values of fl and r/ the probability of survival of the item under test to any time t can be found using (3). For a three parameter distribution the procedure is similar except that the value of to is found in accordance with Pratt and Whitney report L and then replotted with l , ( t - t o) as the principal abscissa.

A plot on the basis of true rank, in general, fails to accurately represent the population because the first failure is normally further away from zero than the last is from 100%. This also fails to produce a good fit. To overcome this, and thus to improve the accuracy, mean or median ranking of cumulative percentages is used for plotting on the Weibuil or probability graph paper. In the following sections these two rankings and the procedure for selecting the correct ranking for the known failure data are described. An example is worked out to explain the procedure and to indicate the impact of using inappropriate ranking.

Effect of ranking on reliability prediction using Weibull analysis 259

MEAN AND MEDIAN R A N K I N G

While both estimates are statistically unbiased, the mean rank would give more pessimistic results at low values of fraction failing and more optimistic at high values than would the median ranking. If i is true rank of the failure and N the sample size then the mean ranks are given by (4).

i Mean r a n k - (41

N + I

Median ranks are determined from the binomial expansion of (R + Q)N= 0.5, R and Q being the probabilities of survival and failure. However, a good approximation can be obtained by using A. Bernard's formula given by (5)

i - 0 . 3 Median r a n k - (5)

N+0"4

Although expression (4) is easier to use, engineers tend to prefer (5) because for different values of N, in general up to 50, standard tables are available for median ranks. Probably, this is the only reason why most engineers tend to use median ranking instead of the mean ranking without ensuring its validity. It must be pointed out in this respect that engineers, invariably poor in statistics, are left in the cold without any guidelines. The only available reference to this is given by O'Connor 2 that mean ranking is appropriate for symmetrical distribution and for skewed distributions median ranking provides a better correction. Based on this it is suggested that the data should first be tested for skewness and then depending upon the outcome of the test the appropriate ranking should be used.

PROPOSAL FOR CHOOSING CORRECT R A N K I N G

It is proposed that the appropriate ranking suitable for the data should be chosen on the basis of test for skewness. The third moment (rn3) of probability density distribution about the average gives the measure of skewness, i.e. lack of symmetry. The value of rn 3 is given by (6) (Ref. 3).

K3 m3 = S 3 (6)

where N

K3 = (N - 1)(N - 2) ( t i - I-')3

I = I

260 K. K u m a r

Symmetrical Skewed to the Right (Normal)

(a)

Fig. 1.

rn 3 < 0

Skewed to the Left

(bl (¢1

Different character i s t ics o f fai lure d i s tr ibut ions .

7)'gig." rgl/lk

1i)

and N

s =LFi= I ("- w] t'2 =i The symmetry or lack o f it is determined by the numerical value o f m 3. Three different types o f distributions as shown in Fig. 1 are obtained for m 3 = 0, m3 < 0 and m 3 > 0. The fo l lowing conclusions are drawn from this test:

(i) for m 3 = 0 , symmetrical , i.e. Gauss ian distribution: use mean ranking;

(ii) for m3 > 0, distribution skewed to the right: use median ranking; (iii) for m 3 < 0, distribution skewed to the left: use median ranking.

T A B L E I

Test for S k e w n e s s

T i m e t o . l a i l u r ( , t, -- I- I t i -- 7) 2 (t, - T) 3

(t i)

I 61) - 188-5 35 5 3 2 2 5

2 95 - 153"5 23 562"25

3 120 - 128"5 16512.25

4 170 - 78-5 6 162.25

5 220 -- 28"5 812.25

6 261) + I 1"5 132.25

7 290 + 41.5 1 722.22

8 350 + 101-5 I0 302"25

9 400 + 1515 22952"25

I0 520 ~-271 5 73 712-25

~ l i = 2 485 ~ ( l , - - t) e

= 191 402.5

i - 248.5 .5" = 145.831 9

,S ̀3 = 3 101 400

. . m.~ = 5 2 2 4

- 6 6 9 7 8 2 9 . 1

- - 3 6 1 6 8 0 5 3

- 2 121 824"1

- 483 736.62

+ 2 3 149.12

+ I 520.87

+ 71 473.37

+ 1 045 678.3

+ 3 4 7 7 2 6 5 . 8

+ 2 0 0 1 2 8 7 5

Y~(t,--/-)~ = 1 I 665 469

K.~ = 16 202 040

Effect of ranking on reliability prediction using Weibull analysis 261

TABLE 2 Data for Plotting the Weibull Line

True Mean rank Median rank Time to failure rank (%) (%) (h)

~i)

1 9'09 6"6 60 2 18'18 16'2 95 3 27'27 25'8 120 4 36-36 35'5 170 5 45'45 45-1 220 6 54"54 54'8 260 7 63"63 64.4 290 8 72.72 74'1 350 9 81"81 83'7 400

10 90'90 93'33 520

Example

In a life test of a printed circuit board containing discrete, integrated and passive components the failures occurred at 60, 95, 120, 170, 220, 260, 290, 350, 400 and 520 h. We need to determine the probability of its survival to 190h.

