n I Kesea(rcn Note No. 1983 ISSN 0226-9368

Some sample size tables for forest sampling

by H.B. Stauffer

Province of British Columbia

istry of 634.9097 e& I1 BCMF R E S RN 90

Some Sample Size Tables for Forest Sampling

by Howard B. Stauffer

RESEARCH NOTE#90

Province of British Columbia Ministry of Forests

1983

Canadian Cataloguing in Publication Data S t a u f f e r , Howard B . , 1941-

Some s a m p l e s i z e tables f o r f o r e s t s a m p l i n g

( R e s e a r c h n o t e , ISSN 0226-9368 ; no. 90)

B ib l iog raphy : p. ISBN 0-7719-9165-7

l . ' F o r e s t s a n d f o r e s t r y - Mensura t ion . 2. Sampling. 3. I t e r a t ive methods (Mathematics) 4 . Sampling - Tables. I. B r i t i s h C o l u m b i a . M i n i s t r y o f F o r e s t s . F o r e s t r y D i v i s i o n . Research Branch. 11. T i t l e . 111. S e r i e s : Research no te (Province of Br i t i sh Co lumbia , M i n i s t r y of F o r e s t s ) ; no. 90.

@276.6.s72 634.9'285 C83-092093-5

Published by: Information Services Branch 5.C. Ministry of Forests 1450 Government S t r e e t Victor ia , 5.C.

V8W 3E7

!<!1983 Province o f British Columbia '\ ,

Ministry of Forests Publication No. S28-82058

ABSTRACT

Th is repo r t p resen ts a c o l l e c t i o n o f sample s i z e t a b l e s f o r f o r e s t

sampling. One group o f t ab les exp resses sample s i ze as a f u n c t j o n o f

e s t i m a t e d c o e f f i c i e n t of v a r i a t i o n and prescribed percentage sampling error

for s imple random samp l ing us ing va ry ing l eve l s o f con f i dence . A second group

o f t ab les exp resses sample s i ze as a func t i on o f es t ima tes o f dens i t y and

spac ing and p rescr ibed a l lowab le percentage samplbg e r ro r fo r dens i ty

sampling with f i x e d - a r e a p l o t s ( q u a d r a t s ) u s i n g v a r y i n g l e v e l s o f c o n f i d e n c e .

These t a b l e s have been const ructed us ing an : i terat ive scheme nn t h e

standard s imple random sampl ing formula for sampl ing er ror . It i s

d e m o n s t r a t e d t h a t t h e t t c l a s s i c a l " i t e r a t i v e methnd f o r c a l c u l a t i n g sample s i z e

does not always converge t o a so lu t i on . A c o r r e c t e d ( c o n v e r g e n t ) i t e r a t i o n

scheme i s described and used t o c a l c u l a t e t h e sample s i z e s f o r t h e s e t a b l e s .

Examples a r e i n c l u d e d f o r i l l u s t r a t i o n .

TABLE OF CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i j

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . i v

L I S T OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

THE CLASSICAL ITERATIVE ALGORITHM FOR CALCULATING SAMPLE SIZE . . . . . . 3

A CORRECTED ITERATION SCHEME . . . . . . . . . . . . . . . . . . . . . . . 7

SAMPLE SIZE TABLES FOR SIMPLE RANDOM SAMPLING . . . . . . . . . . . . . . a

SAMPLE SIZE TABLES FOR DENSITY SAMPLING USING FIXED-AREA PLOTS . . . . . . 10

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

- i v -

LIST OF TABLES

Page

Table 1. Examples o f convergence, using the . . . . . . . . . . . . . . 17 c l a s s i c a l i t e r a t i v e a l g o r i t h m t o c a l c u l a t e sample s i z e (95% con f idence l eve l ) .

Table 2. Examples o f d ivergence, using the . . . . . . . . . . . . . . 18 c l a s s i c a l i t e r a t i v e a l g o r i t h m t o c a l c u l a t e sample s i z e (95% c o n f i d e n c e l e v e l ) .

Table 3. Examples from Table 2, us ing t he . . . . . . . . . . . . . . 19 c o r r e c t e d i t e r a t i o n scheme t o c a l c u l a t e sample s i z e (95% con f idence l eve l ) .

Table 4. Sample s i z e t a b l e (99% conf jdence): . . . . . . . . . . . . . 20 sample s i zes (n) requ i red f o r es t ima ted c o e f f i c i e n t s o f v a r i a t i o n (CV) t o ensure prescr ibed al lowable percentage sampl ing e r r o r s (PE) .

