a new method for determining the type of distribution of plant

A New Method for determining the Type ofDistribution of Plant Individuals

BY

BRIAN HOPKINS{Botany Department, Unmerrity College, London)

WITH AN APPENDIX BYJ. G. SKELLAM

(The Nature Conservancy, London)

With one Figure in the Text

A B S T R A C T

The method depends on linear measurements between random points andadjacent individuals, and between adjacent pairs of individuals. Its results com-pare favourably with those of the current methods when tested on synthetic andnatural populations. The method is quicker than the quadrat methods and isespecially useful for analysing the distribution of trees.

i . I N T R O D U C T I O N

A CONSIDERABLE amount of work has been published on the typesJL~\. of distribution of plant individuals. Current methods for determiningthe type of distribution require the use of quadrats. The quadrat, thoughextremely valuable to the plant ecologist, is not always very practical to use.The method described in this paper does not require the use of quadrats.

2. CONCEPTS AND TERMS

2.1. Population is a term with several distinct statistical and biologicalmeanings. A population will be defined as the individuals of a species livingtogether on an area. Any discrete unit (shoot, tiller, frond, &c.) may be usedas a plant individual.

The density of a species is 'the number of individuals per unit area'(Goodall, 1952).

Sample areas are essential for the quantitative analysis of vegetation. Thequadrat will be defined as a sample area of given shape and size used foranalysis within a plant community. Objection may be made to this definitionon the grounds that the term is derived from quadratum (a square) and wasfirst defined (Clements, 1905) as 'a sq. m. of vegetation marked off for count-ing, mapping, etc.'. Numerous authors have used the term loosely to include

[Annals of BjUnj, N.S. Vol. XVIII, No. 70, April, 1954.]

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214 HopkinsA New Method for determiningall sizes (Clements states that the area need not be i sq. m.) and shapes, sothat it would seem justifiable to define it as above.

2.2. Distribution is static arrangement in space; dispersion is a dynamicconcept, implying dissemination. Dispersed, being the past tense, may beapplied in some cases (e.g. in the terms overdispersed and underdispersed) tothe results of dispersion.

Three types of distribution have been distinguished, random, aggregate,and regular. A random, or 'normal', distribution is one in which the individualsof a population are distributed according to chance. There has been con-siderable confusion in the terminology of non-random distributions. Thishas been mainly due to the use of the term overdispersed in two opposingmeanings. The statistical term of Svedberg (1922) refers to the distributionof quadrats containing o, 1, 2, 3, . . . individuals; his overdispersed populationhas more empty quadrats than would be expected by chance, and vice versafor his underdispersed population. Also there is a physical meaning referringto the actual distribution of the individuals on the area; an overdispersedpopulation has its individuals evenly distributed, and an underdispersedpopulation has its individuals in groups. Because of this confusion, the termsoverdispersed and underdispersed will not be used in this paper.

The term overdispersed used in its statistical sense (grouping of individuals)will be replaced by the term aggregated as suggested by Goodall (1952), andthe term underdispersed used in its statistical sense (evenly distributedindividuals) will be replaced by the term regular.

3. CURRENT METHODS

.3.1. All the current methods involve the counting of the number ofindividuals present in quadrats which are either thrown at random or arrangedin a regular manner.

3.2. If the individuals are randomly distributed, then the proportion ofrandom quadrats containing o, 1, 2, 3, . . . individuals will be given by thevalues of the terms of a Poisson series

e~m, mer

m, m

%e-

m\z\, OT3^-"1^ !,...,

where m is the mean number of individuals per quadrat. Observed andcalculated values of these terms are compared, and the significance of thedeviations is measured by a x2 test.

