are numerically greater than 3 for all pairs of values of (λ, κ) lying
above the line
27λ - 6κ + 1 = 0
As statistically promising within the discriminant of the cubic we then
have the shaded area of the (λ, κ) plane.
=The Origin.= The generating curve is the symmetrical normal
probability curve with origin at its center. Since x = 0 when t = 0,
the origin of the translated curve coincides with that of the base or
generating curve. The translated curve may not be symmetrical so that
the mean ordinate may not coincide with the modal ordinate. Because of
the relation between corresponding areas the ordinate at the origin
must continue to divide the area under the curve into equal parts, that
is, the origin and median always coincide.
=Determination of the Constants.= Since the exact position of
the median can not ordinarily be determined by inspection or direct
computation there are in reality four constants to be determined: the
distance between the median and the mean, a, κ and λ.
In determining the constants it is usual to compute the value of the
first four moments. The third and fourth moments are extensions of
the idea of the well known formulas for the first and second moments.
Denoting the moments about the median by μ, we have
1 ⌠+∞
μ_{1}^´ = ——— │ xy dx
N ⌡-∞
1 ⌠+∞
μ_{2}^´ = ——— │ x^{2}y dx
N ⌡-∞
1 ⌠+∞
μ_{3}^´ = ——— │ x^{3}y dx
N ⌡-∞
1 ⌠+∞
μ_{4}^´ = ——— │ x^{4}y dx
N ⌡-∞
where N is the total area under the curve.
The values of the μ’s are computed from the data[8] and equated to the
corresponding integrals which of course involve the four constants.
In this way four equations are obtained from which the values of the
constants may be determined. Since it is our present object to discuss
the solution only of these equations, merely the principal results will
be given.
The general form for the moments about the median of the
area under the translated curve is
1 ⌠+∞
μ_{n}^´ = ——— │ x^{n}y dx
N ⌡-∞
1 ⌠+∞ a^n(t + κt^2 + λt^3)^n
= —————— │ ———————————————————————— e^{-t^2/2} a(1 + 2κt + 3λt^2) dt
√(2π)N ⌡-∞ a(1 + 2κt + 3λt^2)
1 ⌠+∞
= —————— │ a^n(t + κt^2 + λt^3)^ne^{-t^2/2} dt.
√(2π)N ⌡-∞
On applying the two well known formulas:
⌠+∞
│ x^{2n + 1}e^{-x^2} dx = 0
⌡-∞
⌠+∞ 2n + 1 ⌠+∞
│ x^{2n + 2}e^{-x^2} dx = —————— │ x^{2n}e^{-x^2} dx,
⌡-∞ 2 ⌡-∞
the determination of μ_1´, μ_2´, μ_3´ and μ_4´ is reduced to a matter
of algebraic detail. Then on transferring to the arithmetic mean as
origin the values of μ_2, μ_3, and μ_4 can be determined in terms
of a, κ and λ. It is most convenient however, to make use of the
quantities β_1 = μ_3^2/μ_2^3 and β_2 = μ_4/μ_2^2 or rather β = β_1/8
and ϵ = (β_2 - 3)/12 and express the constants in terms of these
quantities. It is to be noted that both ϵ and β are zero for a normal
distribution, that is, for λ = κ = 0.
Omitting the detailed reduction[9] which is straightforward and direct,
we have
(1) μ´ = aκ
(2) μ_2 = a^2(1 + 6x + 15x^2 + 2κ^2)
2κ^2(2κ^2 + Q)^2
(3) β = ——————————————————————
(2κ^2 + R)^3
4κ^4 + 4κ^2S + T
(4) ϵ = ——————————————————————
(2κ^2 + R)^2
where the symbols, S, R, Q and T are defined as follows:
S = 1 + 18λ + 90λ^2,
R = 1 + 6λ + 15λ^2,
Q = 1.5 + 18λ + 135/2λ^2,
T = 2λ + 36λ^2 + 270λ^3 + 810λ^4.
Obviously no algebraic solution can be obtained from equations (3) and
(4) for κ and λ in terms of the computed values β and ϵ, and hence a
resort to tables is necessary. The values of β and ϵ for values of κ
from 0 to 0.0335 and of λ from -0.040 to +0.100 have been computed.[10]
The process of determining the constants of the translated normal curve
consists first in computing β and ϵ from the given data, and then in
entering the table and interpolating for the corresponding values of
κ and λ.[11] On substituting these values in (2) the value of a can be
found and thence on multiplying a by κ the position of the median of
the distribution is obtained.