Table 5 . Sample s i z e t a b l e (95% conf idence) : . . . . . . . . . . . . . 22 sample s i zes (n ) requ i red f o r es t ima ted c o e f f i c i e n t s o f v a r i a t i o n ( C V ) t o ensure prescr ibed al lowable percentage sampl ing e r r o r s (PE).

Table 6. Sample s i z e t a b l e (90% confidence) : . . . . . . . . . . . . . 25 sample s izes (n> requ i red fo r es t imated c o e f f i c i e n t s o f v a r i a t i o n ( C V ) t o ensure prescr ibed al lowable percentage sampl ing e r r o r s (PE) .

Table 7. Sample s i z e t a b l e (80% conf idence): . . . . . . . . . . . . . 28 sample s izes (n ) requ i red f o r est imated c o e f f i c i e n t s o f v a r i a t i o n (CV) t o ensure prescr ibed al lowable percentage sampl ing e r r o r s (PE) .

- v -

LIST OF TABLES cont 'd .

Table 8. Sample s i z e t a b l e f o r d e n s i t y s a m p l i n g . . . . . . . . . . . . 31 (99% conf idence) : sample s i z e s r e q u i r e d f o r e s t i m a t e d s t a n d d e n s i t i e s ( s te rndhec ta re ) and t ree spa t ia l p a t t e r n s (v/m = variance/mean r a t i o s ) t o ensure prescr ibed a l lowable percentage sampling errors (PE).

Table 9 . Sample s i z e t a b l e f o r d e n s i t y s a m p l i n g . . . . . . . . . . . . 36 (95% conf idence) : sample s i z e s r e q u i r e d for es t ima ted s tand dens i t i es (s tems/hec tare) and t ree spa t ia l p a t t e r n s (V/m = variance/mean r a t i o s ) t o ensure prescr ibed al lowable percentage sampling errors (PE).

Table 10. Sample s i z e t a b l e f o r d e n s i t y s a m p l i n g . . . . . . . . . . . . 41 (90% confidence) : sample s i z e s r e q u i r e d f o r e s t i m a t e d s t a n d d e n s i t i e s ( s te rndhec ta re ) and t ree spa t ia l p a t t e r n s (V /m = variance/mean r a t i o s ) t o ensure prescr ibed al lowable percentage sampl ing errors (PE).

Table 11. Sample s i z e t a b l e f o r d e n s i t y s a m p l i n g . . . . . . . . . . . . 46 (80% cnnfjdence) : sample s i zes requ j red f o r e s t i m a t e d s t a n d d e n s i t i e s ( s tems /hec ta re ) and t ree spa t ia l p a t t e r n s (V/m = variance/mean r a t j o s ) t o ensure prescr ibed al lowable percentage sampling errors (PE).

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INTRODUCTION

T h i s report presents a collection of sample s i z e t a b l e s f o r f o r e s t sampling. There a r e two groups of tables, each l ist ing the sample s i z e s required for prescribed percentage sampling errors at g i v e n l e v e l s of confidence. The f i r s t group is f o r simple random sampling i n general and uses estimated coefficients of variation. The second group i s more s p e c i f i c a l l y f o r d e n s i t y sampling w i t h f ixed-area plots (quadrats) and uses esti-mates o f

density and spacing.

I t will be demonstrated tha t the "c lass ica l" i t e ra t ive a lgor i thm used by

f o r e s t e r s t o c a l c u l a t e sample s i z e does not always converge t o a solut ion. A

corrected (convergent) i teration scheme will be described and used t o construct the tab les . 1

It should be noted that i t is of ten not necessary to use i terat ion to estimate sample s i z e . If the sample s i z e o r extent of var ia t ion o f the data is f a i r l y l a r g e , if the prescribed allowable percentage sampling error i s r a the r a rb i t r a ry , or i f t he e s t ima te of t he coe f f i c i en t of var ia t ion i s very imprecise, it is usua l ly su f f i c i en t t o l e t the Student 's t value t = 2 (95%

confidence level) i n the standard formula for sampling error ( 9 ) and perform one calculat ion to determine sample s i ze (Schae f fe r e t a l . 1979) . For la rge sample sizes or coeff ic ients of var ia t ion, this estimate f o r t w i l l be close