3.3. Clapham (1936) pointed out that for a Poisson series the variance (V)equalled the mean (M), and hence the relative variance (V/M) was unity.For aggregated populations the relative variance is greater than unity, andfor regular populations it is less than unity. The significance of the deviationof V/M from unity is measured by a 't' test or a %* test. When a 't' test isused, the value of the standard error (5) is given by S = *J[2/(ni)], wheren is the number of observations (Greig-Smith, 1952). The value of x2 inthe x2 t e s t is obtained by multiplying VIM by the number of degrees of

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the Type of Distribution of Plant Individuals 215freedom (ni) (Skellam, 1952). When the number of degrees of freedom islarge ( > 30), the normal deviate [>J(2^) i, 1Ix) is the probability of a chance departure from

expectation () exceeding that observed and in the same direction.(To ease this computation an abac (Fig. 1) has been constructed. For n

observations, the probability (/>) of exceeding the calculated value of x bychance can be read off quickly.)

For large numbers of observations (n > 50).v. Calculate X from the equation X = \x0-5! .2A/(2+I) , where |acO'5|

is the absolute value of x0-5.vi. Find the value of 0-5(1+0) from Pearson's (1914) tables of the normal

curve probability integral.vii. Calculate the probability (p) of exceeding the observed value of A by

chance from the equation/) = 10-5

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216 HopkinsA New Method for determining

APPLICATION OF THE NEW METHOD TO SYNTHETIC POPULATIONS

5.1. Methods. Plant individuals were represented by map pins on a squaremetre of millimetre graph paper. The position of any point on this area couldbe given (to the nearest millimetre) by a six-figure grid reference; henceby using six random figures an individual or point could be placed at random.

0-7

0-6

0-5

0-4

0-3

20 30n

40 50

FIG. 1. Abac for determining the probability (p) of exceedingthe calculated value of x for n observations by chance.

Nine different populations were used. The details of these are given inTable I.

In the aggregated populations the individuals were placed in hexagons ofside 25 mm. containing seven individuals, the central one being placed atrandom. In the regular distributions the individuals were placed at randomexcept where the position of an individual fell within 25 mm. of one whichhad already been placed. When this happened, a new random grid referencewas selected, and the process repeated until the required number of individualshas been placed.

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the Type of Distribution of Plant Individuals 217TABLE I

Details of Synthetic PopulationsPopulation.

A iAzA3B iBzB 3C iC zc 3

Type of distribution.Random

,,

,,

Aggregated,,

MRegular

,,

,,

Total density.IOO2 0 04 0 0IOO2 0 04 0 0IOO2 0 04 0 0

Density per (0-250 5 0i-oo0 2 50 5 0i-oo0 2 50-50i-oo

As the individuals were numbered, random figures were used to obtainrandom individuals. The distance from a random individual, or point, to itsnearest individual was measured by a millimetre scale. In each- population100 measurements of P and / were made. Later it was decided to leave outthe values where the distance measured was greater than the distance from therandom individual, or point, to the edge of the area. Hence, in order to keepthe number of observations of P and / the same, some other values had to bediscarded.

In order to compare this method with the current methods, the 1 sq. m.area was divided into a grid of 400, 25 sq. cm. quadrats, and the number ofindividuals present in each was recorded. Contiguous quadrats were groupedinto 100 quadrats of 100 sq. cm., 25 quadrats of 400 sq. cm., 16 quadrats of625 sq. cm., 4 quadrats of 2,500 sq. cm., and the total 10,000 sq. cm. Thenumber of individuals present in these quadrats was obtained from the 25sq. cm. quadrat data. The relative variance test was done on the valuesobtained for quadrats of 25, 100, 400, 625, and 2,500 sq. cm. (1, 4, 16, 25,and 100 grid units) and the values for the 25 and 100 sq. cm. quadrats (1 and 4grid units) were fitted to a Poisson series and tested by a x2 test. (It was notpossible to fit a Poisson series to quadrats of 400 sq. cm. and over as therewere not enough observations. In fitting data to a Poisson series, all thequadrats which have less than five individuals are grouped together.)

