These techniques should be used when the actual shape of the distribution of fatigue life values for a given material is unknown or sketchy and the number of specimens tested at each applied stress level is too small, say less than 50, to estimate the shape of the distribution. In such cases, these tech- niques give conservative results.
(a) One Group at Each Stress Level.—Usually the first step in the analysis of fatigue data is to draw the S-N curve for 50 per cent survival; it is the curve fitted to the medians of the groups at the several applied stress levels.
The median, an "order statistic," is the middlemost value when the observed values are arranged in order of magnitude, or the average of the two middle- most values if the group size is even.
Other S-N curves, those for p per t survival (where p is not 50), may be fitted to other order statistics if the group size is greater than 1. If the group values are arranged in order of magnitude, NI is the minimum cycle life value, or the first order statistic, A7 2 is the second observed value, or the second order statistic, and so forth.
The estimated percentage of survivors for the population at cycle life
11 In the Normal distribution, the median and the mean are equal.
12 See Appendix IV, p. 71.
24 FATIGUE TESTING AND STATISTICAL ANALYSIS or DATA
values of Ni, or 7V2, depends upon the group size. Table 8 gives the median percentages at Ni and Nz for several group sizes.13 Some of these percent- ages also are given in Table 25 for one group, m = 1, and the 50 per cent confidence level.
The median percentage of survivors at the maximum value of the sample, Nn , is 100 — (per cent for N\), etc. Examples of how Table 8 may be used follow:
1. The 50 per cent survival curve may be estimated from the median of any sample size.
2. If three specimens are tested at each applied stress level, the 79, 50, and the 21 per cent survival curves may be estimated from the entries in Table 8 and their complements. The value 79 per cent is found opposite sample size 3 in the second column, the value 50 per cent is taken from the median
TABLE 8.—MEDIAN PERCENTAGE OF SURVIVORS FOR THE POPULATION.
c ™~iô, c;
M - At the Lowest At the Next Lowest
Sample Size, n Vajue ^ ValuCj ^ 21
34 56 78.".
:' 9.
"10 :• 11.-.
1213 1415.' 16,
. 50 7079 8487 8990 9192 9394 9495 9595 96
3050 6169 7377 8082 8485 8687 8889 90
S-N curve, and the value 21 per cent is obtained by subtracting the value in the second column from 100 per cent.
3. If 7 specimens are tested at each applied stress level, the 90, 77, 23, and 10 per cent survival curves may be estimated from the entries in Table 8 and their complements. The 50 per cent survival curve may be estimated from the median.
At least 13 specimens must be tested at each applied stress level to esti- mate the 95 per cent survival curve.
In practice, values of per cent l less than? 50 usually are not wanted.
Hence, if all of the specimens in a sample are tested simultaneously, the tests may be stopped as soon as the specimen having the median value of fatigue life for the sample has failed, unless the data are required for other purposes.
13 These are called "median percentages" because, half of the time, the true*percentage will be larger, and for the other half of the time, smaller. They are close to, but usually not equal to, the "expected" percentage of survivors, which is equal to 1 — i/(n + 1), where i is the number of the order statistic and n is the sample size. The confidence level associ- ated with expected percentages varies with the sample size, whereas it is constant for median percentages.
ANALYSIS or FATIGUE DATA 25 As mentioned previously, the percentage survival values given in Table 8 are median values; they are based on a "confidence level" of 50 per cent.14 Percentage survival values corresponding to higher confidence levels, such as 95 or 99 per cent, are given in Table 25 for a single sample when m = 1.
For example, if three specimens are tested at an applied stress level, 79 per cent of the population are expected to survive N\ cycles (50 per cent confidence level), but the statement that at least 37 per cent of the popula- tion will survive N\ cycles may be made with greater confidence (confidence level = 95 per cent). If estimates of the population percentage are made from a series of samples tested at one applied stress level and the statement is made that at least 79 per cent of the population will survive N\ cycles, 50 per cent of such statements are expected to be incorrect. If the statement is made each time that at least 37 per cent will survive N\ cycles, only 5 per cent of such statements are expected to be incorrect. However, S-N curves corresponding to a 50 per cent confidence level are usually shown.
