1. Normal Distribution:
It is assumed here, as well as in Section V B that the fatigue data can be transformed so that they will be approximately Normally distributed. A Normal distribution is assumed in all cases. Each sample is assumed to be drawn at random from its population.
2. S-N Curves:
Table 33 gives k factors for computing points on 75, 90, 95, 99, and 99.9 per cent survival curves for four values of confidence level, including 50 per cent, and for n = 3 to 25. The minimum number of specimens should in- crease as the per cent survival increases, but there is no definite criterion for choosing a particular group size except for the relative magnitudes of the k values. (Note that the rate of decrease is less as,ô increases.) The num- ber of specimens tested at each stress level can be smaller than the group sizes needed when the life distribution is not assumed (Section V A).
TABLE 2.—MINIMUM NUMBER OF SPECIMENS0 NEEDED FOR DE- TERMINING 95 PER CENT CONFI- DENCE INTERVALS OF STATED
WIDTH FOR A POPULATION
MEAN, /*.
Standard Deviation, a, Assumed Known.
Width of Interval
0 2<r 0. 4 0 60 8 1 0 1.21.4 1.61.8 2. 0
Confidence Number of Limits Specimens, n . . . X ± 0. 1 <7
± 0.2
=t 0.3
± 0.4
± 0.5
± 0.6
± 0.7
± 0 . 8
± 0.9
± 1 .0 38496
4324 1511 86 54
._Where: n =
E = width of interval x = sample mean.
"ASTM Designation E 122 (see foot- note 9).
TABLE 3.—MINIMUM NUMBER OF SPECIMENS" NEEDED FOR DE- TERMINING 95 PER CENT CONFI- DENCE INTERVALS OF STATED WIDTH FOR A POPULATION STAND- ARD DEVIATION, a.
Some Estimate of <r Available.
Width of Interval Number of Specimens, ằ
0 Based on Fig. 1 of Greenwood and Sandomire (13).
* The values of re given in Table 30.
3. Confidence Intervals:
For the Mean.—If a good estimate of the population standard deviation, 0, is available, Table 2 gives the minimum number of specimens needed for confidence intervals of stated width for the mean, p, of the population.
If the sample is used to estimate a as well as /*, the sample sizes should be
0.14<7 385
0.2 190
0.3 84
0.4 47
0.5 30
0.6 21
0.7 16
0.8 13
0.9 10
1 .0 8
2
20 FATIGUE TESTING AND STATISTICAL ANALYSIS or DATA
larger, since ^0.975 values from Table 29 should be used instead of the 1.96 in the equation for n (Table 2).
For the Standard Demotion.—In order to find the minimum number of specimens needed for determining confidence intervals of stated width for the standard deviation, <r, of a population, some estimate of a must be avail- able, since the width of the interval is measured in units of a. However, Table 3 can be used as a guide even if no good estimate of a is available.
For example, if n = 8, the sample-standard deviation, used to estimate the population standard deviation, may be above or below a by 0.5 <r, whereas an estimate based on n = 30 will not be expected to deviate from the true value by more than 0.25<r.
TABLE 4.—MINIMUM NUMBER OF SPECIMENS" NEEDED TO DE- TECT IF THE STANDARD DEVIA- TION OF A POPULATION IS A STATED PERCENTAGE OF A FIXED VALUE.
Percentage of Fixed Value
4045 5055 6 065 7075
Number of Specimens, n Chance80%
Detectionof 78 129 1520 4228
Chance90%
Detectionof 89 1214 1926 3855
TABLE 5.—MINIMUM NUMBER OF SPECIMENS0 NEEDED IN EACH SAMPLE TO DETECT IF A STAND- ARD DEVIATION OF ONE POPULA- TION IS A STATED MULTIPLE OF THE STANDARD DEVIATION OF ANOTHER POPULATION.
Multiple
1 .5.
2.52.0 3.03 .5 4.0
Number of Specimens, ô Chance80%
Detectionof 3915
97 65
Chance90%
Detectionof 5220 139 87
a Based on Fig. 2 from Ferris et al (14). 0 Based on Fig. 3 from Ferris et al ằ(14).
4. Tests of Significance:
Difference Between Two Standard Deviations.—The sample sizes for testing the difference between two means are given in Tables 6 and 7. In some cases, the principal interest is in the difference between standard deviations.
1. One Standard Deviation a Fixed Value.—If one standard deviation is a fixed value—for example, the long-time standard deviation of data based upon an old procedure—and if the other standard deviation is to be com- puted from data based upon a new procedure that may reduce the varia- bility, Table 4 gives the minimum number of specimens needed to detect a reduction of a stated amount. These e sizes apply when the observed standard deviation, s, for the new procedure is indeed smaller than the fixed value, and the ratio s2/(fixed value)2 is compared with 1 /F0.95, corresponding to °o and n — 1 degrees of freedom for numerator and denominator re- spectively. (See Section V B4(a) and Table 32.)