5.2. Results. The results of the analysis of these synthetic populations areshown in Table II. They do not require explanation except for populationC 1. That this regularly distributed population does not differ significantlyfrom a random distribution is due to the method of placing regular individualsand the low density. Had the value of the minimum distance between twoadjacent individuals been larger, then the distribution of this population wouldhave been significantly different from random.

6. APPLICATION OF THE NEW METHOD TO NATURAL POPULATIONS

6. I . Methods. The method used for natural populations was similar tothat used for the synthetic populations. A uniform area of a plant population

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Quadratsize(gridunits).

Populationi

41625

1 0 0

Population1

41625

1 0 0

Population1

41625

1 0 0

Population1

141625

1 0 0

VjMA i

1-05280-98990-81250-86401 -0667

A 21-16301-05050-96881-45072-2534

A 31-05761-04041-04170-92800-5600

B i269685-7980

10-27086-773327467

t

0 7 50-070 6 50-370 8 2

2 3 00 3 6O-II1 2 3i-54

0 8 10 2 80-14O-2O0 5 4

2397337632-n15812 1 4

TABLE II

Analysis of Synthetic PopulationsRelative variance.

P

0-4-0-5> 09

0-5-0-60-7-0-80-4-0-5

0-02-0-050-7-0-8

> 0-90-2-0-30-2-0-3

0-4-0-507-0-80-8-0-90-8-0-90-6-0-7

< o-ooi< o-ooi< o-ooi< o-ooiO-I-O-2

X*

4 2 0981951 3 0

3 2

464104

2321-8

6-8

422103251 3 91 7

1,076574246I O I

8 2

N rmaldeviate.

07520036

22320386

0-8210-317

18-14919-846

1

P

0-05-0-060-96-0-97

0-2-0-30-3-0-50-3-0-5

0-02-0-03070-071

0-3-0-5O-I-O-2

0-05-0-10

0-41-0-42075-0760-3-0-50-3-0-50-3-0-5

Populationi

41625

IOO

Population1

41625

IOO

Population1

41625

IOO

Population1

41625

IOO

Population1

41625

IOO

Ba3-8472551529-22925653429467

B 32-64164-4849525523 52533-8867

C i0-8320O7474081250-90670-4800

C 3O75180-6263054170-9600O-4934

c3058150-4344032290709301333

2609317728501275

2 3 8

2319245214746-92354

2-371 7 80 6 50-260-64

3 5 12-63i-59O-II0-62

5 923982-35o-8o1 0 6

< o-ooi< o-ooi< o-ooi< o-ooi0-I-O-2

< o-ooi< o-ooi< o-ooi< o-ooi

0-02-0-05

0-0I-0-O20-05-0-100-5-0-607-0-80-5-0-6

< b-ooiO-OOI-O-OI

o-i-o-a> 09

0-5-0-6

< o-ooi< o-ooi

0-02-0-05 0-4-0-5

0-3-0-4

1,136. 546

2 2 2

858-8

1.054444126531 2

332741 9 51 3 6

i-4

3 0 062131 4 4

i-5

23243

7 7io-60-4

1943519-010

1768215-763

-2463-1-870

-37362-900

6690- 4 7 6 2

< o-ooi-< o-ooi< o-ooi< o-ooi

0-02-0-05

22o HopkinsA New Method for determiningwas chosen and a 20 X 20 square grid set up, the length of the side of the gridunit (L.G.U.) depending on the density of the individuals. The selectedarea was the site within which all sampling was done. Every fifth individualencountered, whilst traversing the rows to record the number of individualsin each grid unit, was used as an individual from which the distance to itsnearest neighbour was measured.

The setting up of a grid is not necessary in order to determine the type ofdistribution of a population. A series of lines are placed across the site andevery/th individual occurring within a short but fixed distance of one or bothsides of the line is used as an individual chosen at random.

Random points were chosen by throwing a cane at random. All distanceswere measured at ground level in the herbaceous populations and at breastheight in the tree populations. No allowance was made for the size of theindividuals; the distance between them was measured to the nearest centi-metre.