The effect of fitting a curve to the same order statistics at several stress levels probably increases the confidence level; how much is not known. If S-N curves are based on other confidence levels, the fact should be plainly in- dicated on the chart.
(Z>) Several Samples, or Groups, at Each Stress Level.—If it is not possible to test all the specimens in a sample simultaneously and if stopping the tests before all the specimens have failed is desirable to save time, the required sample may be divided, at random, into two or more groups (see references 17 and 18). Then the median of the particular order statistics (the first, second, and so forth) for the several groups may be used for constructing the S-N curve. Table 25 gives values of percentage survival for several num- bers of groups and several confidence levels.
EXAMPLE.—With five testing machines available, 15 specimens were tested at a constant applied stress level in three groups of 5 each. For each group, all machines were assumed to be stopped after the second failure. (Actually, all machines were allowed to run until fracture occurred or until 10 million cycles of fatigue stressing had been applied, so that the time saved could be estimated for this particular set of tests.)
The test data are:
Entering Table 25, under "Lowest Ranking Points," in the column for m = 3 groups, opposite n = 5 in each group, and at a confidence level of 50 per cent,
14 Technically speaking, the S-N curves based on order statistics are "nonparametric tolerance limits," which are described by Murphy (16). The probability that at least p per cent of the population lies above Ni cycles, where Ni is the ith order statistic of the sample, is properly called a "tolerance level"; but the term confidence level appears to have been used more frequently.
Group 21 3
162,105, 275,
Life, kilocycles 229, (261, 668, 2 281) 131, (140, 245, 10 000+) 373, (5 503, 8 695, 10 000+) From these data we have:
2Vi.
#2
Lowest Ranking Points 162, 105, 275 229, 131, 373
Median 162229
26 FATIGUE TESTING AND STATISTICAL ANALYSIS or DATA
read 87.05 per cent. This value is an estimate of the percentage of the population from which the original 15 specimens were selected that will survive 162 kilocycles.
Similarly, at a confidence level of 95 per cent, 67.03 per cent or more of the popula- tion are estimated to survive the 162 kilocycles. Again, for the "Second Ranking Points," at a confidence level of 50 per cent, 68.61 per cent of the population are estimated to survive 229 kilocycles and, at a 95 per cent confidence level, 45.40 per cent or more of the population are estimated to survive 229 kilocycles.
Additional information can be obtained from the preceding test data by con- sidering all 15 specimens as one "group" and determining the percentage of the population expected to survive 105 kilocycles, which is the lowest ranking point for m = 1 and n = 15 in Table 25. For a 95 per cent confidence level, straight-line interpolation between 74.11 per cent for n = 10 and 86.09 per cent for n — 20
TABLE 9.—CONFIDENCE INTERVALS FOR THE MEDIAN."
Confidence Level =£ 0.95.
Sample Size, n
67 8 9 1011 1213 1415 2025 3540.
4550
Confidence Limits
Lower Upper
#
#saa
#
#2
#2#2
#3
#3
#
#
#1
#1
#1
#1
#6#7
#8
#9
#W
#11
#12
#W
ằ
1 #24
. #27
S #30
ằ #33
" Based on a table in Nair (19).
gives about 80 per cent. From this, it is estimated that at a 95 per cent confidence level about 80 per cent of the population will survive 105 kilocycles.
2. Estimates of Parameters—Single Stress Level:
(a) Median Fatigue Life:
1. Point Estimate. —A point estimate of the population median is the sample median, described above in n V Al(a).
2. Confidence Interval Estimate. —A confidence interval for the median that does not assume a particular frequency distribution for the population may be computed if the sample size is larger than five.
The n observed values of fatigue life, N, are arranged in order of magnitude as follows:
Ni£N*gNf- £ N.
