NOV 4; 1945ET-2e5
United States Department of Agriculture
Agricultural Research Administration
Bureau of Entomology and Plant Quarantine
A MM0D OF ESTIMATING BUT LZAFlPPE POPULATION
FROM MM PROPORTICM OF IJNIIFTED PLANTS _/
By M. F. Bowen, 2/
Division of Track Crop and Garden Insect Investigations
In the course of ecological studies of the beet leafhopper
(Eutettix tenellus (Bak.)), a large number of population counts
vere taken in beet fields within the range of this insect. The pur-
pose of these counts was to determine the time and magnitude of the
spring movement from the breeding areas to the beet fields, and also
to follow the trend of populations during the period of the spring
Statistical analyses of these data demonstrated that the popup.
lations that move Into the beet fields are distributed in a Poisson
series. This indicated that the mean number of leafhoppers per
plant can be estimated from the proportion of plants not infested,
or from the portion of plants infested with a given mvnber of the
Insects. Since it appeared that this method might be more rapid
than the conventional method, a study was made of its reliability
and It practical application.
In a Poisson distribution the mean determines the frequency of
occurrence of a given number of individuals in the sampling units.
In other words, the mean may be calculated from the proportion of
the total numer of sampling units of a given class, that is, from
the proportion of sampling units in which a given number of indi-
viduals are found. For example, if beet leafhoppers are distributed
in a Poisson distribution and 37 percent of the plants are not
1/ This paper was read at the joint meetings of the American
Association of Economic Entomologists and the Entological Society
of America at San Francisco, Calif., Dec. 29, 1941, to Jan. 1, 1942.
2/ Now with the Ninth Service Command, Army Service Forces.
infested, then it is possible to equate the mean of the distribution
from the equation e-m Z 0.37. The mean is I leafhopper per plant.
This relation between the mean and the proportion of O's is shown
graphically by curve A in figure 1. Curves B and D in this
figure show the relation between the man and the proportion of l's,
2's, and 3's, respectively. The values shown in the curves may be
obtained by direct calculations or from tables prepared by Pearson 3/
The curves for all except the proportion of O's have two values
of the mean corresponding to every relative frequency. For tks
reason classes greater than 0 must be used with caution. For ezauplq,
if the percentage of l's is used and this number appears in 20 per-
cent of the sampling units, it my indicate a mean of either 0.25 or
2.53 insects per unit area. Similar difficulties in determnng
mean values will appear when the 2's, 3's, or higher numbers are use4.
Therefore, the relative frequency of classes higher than 0 can Wafell
be employed only when the mean is approximately knovn, and preferablS
when it is equal to or greater than the size of the class being used.,
Theoretically, the mea will be estimated fron a Oas fre-
quency with the least absolute error by using the *se having
thei atest i~tire f uen, The error of eet1 tIon _
eaes raid j tii departre from_ this ixum value. Itca
be ahuwn iuie of 't calculus that for a Poseon datri-
bution auz class frequency will be at a mxrl when the man
is equ o te z sIze of the class. This fact is gaphially I-
lustrated by the curves in figure l, vhich indicate mxim of Ole*
l's, 2's, or 3'6 whea the means are, respeotively, 0.0, 1.0, 2.O,
The application of fquencies of bigher-thwa-0 classes in est
mating beet leafhopper populations is of doubtful utitTi. Sine
the insect is small and very active, the effort required to detern
the papoxtion ot e eple ic eated with I, 2, or a higher number of
leafnahpers becoes almost as great as to uake a complete census of
the ssmple unite, This difficulty becomes increasingly pronounced
as the 8ize -,e cas i increased. On the other hand, the ue~e
pesee x &bsenee of the ise~t iy be determined with speed and
_/ 2 ub+ or ,tatisticiane and biometrioians.
i i, ed. 3, Vble i, pp. 113-121. 1930.
aaoiwaoyp and a large number of samples my be taken in a short time.
Data obtained in the Grand Valley area of Colorado in the spring
ad early sg r of 1937 are used here to illustrate the application
of the peroentage-frequenoy method to the estimation of populations
of the beet leafhopper. The data are eummarized in figure 2. ach
date mean plotted in figure 2 represents the average of the means
obtained f 10 fields scattered over the area. The mean for each
field on each date was based on 50 sampling units.
Clearly there is a striking agreement between the mean popu-
latioms per beet calculated by the two methods. This agreement sup-
ports the conclusion that migrant beet leafhoppers are distributed
in a Polson series, Inasmuch as the calculated values were obtained
by a formla derived from this series.
The large increases from June 21 to 28 and June 28 to July 15
represent the beginning of the appearance of adults of the first
seer generation. The leafhopper population estimated by the two
methods agrees very closely but the calculated values tend to be
slightly lower than the observed. The oorrelaticn coefficient for
the 15 pairs of means is r : 0.991, which denotes a high degree of
association between the two sets of values.
The foregoing data demonstrate that the proportion of uninfeeted
plants my be used to estimate the density of migrant beet leafhopper
populations. However, the method has limitations, of which the
following should be noted:
(1) It is applicable only when the distribution of the insect
accords with the Poisson law. As the distributing deviates from this
law there will be an increased tendenci for the method to underesti-
mate the true population.
(2) The mean determined from the proportion of O's will be esti-
umted with less precision than that obtained froi the total sample,
'but the larger number of sampling units that my be taken compen-
sates for this defiolency, at least within certain ranges of popu.
(3) The method is adapted primarily to population of low
density, say between 0 and 3.00 insects per unit area. For a Poidon
distribution with a mean of 5.00, 00s will appear only about 7 times
In 1,000 samples; and for a mean of 7.00 only once in ,O00 amplas.
Obviously such wall probabilities render imp tIcable the us. of
the 0 class in estimting dense populations because of the lag
number of sampling units that vould be required. When dens. popu-
lations are studied, perhaps equal inform tion could be obtained
vith les effort by taking a smaller number of sampling uaits =A
estimting the mean in the usual manner. This In a point that re-
quires further study.
Figure I.--Relatiou between the lean in a Poisson distri-
bution and the relative frequency of O's (curve A), of
l's (w 1),, of 2's (curVe C), and of 3's (curve D).
UNIVERSITY OF FLORIDA
3 1262 09240 8706
1.0 CALCULATED MEANS
0 p p I p p p p p p p p
15 20 25 30 4 9 14 19 24 29 4 I 14 I9
MAY JUNE 4ULY
Figure 2.--Populations of the beet leafhopper on sugar 1 eets
in the Grand Valley area of Colorado in the spring and
early summer of 1937. Observed means are shown by the
solid line, and the means calculated from the, relative
frequency of the 0 class by the broken line.