6.2. Populations studied. Twelve natural populations were analysed. Mostof these were from areas known to have been little altered by man, and thoughtto have remained unchanged for hundreds of years. Hence the distribution ofthe individuals was assumed to have attained equilibrium with the environ-ment and to be entirely natural. They were selected to obtain a wide rangeof type of distribution, habitat, and vegetation. However, as the new methodwill potentially have its greatest use in wooded areas, more than half theexamples were chosen from tree populations.

A brief comment on each population will now be given.P. I. MAYO BOG (Eriophorum angustifolium) (July 1951). The site was

4 miles south of Louisburgh, near Westport. It was part of a vast expanse ofblanket bog which had presumably been growing throughout the subatlanticperiod and had apparently been little influenced by man. There was nodominant species; Narthecium ossifragum, Eriophorum, angustifolium, Rhyn-chospora alba, Schoenus mgricans, Molinia caerulea, and several species ofSphagnum were very frequent. There were occasional pools and hummockson the site, which was similar to, but wetter than, those described by Pearsalland Lind (1941). Fruiting heads of Eriophorum were chosen as individuals.

P. II. MAYO BOG (Utricularia intermedia) (July 1951). The site wasidentical with that of P. I; the two analyses were done simultaneously. Flower-ing heads of Utricularia were chosen as individuals. The Utricularia growsin bog pools, but only four of the pools on the site contained it, hence itoccurred in four clumps. This population was chosen as being an example ofapparent extreme aggregation.

P. III. RANNOCH BOG (Eriophorum angustifolium) (August 1951). Thesite was quite close to Rannoch Station, Perthshire, on an area of bog at analtitude of 950 ft., which had probably been burnt a few years before it wassampled. There was no dominant species, and the most frequent species wereCalluna vulgaris, Erica tetraUx, Narthecium ossifragum, Trichophorum caespito-sum, Eriophorum angustifolium, several species of Sphagnum, several Hepaticae,

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the Type of Distribution of Plant Individuals 221Cladonia impexa, and C. unciatis. Fruiting heads of Eriophorum were chosenas individuals.

P. IV. HILLOCK WOOD (Fagus sylvatica) (March 1952). The site was partof a plateau beechwood about 2 miles east of Princes Risborough in theChiltern Hills. The area had presumably been beechwood since prehistorictimes and had been managed (felling and clearing) since the sixteenth century.There were occasional oak trees present and the ground flora was mainlycomposed of brambles. It would be type 'B1 in Watt's (1934) classification.Beech trees were chosen as individuals.

P. V. RIDLEY WOOD (Fagus sylvatica) (April 1952). The site was part ofa beechwood, with some oak and holly, about 3 miles east of Ringwood in theNew Forest. Local records show that the area had been beechwood for atleast 350 years. Beech trees were chosen as individuals. (This was the onlypopulation of trees in which all the stages from seedlings to trees were present.Beeches with a girth at breast height of 20 cm. or more were regarded astrees.)

P. VI. ROTHIEMURCHUS WOOD (Pinus sylvestris ssp. scotica) (July 1952).The site was about 4 miles south-east of Aviemore, Inverness-shire. It was aremnant of the old Caledonian pine forest and showed no signs of having beenfelled for several hundred years. The ground flora consisted mainly of Vac-cirriian vitis-idaea, V. myrtiUus, Calluna vulgaris, and Hylocomium splendent.The vegetation is more fully described by Tansley (1949). Pine trees werechosen as individuals.

P. VII. CRESSBROOKDALE WOOD (Fraxinus excelsior) (August 1952). Thesite was about 2 miles east of Miller's Dale, Derbyshire. It was part of aregenerating ashwood on a very steep, unstable, limestone scree slope facingwest. Occasional rowan trees were present, and where there was a groundflora it was mainly Deschampsia caespitosa or Mercurialis perenms. The vegeta-tion of these ashwoods is described by Moss (1913). Ash shoots were chosenas individuals.