N4 N6 N2
N13
N8 N10
N12N15
30
ANALYSIS OF FATIGUE DATA 27 The confidence limits corresponding to a confidence level of at least 0.95 are given by the order statistics designated in Table 9, p. 26.
EXAMPLE. — Assume that ten specimens are tested at a particular stress level and the observed values of fatigue life in kilocycles are 201, 224, 226, 230, 232, 238, 24 244, 245, and 248. The point estimate of median fatigue life is the average of the two middlemost values, namely 235 kilocycles. The interval estimate is defined by A/2 and Ng (see Table 9), which are 224 and 245 kilocycles, respectively.
The population median may be above or below the sample median — 235 kilocycles — but the chances are at least 95 in 100 that the statement, "the median lies between 224 and 245 kilocycles," is correct if the sample came from one population.
(b} Mean Fatigue Life:
1. Point Estimate. — A point estimate of the population mean is the sample average.
TABLE 1 0— APPROXIMATE CONFIDENCE INTERVALS FOR THE MEAN.
Confidence Level = 0.95.
Sample Size, n Procedure". 6 Length of Interval
3 ... add the range of the observed values to the largest 3 X range value and subtract it from the smallest value:
that is, Ni - (N3 - JVi) and N3 + (N3 - Ni).
4 ... add (range) /4 to the largest value and subtract it lj£ X range from the smallest value:
5 ... use the range: N\ and Ns I X range
0 See Youden (20) for n = 3.
6 Private correspondence from W. J. Youden, for values of n greater than 3.
2. Approximate Confidence Interval Estimate. — An approximate confidence interval estimate for the mean that does not assume a particular frequency distribution for the population may be computed as shown in Table 10, if the sample size is 3, 4, or 5.
(c) Per Cent Survival for a Stated Value of Fatigue Life:
1. Point Estimate. — A point estimate of the percentage of the population that has fatigue life values equal to or above a stated value is the sample percentage of observed values equal to or above the same stated value.
2. Confidence Interval Estimate. — Confidence limits corresponding to possible values of sample percentage, p, for four sample sizes are given in Table 11. Values for other sample sizes may be read from a chart from Dixon and Massey (9), p. 415, from which many values in Table 11 were taken.
EXAMPLE. — Using the data given in the above example of this Section and 230 kilocycles as the stated value of fatigue life, the following estimates of the popu-
28 FATIGUE TESTING AND STATISTICAL ANALYSIS or DATA
lation value of per cent survival are obtained: (1) point estimate: 70 per cent and (2) interval estimate: 34 to 94 per cent.
A.larger sample size will give a shorter interval estimate (see Table 11).
(d) Fatigue Life for a Staled Value of Per Cent Survival:
1. Point Estimate.—A point estimate of the population value of fatigue life for a stated value of per cent survival is based on order statistics as
TABLE 1 1 .—CONFIDENCE INTERVALS FOR PERCENTAGES."
Confidence Level = 0.95.
Sample Size
ằ = s
Limits
#,per
cent Lower Upper 100 48 100
80 29 99
60 15 95
n = 10 Limits P, per
cent Lower Upper
100 68 100
90 54 100
80 43 98
70 34 94
60 25 89
50 18 82
P, per cent
100 95 90 85 80 75 70 65 60 55 50
ằ = 20 Limits Lower Upper
82 100 75 100 68 . 98
62 97
56 94
51 92
45 88
40 85
36 81
32 77
27 73
P, per cent 10097.5
9592.5 9087.5 8582.5 8077.5 7572.5 7067.5 6562.5 6057.5 5552.5 50
ằ = 40 Limits Lower Upper
91 100 87 100
83 99
79 98
77 97
73 96
70 94
67 93
64 91
61 89
58 87
56 85
53 83
51 82
48 79
46 77
43 *75 41 73
38 71
36 68
34 66
Where: p = sample percentage (for example, percentage surviving). Confidence lim- its corresponding to (100 — p) per cent are: lower: 100 — (tabular value for upper limit corresponding to p, per cent); upper: 100 — (tabular value for lower limit cor-' responding to p, per cent).0
Based on chart from Dixon and Massey (9), p. 415, and, for n = 40, on chart from Pearson and Hartley (2), p. 204.
outlined in the Section on S-N curves: "One Group at Each Stress Level"
(Section V A). A particular value is e median, corresponding to 50 per cent survival.