2. Two Sample Standard Deviations.—If the problem is to test whether the variability of procedure 1, say, is greater than the variability of pro- cedure 2 (the numbers having been assigned prior to taking the data), Ta-
NUMBER or SPECIMENS AND THEIR SELECTION 21 ble 5 gives the minimum number of specimens needed in each sample to detect that si is a stated multiple of s%. If the observed value of si is indeed larger than the observed value of sz, compare s?/s<? with ^0.95 corresponding to (HI — 1) degrees of freedom for numerator and denominator (since n\ = HZ). (See Section V B4(a) and Table 32.) In this case it is not correct to make the test if s22 is greater than Si2.
Difference Between Two Means:
1. One Mean a Fixed Valise.—If one mean is a fixed value—for example, the long-time mean of data based on an old procedure or a commonly used material—and the other mean is to be computed from data based upon a new
TABLE 6.—MINIMUM NUMBER OF SPECIMENS0 NEEDED TO DETECT A STATED DIFFERENCE BETWEEN A MEAN AND A FIXED VALUE.
a = Unknown Standard Deviation of the Population Being Estimated.
Difference
00 11 11 2 2
50<r 7500 2550 7500 50
Number of Specimens, ằ Chance80%
Detectionof
3416 107 65 43
Chance90%
Detectionof
4421 139 76 54
TABLE 7.—MINIMUM NUMBER OF SPECIMENS0 NEEDED TO DETECT A STATED DIFFERENCE BETWEEN THE MEANS OF TWO POPULATIONS.
<7 = Unknown Standard Deviation of Each Population; <n = az •
Difference
00 11 11 22
50<r 75 0025 5075 0050
Number of Specimens, n 80% Chance
of Detection 6429 1712 79 64
90% Chance of Detection
8639 2315 119 75
0 Taken from Table E.I of reference
0 Taken from Table E of reference (11). (11) •
procedure that may shift the mean, Table 6 gives the minimum number of specimens needed to detect a shift in either direction, measured in terms of the population standard deviation of the new procedure. These sample sizes apply when the computed value of
is compared with £0.975 in Table 29. e Section V B4(6).) No F-ratio test is needed.
2. Two Sample Means.—The minimum number of specimens needed in each sample to detect a difference in two population means, stated as a multiple of their equal universe standard deviations, is given in Table 7.
These sample sizes apply when (1) the two sample standard deviations are not significantly different and (2) the computed value of / (see Section V B4(6)) is compared with /0.9?5 in Table 29.
22 FATIGUE TESTING AND STATISTICAL ANALYSIS or DATA V. ANALYSIS OF FATIGUE DATA
A bask concept of statistics is that a group of one or more specimens is a sample taken from a larger body or population. Such a sample is considered to be just one of a "number," often very large, of samples that could have been taken. The sampling procedure used delimits the-population being estimated. The results obtained from tests on a random sample from the population can be used to estimate the characteristics of the whole popula- tion and to measure the precision of the estimates.
In the case of fatigue tests the data observed are usually the lives of specimens tested at a constant applied stress (strain or deflection) amplitude.
Since the cycle life varies from specimen to specimen, this measurable char-
FIG. 6.—"Normal" or Gaussian Distribution Curve.
acteristic is not a fixed value and is best described by a frequency distribu- tion. The graphical presentation of the distribution of cycle lives for the population of specimens that have lives between certain limits is known as a frequency distribution curve. Such a distribution curve may be estimated from the raw test data or from transformed test data, that is, either from values of N or from values of log N, log log N, N1'2, and so forth.
When the frequency distribution curve has a particular kind of bell shape, as shown in Fig. 6, the data are said o have a "Normal" or Gaussian dis- tribution. This Normal probability distribution curve, f(x), is represented by the equation:
The constants in the formula are the population mean, /*, and <r, the popula-
ANALYSIS or FATIGUE DATA 23 tion standard deviation (a measure of the dispersion).11 It should be empha- sized that values of the parameters of the population can only be estimated from tests on the specimens in the sample; to obtain exact values would re- quire that the total population be tested.
While some fatigue tests, particularly those made in the finite life range of an S-N curve, may yield approximately Normal distributions of cycle life, generally a transformation to log cycle life is required. Others do not yield Normal distributions, even after various transformations are performed on the data. This is particularly true in the case of tests made at applied stresses near the fatigue limit where runouts are observed. Hence, other distributions, such as the Weibull distribution,12 the "extreme value" dis- tribution with and without lower limits, as used by Freudenthal and Gumbel (IS), and other distributions, that are just as normal in the usual sense, as the Normal or Gaussian distribution, have been applied to the analysis of fatigue data. While references to some ,of these distributions are included in this Guide, analysis of the fatigue data has been confined mostly to methods that require no assumptions of distribution shape or to the methods based upon the assumption that the raw data or the transformed data have a Normal distribution.
As stated previously, however, any set of observations to which these statistical methods are applied is assumed to come from a random sample from the population of interest. If a series of samples is drawn, procedures for testing for statistical control are given in the ASTM Manual on Quality Control of Materials (see footnote 5). Lack of statistical control in data in- dicates that the series of samples does not come from the same population.