P. VIII. WATENDLATH WOOD (Betula pubescens) (September 1952). Thesite was about 5 miles south of Keswick in the Lake District. It was part of abirchwood on a slope facing north at an altitude of 1,600 ft. Pearsall (1950)suggests that originally the area was covered by oakwood which changed tobirchwood several hundred years ago. Since then there has been considerabledegeneration of the woodland to moorland. Watendlath Wood is a small relicof this once extensive birchwood. Regeneration is now prevented by sheepgrazing, and signs of degeneration to bog can be seen in places. The groundflora mainly consisted of grasses and bracken. Birch trees were chosen asindividuals.

P. IX. WATENDLATH GRASSLAND (Pteridium aquilinum) (September 1952).The site was next to that of P. VIII. It had probably had a similar historybut had reached a further stage of degeneration; the wood had been completelyreplaced by bracken. The bracken was at the 'mature' stage in Watt's (1947)terminology. Bracken fronds were chosen as individuals.

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222 HopkinsA New Method for determiningP. X. SOUTH BENTLEY WOOD (Quercus petraea) (October 1952). The site

was part of an enclosed oakwood, with a considerable amount of beech, about5 miles north-west of Lyndhurst in the New Forest. Local records show thatit had been oakwood for several hundred years. The site was chosen in anarea of considerable regeneration, and oak saplings (there were no trees on thesite) were chosen as individuals.

P. XI. CoTHiLL FEN {Schoenus nigricans) (October 1952). The site was ina Schoenus dominated area of Cothill Fen, about 5 miles south-west of Oxford.The site was calcareously flushed and all the Schoenus occurred as largetussocks, which were chosen as individuals. It was the area where Dawkins(1939) analysed the distribution of Schoenus. As the size of the tussocks wasnot negligible compared with the distance between them, the distances weremeasured from the centre of tussocks and at tussock height.

P. XII. KTNGLEY VALE WOOD (Taxus baccata) (November 1952). The sitewas part of a yewwood in Kingley Vale National Nature Reserve, about 5miles north-west of Chichester in the South Downs. The wood was at thehead of a chalk coomb on a steep slope facing south. The area has beendescribed by Watt (1926), who showed that it had been yewwood for severalhundred years. Yew trees were chosen as individuals.

6.3. Results. The results of the analysis of these natural populations areshown in Tables III and IV.

TABLE III

Details of Natural Populations

No.IIIIIIIVVVIVIIVIIIIXXXIXII

Locality.Mayo BogMayo BogRannoch BogHillock WoodRidley WoodRothiemurchu3 WoodCressbrookdale WoodWatendlath WoodWatendlath GrasslandSouth Bentley WoodCothill FenKingley Vale Wood

Species.Eriophorum angustifoliumUtricularia intermediaEriophorum angustifoliumFagus sylvaticaFagus sylvaticaPinus sylvestrisFraximu excelsiorBetula pubescentPteridium aquilinumQuercus petraeaSchoenus nigricansTaxus baccata

L.G.U.(m.)i -oi-oi-oS-o

io-o3-01-54 00-150-5o-si-5

Areaof

site(sq. m.)

4 0 04 0 04 0 0

10,00040,000

3,600900

6,4009

1 0 01 0 0900

Totaldensity.

510IOI

91382193347SS510621324035i141

Densitypergridunit.

1-2750-2530-228-955048008681-3880-127-533o-6oo0-878-353

The results do not require much comment. The Schoenus tussocks (P.XI) were regularly distributed. The beech trees (Ridley Wood, P. V),bracken fronds (P. IX), and oak saplings (P. X) were distributed at random.The remaining eight populations were aggregated. They were, in order ofincreasing aggregation, beech trees (Hillock Wood, P. IV), pine trees (P. VI),birch trees (P. VIII), yew trees (P. XII), fruiting heads of Eriophorum angusti-

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the Type of Distribution of Plant Individuals 223folium (Rannoch Bog, P. I l l , and Mayo Bog, P. I), ash shoots (P. VII), andfinally, flowering heads of Utricularia intermedia (P. II).