Another point estimate may be derived from the. cumulative frequency distribution of the observed values. In general, the two point estimates would not be exactly equal.
2. Confidence Interval Estimate.—Interval estimates for medians '(50 per cent survival) are described in Section V A2(a). Interval estimates for fatigue
ANALYSIS or FATIGUE DATA 29 life values corresponding to other percentage points may be computed by using reference (21) ,15
3. Tests of Significance:
(a) Differences of Group Medians—Single Stress Level.—-If two or more groups of specimens are tested, the question of whether the observed differ- ences in the values are due to chance or to some differences in the popula- tions from which the groups were drawn often arises. The observed differ- ences, for example, could arise because of differences in material lots or differences in the characteristics of the testing machines.
The rank tests given in this section assume that the several groups are in- dependently and randomly drawn from populations that are of the same shape but may differ with respect to their medians. All the observed values in one group are assumed to come from one population. Since the populations are assumed to be of the same (though unknown) shape, only those groups that are tested at the same stress level should be compared, since the form of the distribution tends to change with change in stress level.
1. Rank Test for Two Groups.—In the rank test for two groups the rank of each observation in the two groups combined is determined. The lowest value is given the rank of 1, the next higher observed value is given the rank of 2, and so forth. If one value appears several times, that is, there is a tie, the average of the ranks for those numbers is assigned to each one. For ex- ample, if the llth, 12th, 13th, and 14th values are all equal, they are each given the rank of (11 + 12 + 13 + 14)/4 = 12.5. The ranks for the two groups are totaled separately and the total for one of the groups (the one with the smaller number of observations if the group sizes are unequal) is compared with the critical values given in Table 26 for sample sizes equal to the group sizes.
If the observed value falls within the range of values given in Table 26 for the chosen significance level (5 or 1 per cent), the groups may be considered to have come from one population. If the observed value falls outside the range of values given in the table, the two groups are said to be significantly different, that is, to have come from two populations with different medians.
The use of the 1 per cent significance level gives a smaller risk of calling the
where
ô = 1 — (confidence level), and
p = (stated value of per cent survival)/100.
15 The interval Nk to Nm may be computed as follows:
(1) k is chosen so that
(2) m is chosen so that
30 FATIGUE TESTING AND STATISTICAL ANALYSIS or DATA
groups significantly different when they are actually drawn from one popu- lation and the observed difference is due to chance.
EXAMPLE.—To compare two machines, the rank test was applied to the data from 27 specimens randomly assigned to two testing machines. (See Table 12.) According to Table 26, the rank total for Machine A in Table 12, which has "the smaller number of measurements," should be between 101 and 179 (Ni = 10, NZ = 17) for the 5 per cent level of significance, and between 89 and 191 for the 1 per cent level of significance. This means that the actual total, 87, would not be expected to occur as often as once in a hundred samples due to chance alone, if the two machines were completely interchangeable. Thus, on the average, the machines give significantly different fatigue life values.
TABLE 12.—FATIGUE TEST DATA.
Rank 1 24 56 89 1018 24 87 Total
Machine A
Kilocycles 624662 681688 7329 9 774781 865948
Rank 37 1112 131 4 15.515.5 1719 2021 2223 2526 27 291 Total
Machine B
Kilocycles 667715 811822 833841 842842 849869 892903 944946 1 , 032 1,067 1,092
2. Rank Test for More than Two Groups. — The method of assigning ranks is the same as for the two-group test, ranking the observations for all the groups combined. The ranks are totaled separately for each group and the following test-statistic, H, is computed from the rank totals (22):
where:
k = number of groups,
Hi = number of observations in the ith group,
N = y^, ni, the number of observations in all groups combined, and Ri = sum of the ranks in the ith group.