First ofall the value o fm 3 is found by constructing Table 1. The calculated value o f m 3 indicates that the failure data are heavily skewed to the right (as in Fig. l(b)) and thus the median ranking should be used. The data for plotting the Weibull line are given in Table 2 which also includes the mean ranks and the time to failure. To show the effect of choosing mean ranking instead of median Weibull lines are plotted in Figs 2 and 3. The values of fl and r/are determined using the established procedure. These values for mean and median ranking are 1.4, 290 and 1.65, 285 respectively.

D E T E R M I N A T I O N OF WEIBULL LINE USING LEAST-SQUARE FITTING P R O C E D U R E

In reliability plotting the Weibull line is visually fitted through the sample points to obtain the required parameters. Kapur and Lamberson 4 suggest the possibility of using the least-square fitting procedure to obtain the best linear fit. This procedure was applied to both mean and median rank data of Table 2. The correlation coefficients, as defined in (2), and the best fit regression lines are given in Table 3.

262 K. Kumar

.Sample S,ze

Shape

ChazacteristLc Life

Min imum Die

70" r/ E$11m+t~-

0.1

IGO 1009

HOURS

Fig. 2. Weibull plot for mean ranked data.

Effect of ranking on reliability prediction using Weibull analysis 263

C h a r a c t e r i s t i c Lile ~1 295

M i n i r ~ m b|e ~ 0 9%9

. ' , , , I

\ , 1 = " \ ~ 1 70 \ / / E,,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . g : . . . . . . . . . . . . . -t-

7 = 1 - ® < I/. 2o

~'z 0 / / u /

. , , /

o / >_ I.-

5

U

!

o°| 2 3 4 ~ 6 7 g 9 I 2 ] 0 5 6 7 g ? I

100 I000

HOURS

Fig. 3. Weibull plot for median ranked data.

264 K. Kumar

TABLE 3

Ranking Correlation Regression line used coefficients

Mean 1.0 Q(t) = 0.188 9t + 5.099 5 Median 0.98 Q(t) = 0.195 9t + 3.387 9

The values of the correlation coefficients close to one clearly show that the data fit a straight line. The mean least-square errors for the visually plotted and the regression lines for the mean and median ranked data are 3.5, 25.22 and 9"48, 27.93 respectively. This suggests that for such data the visually plotted lines yield better result than the best-fit regression line obtained on the basis of least-square fit.

EFFECT OF CHOOSING MEAN R A N K I N G INSTEAD OF MEDIAN R A N K I N G

The expression (3) is used to calculate the probability of survival of the boards under test for 190 h. The values for mean and median ranking are expressed as R(190)mean and R(190)m~a~a, respectively. Table 4 shows that the predicted probability of survival is 0.575. However, if the mean ranking was used this value for the same failure data would be 0.59, i.e. 2.5% better (higher) than for median ranking. The percentage error could be much higher for a different set of data.

TABLE 4 Calculation of Probability of Survival to 190 h

Rankmg Weibull parameter.~ R(190) R(190),.~a,~. - R(190) ...... x 100 u s e d - -

[~ ~1 R{ 190),,,.,1,,,,

Mean 1.4 290 0.590 2.5 % Median 165 285 0.575

This type of discrepancy will have conflicting effects for demonstration and acceptance reliability testing and in the worst case may lead to a lawsuit.

CONCLUSION

A systematic procedure for ranking the data on the basis of a skewness test has been explained and its effect on reliability prediction has also been

Effect of ranking on reliability prediction using Weibull analysis 265

demonstrated. It is recommended that this procedure should be tested, tried and used to suit various products. The standard software for Weibull analysis could be modified to include the skewness test to improve the accuracy and increase the speed of analysis. The least-square fitting procedure applied to the data does not improve the accuracy over the visually plotted line.

A C K N O W L E D G E M E N T

The author wishes to acl~nowledge the encouragement given by Professor Brian Lee, Dean, School of Electrical and Electronic Engineering, to undertake various projects on reliability engineering.

REFERENCES

1. Mitchell, R. A., Introduction to Weibull analysis. Report No. PWA 3001, Pratt & Whitney Aircraft, East Hartford, CT, Jan, 1967.

2. O'Connor, P. D. T. Practical Reliability Engineering. Heyden, London, 1981, ch. 3, pp. 58-9.

3. Glossary and tables for statistical quality control, 2nd edn. Prepared by the American Society for Quality Control Statistics Division, 1983, pp. 21-3.

4. Kapur, K. C. & Lamberson, L. R., Reliability in Engineering Design. John Wiley, New York, 1977.