7. CONCLUSIONS

As will be seen from Tables II and IV, results obtained by the new method,described in this paper, agree closely with those obtained by the methods incurrent use. The agreement is such that the probabilities of the significanceof the deviation of A from unity are of the same order as those for VjM and x2.Thus the new method may be used as an alternative to the quadrat methods.

When analysing the type of distribution of trees using the current methods,quadrats have to be marked out on the ground; this is tedious and timeconsuming. The new method described in this paper is very quick; onlydistances need to be measured. The line samples used for obtaining randomindividuals take a little time but are much quicker than the use of quadrats,especially where the number of individuals in each has to be counted. Hencethis new method would seem to be a useful substitute for the quadrat methods.

The interest of aggregated populations lies in the reason for and the type ofaggregation. The method described above only tells whether or not a popula-tion is distributed at random, and how different from random is its dis-tribution. Work is proceeding relating P and I to the density of the individualsand to the distribution patterns.

8. SUMMARY

1. A new method for determining the type of distribution of a populationof plant individuals is described.

2. This new method does not require the use of quadrats, but depends onthe mean distance between two adjacent individuals, and the mean distancebetween a random point and its nearest individual.

3. The method has been used on nine synthetic and twelve natural popula-tions. In all cases results were in close agreement with those obtained bycurrent methods.

4. It is suggested that, as it is quicker, the new method might replace thequadrat methods especially in the study of tree distributions where quadratshave to be marked out.

This work was done during the tenure of a Nature Conservancy ResearchStudentship, and was carried out under the supervision of Professor W. H.Pearsall. For statistical advice thanks are due to Mr. J. G. Skellam who is,however, not responsible for any of the above statements, but has kindlycontributed an appendix. Thanks are also due to several people who haveassisted with the fieldwork.

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TABLE IV

Analysis of Natural PopulationsQuadrat

ixe(gridunits).

Populationi

> *

35IOO

Population1

41625

IOO

Population1

1625

IOO

Population1

. 025

IOO

Population1

16"25

IOO

V/M

I. Mayo Bog 11-662529491516058-5501

203582II. Mayo Bog

25975825412629733935-4324243300

III. Rannoch1 806627305679909-4359

13-6117

t

[Eriophontm)9-36

13-72144120-682371

(Utricularia)35377171-8099-53943128-57

Relative variance

P

< O-OOI< o-ooi< o-ooi< o-ooi< o-ooi

< o-ooi< o-ooi< o-ooi< o-ooi

O-OOI-O-OI

Bog (Eriophontm)113912-1820-0923-115-45

IV. Hillock Wood (Fagus)1-08912-74215696478073

199965

1 2612-2616-2718652327

V. Ridley Wood (Fagus)1-1278073130-42860-988921389

1 8 1.801-980-031-39

< o-ooi< o-ooi< o-ooi< o-ooi< o-ooi

0-2-0-3< o-ooi< o-ooi< o-ooi< o-ooi

0-05-0-100-05-0-10O-OS-O-IO

> 0-90-2-0-3

6632 9 1

12861

10,3642,516

7 1 453173

7 2 12 7 0163142

41

43527137" 760

4 5 07310-34-86-4

deviate

818310-089

"57456-900

97429-202

1-2659245

1769 2-OO2

P

"t O-OOI< o-ooi< O-OOI< o-ooi< o-ooi

< O-OOI"< O-OOI< o-ooi< O-OOI< o-ooi

t o-ooi< 0001< O-OOI< o-ooi< O-OOI

0-20-0-21< o-ooi< o-ooi< o-ooi< o-ooi

0-07-0-080-04-0-05

< o-oi0-5-07

0-05-0-10

Fit

X1

85-502138-8340

101-82061447795.

