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0.0198.
:d percentages in selected excellent agreement (see
Pearson Distributions
approximation for a given variate, based on knowledge or estimates of
fii and ps. The expressions for the probability density functions for the
various Pearson distributions are given in References 6-7, 6-9 and 6-10,
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nr TTT'.p
. 00500 0.0550 to to
) 0.0549 0.0599
*f
00600 or WC
o'...*
o ' I Ad
observations for Johnson Sf
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k j ) X. IM
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s 'V
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tho*se proposed by Kadfj|I be generated as asolution If
ty fhnction/(as) by propet gf The solution ofthiseqia*~ lira, including the nornuL * ype III) distributions <&g i the (fix, plane con*? hown in Fig. 6-M. TMf ates the wide diversity if to select the appropnr
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Fig. 6-11 Region in (px,pit plane for various type Pearson distributions. Letters
U and J denote U-shaped and J-shaped distributions. (From E. S. Pearson, Seminars,
Princeton University, 1960.)
The descriptions of the procedures for fitting Pearson distributions to data are lengthy, since each family requires solution of a different set of equations. The underlying principles are reviewed in Reference 6-2 and formulae for each family are given in Reference 6-7. These expressions, together with those for estimating the first four moments and and fi3, **y be used to develop a computer program to fit a Pearson distribution
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Fig. 7
The following work illustrates the use of the present table in deriving an upper 1 per cent point for a Pearson curve having the correct first four moments. We first extract the following entries from Table 32:
^ 14 0-4 3-3297 6-6 3-3239 6-8 3-3177 7-0 3-3112
3* -4 --3
1-0 3*
3-4010 3-3980 -10 3-3894 -5 3-3823
1-7 3*
3-4748 3-4710 -21 3-4051 --12 3-4580
1-8
3-5435 3-5452 3-5426 3-6370
3* -44 -28
Interpolating in each column at fia -- 6-7905, using the simple Bessel formula (319), we find:
VA X
3*
1-5 3-3180 1-0 3-3897 40 1-7 3-4854 18 1-8 3-5427
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(6) the standardised values for the terminal or terminals (0 and 100 per cent points), when
the curve is of finite range;
(c) seven additional percentage points;
(d) a much more extensive range of /?2 values.
Tables 31 and 33, based on D. E. Amos's unpublished computations, extend information
into the J-curve area (see Fig. 7) of the Type I or beta distributions. These two tables will be
referred to later in 19 in comparison with the Johnson SB curves having
/?2 values in
this area.
The earlier table tookfix as argument but it was realized that for interpolation purposes it
was better to use *fPv
The curves of Karl Pearson's system are all derived as solutions of the differential equation
\dy -- fa+x) ydx Cq +CiX+CzX2'
*'
where the origin is at the mean and the parameters c0, cx and c2 are functions of *J/i2 -- cr, fix and /?2. Using a standard notation (see Elderton & Johnson, 1969) the main curves of the system may be expressed as follows:
Type 1
II III IV
V VI
VII
Equation
y=
y = y,xme-x>a y= y = y0x-me~',,`
/ aj\"OT y = y0^w,(i+"j
Origin for x At start
Mean (= mode) At start Mean+d* At start At start
Mean (= mode)
Limits for as
0<*<o
--a^x^a 0 ^x ^ co -- oo < * < 00 0 < x < co 0<z<op
-- 00 < < co
f
+
s
ii s
* d ~ Jo6/(--1).
The regions and their boundaries in the
field With which the different types of solution
are associated have been illustrated in Fig. 7. Because all types derive from the solution of
(168), their shapes may be described as changing continuously across the boundaries of the
regions, although the mathematical forms differ.
17.2 Illustrations of use of Table 32
To derive the fall benefit from Table 32, it will generally be necessary to use second difference
interpolation formulae for both
aQd fit; the procedure needed lends itself readily to a
computer routine. In one example below we illustrate, however, the steps needed in computing
with a desk calculator.
Example 48 (taken from Johnson et al. 1963, pp, 464r~5). Stephens (1963) derived the first
four moments of the distribution of G. S. Watson's goodness of fit statistic, U%. For the case
ofa sample ofsize N *= 10, the fpffowmg results are derived from lus expressions (equation (4)):
mean = 0-08333, s .d . = 0-05000,
- 1-6190, = 6-7905.
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