in a suitable manner, analytic functions are fitted 

 to the data, and tests of significance are applied 

 to determine whether certain statistical predic- 

 tions and conclusions can be reached with con- 

 fidence on the basis of the available data. 



Some basic terms used in statistical work are 

 defined in Table 1. A distribution function indi- 

 cates the relative frequency or probability of oc- 

 currence of a particular event. The best known 

 type of distribution is the "normal" distribution. 

 It is theoretically possible, but not often prac- 

 ticable, to convert distributions which are not 

 "normal" into a normal distribution by a change 

 of variables. One such conversion which will be 

 utilized here is to take the logarithm of the vari- 

 able. Thus the "log-normal" distribution is ob- 

 tained, which signifies that the logarithm of the 

 variable is normally distributed. Commercial 

 chart paper (probability chart) is available which 

 is designed so that the integral of the normal or 

 log-normal distribution will plot as a straight line 

 on the chart. The log-normal law has been 

 found applicable in various diversified fields. 

 See, for example, the extensive bibliography given 

 by V. T. Chow (5). 



The probability P{x) takes on definite values for 

 a given population. An estimate of P{x) is often 

 desired on the basis of samples of the members 

 of the population. Statistics provide the tools for 

 making such estimates. It will be assumed that 

 the sample is taken from a population for which 

 the quantity under discussion is defined by a 

 certain analytical expression. The reliability 

 of this hypothesis is evaluated by the application 

 of statistical tests of significance which enable one 

 to estimate the expected variation of the measured 

 data from the analytically defined values, if the 

 hypothesis is true. 



In order to determine whether the variation of 

 the sample data from the assumed distribution is 

 consistent with the hypothesis, use will be made 

 of a theorem in statistics (6) which states that 

 the sample quantile* Xp is asymptotically normal 

 with mean value ^ and standard deviation 



u 



Til - P) 



N 



where N is the size of the sample, p the probability 

 density, P the cumulative probability at x = Xp 

 and ^ is the corresponding quantile of the popu- 

 lation from which the sample is drawn. Other- 

 wise stated: If a sample of TV- values is chosen 

 at random and if these values are numbered in the 

 order of increasing magnitude, then the magni- 



s A quantile denotes the variate corresponding to a given fraction; 

 i.e., the 0.40 quantile is that value of the variate below which 40 

 per cent of all values lie. 



tude of any particular value, say the rth cannot 

 be predicted in advance of the drawing of the 

 sample; there exists a certain probability function. 

 If TV is large, and if neither r/N nor (TV — r)/TV is 

 too small, then the probability function for the rth 

 value is approximately normal in form. If TV is 

 increased and if r is also increased so as to keep 

 r/N as nearly constant as possible (as for a fixed 

 quantile of the sample) then the approximation 

 to normality increases, becoming perfect in the 

 limit as TV — »- oo ; the width of the probability 

 spread simultaneously decreases, becoming zero 

 in the limit. 



If a curve is plotted on each side of the analytic 

 cumulative distribution curve at a distance of 

 1.65 standard deviation of the distribution of the 

 corresponding quantiles, then on the average nine 

 tenths of the observed quantiles would be ex- 

 pected to fall between these limits and 10 per cent 

 of the measured quantiles would be expected to 

 fall outside these limits. This control or confi- 

 dence curve is therefore designated as the 90 

 per cent significance leveF and indicates the 

 probability of a random value falling within the 

 limits when the hypothesis is true. In order for 

 the deviation between the hypothetical distribu- 

 tion and the sample values to be considered not 

 significant at this level, it would be necessary to 

 have less than 10 per cent of a large number of 

 observed values fall outside these limits. In 

 practice it has been found that wrong conclusions 

 will seldom be drawn if the level of significance is 

 set at 90 per cent or less; that is, the hypothesis is 

 probably true if the deviations are no greater than 

 could be expected, on the average, in one out of 

 10 tests. On the other hand if the deviations he 

 outside the 99 per cent limits, that is, if they are 

 larger than would be expected once in a hundred 

 similar tests, then the hypothesis should be re- 

 jected confidently. A distribution may be of 

 practical significance even though the statistical 

 tests indicate a significant deviation between it 

 and the measured data. 



In many cases a visual inspection of the scatter 

 is sufficient to indicate that the assumed distri- 

 bution is an acceptable representation of the actual 

 distribution. The types of basic distributions 

 that will be considered here are the normal, log- 

 normal, and the Rayleigh distributions. 



The normal distribution of x is defined as fol- 

 lows: 



p{x) = 



a a/2t 



— 00 < X < + 



' Some statisticians prefer to label the limits according to the 

 probability with which a random measurement would fall without 

 the limits; i.e.. the 90 per cent significance limit would then be re- 

 ferred to as the 10 per cent level. 