'The term " i te ra t ion" here re fe rs to t.he repeated use o f the standard formula f o r sampling e r r o r ( 9 ) to obtain increasingly better approximations t o a so lu t ion for sample s ize . If these repeated calculations asymptotically s t a b i l i z e t o a solut ion, the i te ra t ion p rocess is s a i d t o "converge". Otherwise, if these ca lcu la t ions do no t s t ab i l i ze b u t i n s t ead , s ay , o sc i l l a t e back and f o r t h between poor approximations t o a so lu t ion , t he i t e r a t ion process i s said to "diverge". More d e t a i l s and some examples i l l u s t r a t i n g these concepts will be g iven la te r .

- 2 -

enough. For a r b i t r a r y l e v e l s o f p r e s c r i b e d a l l o w a b l e s a m p l i n g e r r o r o r imprecise est imates of c o e f f i c i e n t o f v a r i a t i o n , a precise technique such as

i t e r a t i o n w o u l d c r e a t e a f a l s e i m p r e s s i o n o f p r e c i s i o n i n t h e c a l c u l a t i o n o f

t he sample s i ze .

However, an i t e r a t i v e approach may become necessary i f sample s i zes a re

smal l ( i .e . , G 301, t h e v a r i a t i o n i n the da ta i s smal l , and est imates of

v a r i a t i o n a r e f a i r l y p r e c i s e . I t e r a t i v e s o l u t i o n s a r e o f t e n menti-oned i n t h e

f o r e s t r y l i t e r a t u r e ( F r e e s e 1 9 5 6 and 1962, UBC Forest Club 1971, Avery 1975,

Wensel 1977). When an i t e r a t i v e s o l u t i o n i s n e c e s s a r y , t h e c o r r e c t e d

i t e r a t i o n scheme descr ibed i n t h i s paper should be used. Fo r t hese s i t ua t i ons

and more g e n e r a l l y t o a v o i d t h e n e c e s s i t y c f c a l c u l a t i o n , t h e sample s i z e

t ab les presented here should serve as a handy reference f o r prac t i c ing

researchers and foresters .

Throughout t h i s r e p o r t , s a m p l i n g will be assumed t o he simple random

sampling with replacement or without replacement i n a " la rge" popu la t ion . 2

I f the sampl ing i s w i t h o u t r e p l a c e m e n t i n a s m a l l p o p u l a t i o n , t h e i d e a s o f

t h i s paper can be genera l i zed us ing t he f i n i t e popu la t i on co r rec t i on f ac to r .

The i d e a s o f t h i s p a p e r c a n a l s o b e i n c o r p o r a t e d i n t o sample s i z e c a l c u l a t i o n s

fo r o the r t ypes o f samp l ing such as s t ra t i f i ed random sampling (Cochran 1977,

Freese 1962, Schaeffer " e t a l . 1979). This paper i s an ex tens ion o f ideas

b r i e f l y p r e s e n t e d in Stauf fer (1982) .

2Reca l l tha t sampl ing "with rep lacement" permi ts members o f a p o p u l a t i o n t o be sampled more than once, whereas sampling "without replacement" requires t h a t members o f a p o p u l a t i o n may n o t be sampled more t h a n once.

The t h e o r y r e q u i r e s t h a t t h e p o p u l a t i o n be no rma l l y d i s t r i bu ted bu t i s s u f f i c i e n t l y r o b u s t t o be a p p l i e d t o p o p u l a t i o n s t h a t a r e " m o d e r a t e l y " nonnormal, p a r t i c u l a r l y i f t h e y a r e n o t h e a v i l y skewed.

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THE CLASSICAL ITERATIVE ALGORITHM FOR

CALCULATING SAMPLE SIZE

The c l a s s i ca l i t e r a t ive a lgo r i thm used by f o r e s t e r s t o c a l c u l a t e sample s ize for s imple random sampling proceeds as follows.