33-3673257258

48-977616-6490

2-287289589

to PoUton

d.f.

I

I2

I2

35

23

p

< o-ooi< 0001

< o-ooi< o-ooi

< O-OOI< o-ooi

< O-OOIO-OOI-O-OI

0-3-0-50-02-0-05

Coefficient ofaggregation

A n p

3-3632 97 < o-ooi

262678 20

I *

Population VI. Rothiemuichu* Wood (Ptmu)i I-IO6I 377 O-OOI-O-OI

1-6505 4-58 < o-ooi2-0503 3-64 O-OOI-O-OI

25 3-1917 3-36 O-OOI-O-OIIOO 38780 3-30 O-I-O-2

Population VII. Cressbrookdale Wood {Fraxtmu)1 5-0920 57'8o < o-ooi4 14-9513 98-18 < o-ooi

16 49-0464 166-42 < o-ooi35 63-1343 170-18 < o-ooi

100 161-389 196-41 < o-ooi

Population VIII. Watendlath Wood (Betula)1 3-5755 33-25 < o-ooi4 1-4675 3-29 O-OOI-O-OI

i-66o8 2-29 0-030-05

1 0 01-81133-8550

3-333-37

0-03-0-05O-I-O-3

Population IX. Watendlath Gra*aland (Pteridium)1 1-0261 0-37 07-0-84 1-1618 1-14 0-3-0-3

16 07629 0-82 0-4-0-535 0-5659 119 0-2-0-3

100 0-3318 0-82 0-4-0-5

Population X. South Bentley Wood (Quercus)1 0-9523 0-67 0-5-0-64 07744 i-59 0-1-0-2

16 I-059O O-2O O-8-O-925 0-3111 i-8o 0-05-0-10

100 0-1222 i-oo 0-3-0-4

Population XI. Cothill Fen (Sehoenus)1 0-3799 876 < o-ooi

0-2964 4-95 < o-ooi0-3463 2-52 o-oi-o-oa

25 0-2826 1-06 0-05-0-10100 0-3305 0-82 0-4-0-5

Population XII. Kingley Vale Wood (Taxus)1 1-0479 '68 0-4-0-54 1-4471 3-15 O-OOI-O-OI

16 2-4081 5-09 < o-ooi25 2-1711 3-21 O-OOI-O-OI

100 3-8487 3-49 0-02-0*05

1 0

4771634933

9

2,0321,4801,177

947484

1,028145

39927-2

8-6

409

i8-38-5i-o

3807725-4

470-4

152295 942i*o

418143

593313

3-6564-019

35-5184O-37O

17-1133-993

0-3701-130

- 0 6 6 3 1-651

- 1 0 7 9 5-6-381

0-6832-876

O-OOI-O-OI< o-ooi

o-ooio-01O-OOI-O-OI

0-02-0-05

226 HopkinsA New Method for determiningLITERATURE CITED

CLAPHAM, A. R., 1936: Overdispersion in Grassland Communities and the use of StatisticalMethods in Plant Ecology. J. Ecol., xxiv. 232.

CLEMENTS, F. E., 1905: Research Methods in Ecology, p. 321. Univ. Pub. Co., Lincoln,U.S.A.

DAWKINS, C. J., 1939: Tussock Formation by Schoema rrigricans: the Action of Fire and WaterErosion. J. Ecol., xxvii. 78.

FISHER, R. A., 1950: Statistical Methods for Research Workers, pp. 57-61. n t h edit. Oliverand Boyd, Edinburgh and London.

GOODALL, D. W., 1952: Quantitative Aspects of Plant Distribution. Biol. Rev., xxvii. 194.GREIG-SMITH, P., 1952: The Use of Random and Contiguous Quadrats in the Study of the

Structure of Plant Communities. Ann. Bot., xvi. 293.Moss, C. E., 1913 : The Vegetation of the Peak District, pp. 65-74. Cambridge Univ. PressPKARSALL, W. H., 1950: Mountains and Moorlands, pp. 129-31. Collins, London.

and LIND, E. M., 1941: A Note on a Connemara Bog Type. J. Ecol., xxix. 62.PEARSON, K., 1914. Tables for Statisticians and Biometricians, Cambridge University Press.

