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11.10 Probability Paper 375
magnitude of an array generally offers the most helpful concept of central tendency.
Unlike the median magnitude of an array, which is positionally central irrespective of
the magnitudes of the individual observations composing the array, and unlike the mode
of the array, which is the magnitude having the highest frequency of observations com-
posing the array, the arithmetic mean is a collective function of the magnitudes of all
component observations.

11.9.2 Variability and Skewness
The tendency of a bell-shaped curve to deviate by larger and larger amounts from the cen-
tral magnitude of an array with less and less frequency is a measure of its variability, vari-
ation, deviation, dispersion, or scatter. The tendency of a bell-shaped curve to be asymmet-
rical is a measure of its skewness.
Mathematically, the symmetrical bell-shaped normal frequency curve of Appendix 5
is described fully by two parameters: (a) the arithmetic mean, m, as a measure of central
tendency, and (b) the standard deviation, s, as a measure of dispersion, deviation, or vari-
ation. The equation of the curve, which is referred to as the Gaussian or normal probabil-
ity curve, is

1 x - s 2
1
y>n = F(x)>n = f(x) = c dexpe - > c d f (11.15)
2
s12p s
where y is the number or frequency of observations of magnitude x deviating from the
mean magnitude by x m, n is the total number of observations, s is the standard
deviation from the mean, F(x) symbolizes a function of x, and m is the mathematical
constant 3.1416.
As shown in Appendix 5 and calculated from Eq. 11.15, the origin of the coordinate
system of the Gaussian curve lies at x m 0 where the frequency is y = n>(s12p),
and the distance from the origin to the points of inflection of the curve is the standard
2
deviation s = 2©(x - m) >n. This presupposes that deviations are referenced to the true
mean m, whereas calculations must be based on the observed mean x =©x>n, which is
expected to deviate from the true mean by a measurable although small amount.
However, it an be shown that when the observed standard deviation is calculated as
2
s = 2©(x - m) >(n - 1), it closely approximates s. As n becomes large, there is in-
deed little difference between s and s as well as between and m, and the Greek letters
x
are often used even when, strictly speaking, the Latin ones should be. The ratios y>n and
(x m)/s generalize the equation of the normal frequency function.
Engineers are usually interested in the expected frequencies of observations below or
above a given value of x or falling between two given values of x. The magnitudes of areas
under the normal curve provide this useful information better than do point frequencies.
Appendix 5 is a generalized table of these areas. It is referred to as a probability integral
table because a given area is a measure of the expected fraction of the total area (as unity)
corresponding to different departures from the mean (x m) in terms of the standard devi-
ation s, that is, (x m)>s t.

11.10 PROBABILITY PAPER

As suggested by Allen Hazen, the probability integral (Appendix 5) can be used to de-
velop a system of coordinates on which normal frequency distributions plot as straight
lines. The companion to the probability scale can be (a) arithmetic for true Gaussian
normality, (b) logarithmic for geometric normality, or (c) some other function of the