The sign of κ is determined by the sign of the third moment about the
mean μ_3, that is, by the direction of the skewness or asymmetry. For
positive skewness the mean must lie to the right of the median and
hence μ_1´, the first moment about the mean, must be positive which
necessitates a positive sign for κ. Therefore, the sign of κ is the
same as that of the skewness.
To fit a curve to the given data, after the constants have been
determined it is necessary to find, by solving a cubic equation for
each value, the values of t corresponding to the x’s of the respective
classes. The cubic is
aλt^3 + aκt^2 + at - x = 0.
Any of the various methods of approximating to the solution of a cubic
may be used in solving these equations.
The area of each class can now be obtained by computing the
corresponding areas under the standard normal curve from a table of the
probability integral.
=The Method of Interpolation.= The actual fitting of the curve can
now be readily accomplished.[12] The distinctively geometrical operation
is the interpolation for the values of λ and κ for a given pair of
values of β and ϵ.
Within the limits of the table[13] the curves resulting from the
assignment of a constant value to β are practically straight lines,
β = 0 is the λ-axis; β = 1 is a line parallel to the λ-axis.
Hence we may safely assume that the variation from one column
to the next and from one line to the next is linear for
values of β. That is, ordinary first difference interpolation
methods are applicable.
As regards the system of ϵ curves we have for instance ϵ = .128 at (λ
= .050, κ = 0); again, at approximately (.045, .060) and (.40, .085).
We are therefore warranted in assuming the applicability of first
difference methods to interpolation between the ϵ curves.
As an illustration let us find the values of λ and κ for ϵ = 0.112 and
β = 0.044. On inspection of the table it is seen that λ lies between
0.30 and .035 and κ between .090 and .095. When κ = .090, λ = .033 for
ϵ = .112. When κ = .095, λ = .031 for ϵ = .112. For β = .042 and
κ = .090, λ = .033 and for β = .046 and κ = .095, λ = .031, ϵ = .112 in
each case. Hence, to first differences, λ = .032 and κ = .093 for
ϵ = .112 and β = .044. For interpolation in parts of the table showing
more rapid variations appropriate methods will suggest themselves.
Taken geometrically the table represents two distinct systems of
curves, with each curve of one system intersecting all the curves of
the other system. Therefore, a pair of values for λ and κ can always be
found for values of ϵ and β within the range of the table.
Department of Mathematics, Ohio State University.
TABLE OF ϵ AND β.
(ϵ is the first and β the second number of each pair.)
λ
───┬────────────────────────────────────────────
κ │-040 -035 -030 -025 -020 -015 -010 -005 000
───┴────────────────────────────────────────────
000 -061 -056 -050 -043 -035 -027 -019 -010 000
000 000 000 000 000 000 000 000 000
005 -055 -049 -042 -035 -027 -019 -010 000
000 000 000 000 000 000 000 000
010 -055 -049 -042 -035 -027 -018 -010 000
000 000 000 000 000 000 000 001
015 -049 -042 -035 -027 -018 -009 001
001 001 001 001 001 001 001
020 -048 -041 -034 -026 -017 -008 002
001 002 002 002 002 002 002
025 -047 -040 -033 -025 -016 -007 003
002 002 002 003 003 003 003
030 -046 -039 -032 -024 -015 -006 004
003 003 004 004 004 004 004
035 -045 -038 -031 -023 -014 -005 005
004 005 005 005 005 005 006
040 -037 -030 -022 -013 -004 006
006 006 007 007 007 007
045 -036 -028 -020 -011 -002 008
008 008 008 009 009 009
050 -034 -026 -018 -009 -000 010
009 010 010 011 011 011
055 -032 -024 -016 -007 002 012
011 012 012 013 013 013
060 -022 -014 -005 004 014
014 015 015 016 016
065 -020 -012 -003 006 017
016 017 018 018 019
070 -018 -009 000 009 019
019 020 020 021 022
075 -015 -007 002 012 022
022 023 023 024 025
080 -013 -004 005 015 025
025 026 026 027 028
085 -001 008 018 028
029 030 031 032
090 002 011 021 032
032 033 034 036
095 005 015 025 035
036 037 038 039
100 009 018 028 039
039 041 042 044
(ϵ is the first and β the second number of each pair.)