The test-statistic H is distributed approximately as x2 with k — 1 de- grees of freedom if each Ui is at least five. (For a discussion of x2, see refer- ence (9)). Thus, the value of H calculated from the observed data may be
ANALYSIS or FATIGUE DATA 31 compared with the values of x2 given in Table 27 to determine whether there may be a significant difference among the populations from which the groups were drawn or not. If H is greater than the x2 value for k — 1 degrees of freedom and the chosen significance level, the populations are said to be different; that is, the groups may be said to have been drawn from two or more populations. Inspection of the rank totals will usually show which groups are different from the others if the difference is signifi- cant.
EXAMPLE.—To compare five machines, the rank test was applied to the data from 25 specimens, randomly assigned to the five machines (see Table 13).
TABLE 13.—FATIGUE TEST DATA.
Machines
Sum of ranks, R*
Bfm
A (5)596 (10)640 (11)646 (18)733 (24)807 Ri. . . 68
4624 924.8
B (6)599 (13)661 (21)760 (22)774 (23)781
85 7225 1445.0
c
(3)539 (12)651 (14)662 (15)675 (19)744
63 3969
793.8
D (2)530 (8)624 (9)638 (16)684 (25)889
60 3600 720.0
E (1)477 (4)568 (7)607 (17)719 (20)757
49 2401 480.2
Total
4363.8
H = 80.56 - 78 = 2.56
Entering Table 27 with 4 degrees of freedom, one less than the number of groups, gives x2 = °.49, corresponding to a 5 per cent significance level or a percentile of 95. Since the computed value of H, 2.56, is very much smaller than 9.49, the ob- served values of fatigue life may be considered to be from one population; the ma- chines may be considered to be interchangeable.
(ft) Differences of Two or More Percentages (for example, per cent survival values).—The test-statistic used to test the significance of the differences among percentage values computed from observed data is x2- The formula for x2 may be written in two ways; the second one is usually better for computation purposes.
1. When the sample sizes are unequal:
where:
k = number of samples;
sum over k samples;
where n = sample size and x = ^ Xi/k (23, pp. 175-178). The other terms were defined previously.
The computed value of x2 may be compared with the tabular values given in Table 27 for k — 1 degrees of freedom (d.f.). If the computed value of X2 is larger than the tabular value corresponding to: percentile = 100 — (chosen significance level), the percentages are said to be signifi- cantly different; that is, the samples were drawn from different populations.
If the computed value of x2 is smaller than the tabular value, the samples may be considered to have come from one population.
Another use of the x2 test is to test whether or not the observed per- centage values are significantly different from an arbitrary value, such as 50 per cent. The method of computation is the same as that given previously, except that: (1 ) the first way of writing the formula for x2 is used for the computations, (2) the arbitrary value, which may be called p', replaces p, and (3) d.f. = k.
EXAMPLE.—To compare six lots of phosphor-bronze strip, the x2 test was applied to the data given in Table 14, using a significance level16 of 10 per cent.
16 Significance levels commonly used are 10 and 5 per cent.
32 FATIGUE TESTING AND STATISTICAL ANALYSIS OF DATA
ằ,• = size of ith sample (i = 1, 2, • • • k);
Xi = observed number of "events" in the ith sample; an event may be a failure, a survival, etc. ;
pi = observed fraction for the ith sample: pi = #*/ô;; and P = Z_,Xi/£_,ni = average fraction for all samples combined.
2. When the sample sizes are equal the formula reduce to
TABLE 14.— PERCENTAGES SURVIVING 108 CYCLES.
Stress = ±25,000 psi.
Lot 21 34 56
Sample Size, nt- 2015 2517 1914
Per Cent Surviving, V)0pi
40.060.0 58.848.0 57.950.0
Number Sur- viving, Xi 98 1012 117
Xi*
ằi 5.403.20 5.885.76 6.373.50
Total 110 = 57 = 30.11 =