HAIR BREAKAGE: REPEATED GROOMING EXPERIMENTS 447 In this instance, we obtain a characteristic lifetime (α) of 55.2 million grooming strokes and a shape parameter (β) of 0.48. That is, one may predict that 55.2 million grooming strokes would be required in order to break 63.2% of the fi bers. Meanwhile, a shape parameter less than 1 is indicative of a process wherein the highest rate of breakage occurs early in the experiment. Of course, there is danger in ascribing too much signifi cance to the long-range extrapolation that yields the magnitude of the characteristic lifetime. Specifi cally, a prediction involving the outcome after tens of millions of cycles, based on an experiment involving a few thousand cycles, is obviously dubious. Instead, it is em- phasized that together these two Weibull parameters describe the collected data, and, as outlined earlier, can be used to generate a survival probability plot that predicts the Table II Grouped Weibull Analysis of Repeated Brushing Data for Virgin Caucasian Hair at 60% RH Grooming cycles No. of failures Cumulative frequency Median rank 1/(1-Median rank) ln(ln(1/(1-Median rank) Ln (grooming cycles) 1000 110 110 0.00439 1.004407 -5.4267 6.907755 2000 38 148 0.00591 1.005943 -5.1285 7.600902 3000 26 174 0.00695 1.006996 -4.96583 8.006368 4000 19 193 0.00771 1.007768 -4.86165 8.29405 5000 28 221 0.00883 1.008906 -4.72541 8.517193 6000 25 246 0.00983 1.009925 -4.6176 8.699515 7000 25 271 0.0108 1.010946 -4.5202 8.853665 8000 16 287 0.0115 1.011601 -4.46245 8.987197 9000 24 311 0.0124 1.012584 -4.38157 9.10498 10000 15 326 0.0130 1.0132 -4.33412 9.21034 Figure 5. Weibull plot for repeated brushing data of virgin Caucasian hair at 60% RH.

JOURNAL OF COSMETIC SCIENCE 448 likelihood of fi ber breakage as a function of brushing strokes over the range of the ex- periment (see Figure 6). These results lead us to predict that, under these conditions, it is necessary to brush hair approximately 1,000 times to produce a 0.5% likelihood of fi ber breakage and around 3,500 times to produce a 1% likelihood of breakage. These probabilities sound low, at least until projecting the results onto actual heads, where it is generally accepted that around 100,000–150,000 fi bers are typically present. As such, using 100,000 fi bers as a round number, we obtain the prediction that 500 broken fi bers would be expected after 1,000 grooming strokes and 1,000 broken fi bers after 3,500 strokes. These numbers no longer sound so trivial, which is one reason why it is prudent to use conditioning products. As depicted in Figure 4, use of the exact same testing procedure to generate data for con- ditioned hair gives rise to considerably less breakage—and applying the same Weibull analysis yields a characteristic lifetime of 1.04 billion grooming cycles and a shape param- eter of 0.43. Again we highlight the caveat regarding long-range extrapolation and the magnitude of the characteristic lifetime. Thus, one should not directly compare the enor- mousness of this parameter for the treated and untreated hair and conclude a 20-times improvement in fatigue resistance. Instead, Figure 7 compares survival probability plots for conditioned and unconditioned hair, and demonstrates the benefi ts associated with such products in statistical terms. These results predict that around 5,000 grooming strokes (i.e., fi ve times as many) are now required to produce a 0.5% likelihood of breakage in conditioned hair, while approximately 25,000 strokes are necessary to include a 1% chance of breakage. It is worth spending a moment to consider typical grooming habits and practices and their relationship to the number of grooming cycles used in these experiments. If we as- sume that ten grooming strokes are employed each time a consumer brushes her hair— and that three such grooming experiences are performed each day—then 1,000 grooming strokes are attained in slightly over a month. Obviously, widely different habits and prac- tices exist, but 10,000 grooming strokes by no means sounds excessive over the lifetime of a hair fi ber, with long hair potentially seeing signifi cantly higher repetitions. Figure 6. Survival probability plot for brushing virgin Caucasian hair at 60% RH.