λ
───┬────────────────────────────────────────
κ │ 005 010 015 020 025 030 035 040 045 050
───┴────────────────────────────────────────
000 010 021 033 045 057 071 084 098 113 128
000 000 000 000 000 000 000 000 000 000
005 010 021 033 045 057 071 084 098 113 128
000 000 000 000 000 000 000 000 000 000
010 011 022 033 045 058 071 085 099 113 128
001 001 001 001 001 001 001 001 001 001
015 011 022 034 046 058 071 085 099 114 129
001 001 001 001 001 001 001 001 001 001
020 012 023 034 046 059 072 086 100 115 130
002 002 002 002 002 002 002 002 002 002
025 013 024 035 047 060 073 087 101 116 131
003 003 003 003 003 003 003 003 003 003
030 014 025 036 049 061 074 088 102 117 132
004 004 004 005 005 005 005 005 005 005
035 015 026 038 050 063 076 089 104 118 133
006 006 006 006 006 006 007 007 007 007
040 017 028 039 052 064 077 091 105 120 135
007 008 008 008 008 008 008 009 009 009
045 019 030 041 053 066 079 093 107 122 137
009 010 010 010 010 011 011 011 011 011
050 021 032 043 055 068 081 095 109 124 139
012 012 012 012 013 013 013 013 014 014
055 023 034 045 057 070 083 097 111 126 141
014 014 015 015 015 016 016 016 016 017
060 025 036 048 060 073 086 100 114 129 144
017 017 017 018 018 019 019 019 019 020
065 028 039 050 062 075 089 102 116 131 146
019 020 020 021 021 022 022 022 023 023
070 030 041 053 065 078 091 105 119 134 149
022 023 024 024 025 026 026 026 026 027
075 033 044 056 068 081 094 108 122 137 152
026 026 027 028 028 029 029 030 030 031
080 036 047 059 071 084 097 111 125 140 155
029 030 031 031 032 033 033 034 034 035
085 039 050 062 075 088 101 115 129 144 159
033 034 034 035 036 037 037 038 039 039
090 043 054 066 079 092 105 119 133 148 163
037 038 039 039 040 041 042 042 043 044
095 046 058 070 083 096 109 123 137 152 167
041 042 043 044 045 046 046 047 048 049
100 050 062 074 087 100 113 127 141 156 171
045 046 047 048 049 050 051 052 053 054
[6] Pearson, Karl:—“Skew Variation in Homogeneous Material;” Phil.
Trans. 1895, Vol. CLXXXVI, A, pp. 253 et seq.
“On the Systematic Fitting of Curves to Observations and Measurements,”
Biometrika, I, pp. 265 et seq. and Biometrika II, pp. 1 et seq.
Elderton:—“Frequency Curves and Correlation,” pp. 1-105; C. & E.
Layton, 1906.
[7] Edgerton, F. Y.:—“On the Representation of Statistics by Means of
Analytical Geometry,” Jour. Roy. Stat. Soc., 1914, Feb., Mar., May,
June and July.
[8] Elderton, 1. c.
[9] Compare Edgeworth, “A Method of Representing Statistics by
Analytical Geometry,” Proceedings Fifth International Congress of
Mathematicians, Cambridge, 1912.
[10] Only a part of the original table appears in the accompanying
table. The original values were computed to four places of decimals,
but three place numbers are sufficient to illustrate the method of
approximating to the solution.
[11] Compare “Tables for Statisticians and Biometricians,” Cambridge
University Press, 1914.
[12] For the statistical details see Elderton, 1. c.
[13] As may be seen on examining the Table.
A PRELIMINARY LIST OF THE JASSOIDEA OF MISSOURI
WITH NOTES ON SPECIES.
By EDMUND H. GIBSON and ERIC S. COGAN,
U. S. Bureau of Entomology.
The following preliminary list of the Jassoidae of Missouri is mainly
the result of collections and notes made by the authors during the
summer months of 1915. On account of the lack of records for this
state the authors were prompted to undertake such a survey. As far
as possible collections were made so as to embrace all conditions in
different sections, giving some attention to ecological relations. The
list comprises some 98 species.