1934- Tables of the Incomplete Beta-Function. Biometrika, London.SKELLAM, J. G., 1952: Studies in Statistical Ecology, I. Spatial Pattern. Biometrika, m i x .

346.SVEDBERG, T., 1922: Statistik vegetationsanalys. Svensk bot. Tidskr., xvi. 1.TANSI^Y, A. G., 1949: The British Isles and their Vegetation, pp. 447-450. Cambridge

University Press.WATT, A. S., 1926: Yew Communities of the So ith Downs. J. Ecol., xiv. 282.

1934: The Vegetation of the Chiltern Hills with Special Reference to the Beechwoodsand their Serai Relationships. Ibid., xxii. 237 and 445.

1947: Contributions to the Ecology of Bracken (Pteridium aqutlinum), XV. The Structureof the Community. New Phytol., xlvi. 97.

A P P E N D I X

By J. G. SKELLAM(The Nature Conservancy, London)

1. If a very large number of particles are distributed independently of oneanother in a very large area with the same probability density everywhere,then the number falling in a randomly chosen small sub-region is a Poissonvariate with a mean proportional to the area of that sub-region.

If we chose a point at random to serve as a centre of a circle and count thenumber of particles in that circle, we clearly expect the result to be distributedin repeated trials in the same way as if we had adopted a slightly differentprocedure and had chosen one of the particles at random to act as centre andhad counted the number of other particles in a circle of the same size, for theparticles are all scattered independently of one another with uniform probabilitydensity over a very large area and are very numerous.

Now draw two concentric circles with radii rx and r8 where rx < r2.The probability that the inner one is empty is e'^' where A is a con-

stant. The probability that the annulus between the circles is not empty

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the Type of Distribution of Plant Individuals 227is 1e~Kr**$. The probability that the nearest particle to the centre lies be-tween r2 and rlt is

By letting rx-^>-r% = r and calling r%rx = dr, we obtain the probabilitythat the nearest particle to the centre lies at a distance between r and r-\-dr.This is

*** zXrdr.

The variate z = Ar2 then has the distribution given by

/ (z)dz = e~zdz. (1)J2. We now make use of the following theorems, proofs of which are given

for example in Weatherburn (1947). 'A First Course in MathematicalStatistics', chap. 8 (Cambridge University Press).

THEOREM I. If the variate z has the frequency function / (z) = e~z, thenthe variate Z defined as the sum of n independent values each distributed asz has the frequency function

h(Z) = e-z^ir(n). (2)Here T(n) = (n1) (n2) ... 3. 2. 1.THEOREM II. If the independent variables Zlt Z% both have the distributionshown above in (2), then the variate x defined by

x = ZJ^+Zt)has the frequency function

j{x) = ^(i-xr*IB(n,n), (3)where o < x < 1 and B(n, n) = T(n) I\n)jl\2n).

3. If from n random points the distances to the nearest particle are r1(r2, ..., rn, and if from n random particles the distances to the nearest neigh-bouring particles are pv p2, , pn, then both Zx = A^ 2 and Z2 = A^p2 havethe distribution (2) by reasons of relation (1) and Theorem I.

It follows from Theorem II that

x -

has the distribution (3).4. For values of n less than 50 the probability that the random variable x

is less than a stated value can be obtained from the tables of the integral of(3) prepared by Karl Pearson ('The Incomplete Beta-Function').

The distribution has mean = 0-5 and variance = i/[4(2n+i)] and rapidlytends to normality as n increases. For values of n > 50 it is sufficient to takeX = z(xo-5)V(2n+1) as a normal variate with zero mean and unit varianceand to refer to the standard tables of the normal integral.

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