We s h a l l want to cons ider a to ta l popula t ion o f N values xl, x2 , ... , xN w i t h the following parameters:

(i> mean N c x,

i =1 I

u="-

( ii 1 variance

N

(iii) standard deviation

and ( i v ) coef f ic ien t o f var ia t ion

y = ; * 100%

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From t h i s t o t a l p o p u l a t i o n , we s h a l l cons ider a random sample o f n

measurements from the populat ion xl, x2, ... , xn with t h e f o l l o w i n g

parameter est imates:

( i ) e s t i m a t e o f t h e mean n

( i i ) e s t i m a t e o f t h e v a r i a n c e n c ( x i - Kl2

s2 = i = l n - 1

( i i i ) e s t i m a t e o f t h e s t a n d a r d d e v i a t i o n

s = J s 2

and

( i v ) e s t i m a t e o f t h e c o e f f i c i e n t o f v a r i a t i o n

cv = "- * 100% S

X

The (1 - a ) -100% c o n f i d e n c e i n t e r v a l e s t i m a t e f o r 1-1 i s g i v e n by [R-E,R+EJ where

E = trl-l ,a12 - x s

i s the sampl ing e r ro r . tn-l, n/2 (henceforth t o be denoted by t) i s t h e t

- 5 -

' gl'

value w i t h a12 probabi l i ty and n-1 degrees of free do^.^ Solving for the sample s i z e ,

2 5 2 2 x ( 5 * n = t * - = t . - -

X E (x- E - 100%)2

E cv2 where PE = -=- - 100% i s the percentage sampling error and Q = 7 . Note that the percentage sampling error is just the r e l a t i v e amou% o f

sampling e r r o r ( E ) compared to the es t imated mean (2 ) .

4 x

The c l a s s i ca l i t e r a t ive a lgo r i thm used by f o r e s t e r s t o c a l c u l a t e sample size uses formula (10) and proceeds as follows. Given an estimated coef f ic ien t o f variat ion ( C V ) and a prescribed allowable percentage sampling e r r o r (PE), l e t Q = 2- . S t a r t i n g w i t h an i n i t i a l value for n, say c v 2

PE

c1 is the Type I e r ror , the p robabi l i ty of re jec t ing a t r u e n u l l hypothesis. Since the confidence interval i s two-sided, a t value w i t h a12 probabi l i ty for each ta i l i s the appropriate v a l u e .

4These calculat ions are meant to provide guidel. ines for constructing the sample s i z e t a b l e s and not t o be in te rpre ted from a r i g o r o u s s t a t i s t i c a l p o i n t of view. They are dependent on a I1reasonably" precise estimate for t h e coef f ic ien t of var ia t ion (or s ) . Strictly speaking, n should be viewed a s a random variable w i t h one-sided confidence intervals. An F value should be used i n formula (10) when s is estimated from a sample of s i z e m (Grosenbaugh 1947, L i 1964, Snedecor and Mood 1946). The estimate s i from the sample of s i z e m should he m u l t i p l i e d by Fn-1,m-l u s i n g a prescribed confidence leve l , s ince s{ / s$ is Fn-1,m-I d i s t r i b u t e d (Freund and Walpole 1980). Fn-1 m-1 . s$ then provides an es t imate for sg w i t h sample s i z e n. I n this paper, we l e t F = 1 which corresponds t o a prescribed confidence level of approximately 50!% when s is estimated from a small sample s ize .

Alternatively, a two-stage sampling scheme could be used t o provide an "optimal" design fo r estimating sample s ize , es t imat ing CV (or S I a t the f irst s tage and n a t t h e second stage (Stein 1945, Cox 1952, Seelbinder 1953, S t a r r 1966, S t a r r and Woodroofe 1968, Cochran 1977).

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n = 30, i t e r a t e u s i n g 0

2 “1 = t”r-l-l - 4

1 = 1, 2, ... , u n t i l convergence occurs (i.e., nK+l = nK). A sample s i z e

n = n then ensures no more than a percentage sampl ing er ror PE with

(1- a) ~100% p r o b a b i l i t y . K

Table 1 i l l u s t r a t e s examples o f convergent i te ra t ions us ing fo rmula (11)

above (95% conf idence leve l ) . I n t h e f i r s t example, CV = 20% and PE = 10%. S e t t i n g t h e i n i t i a l v a l u e f o r n a t t h e 0 t h i t e r a t i o n ( I = 0 ) a t n = 30, t h e

0 associated t value wi th 29 degrees of freedom i s t29 = 2.045. The

ca l cu la ted va lue t hen f o r n, us ing formula (111, i s 16.731. Rounding upward

( t o be conservat ive) and moving t o t h e 1 s t i t e r a t i o n (I = 11, we s e t n 17

wi th the assoc iated t value o f t16 = 2.120. The c a l c u l a t e d v a l u e f o r n,

again us ing formula (111, i s 17.976. Rounding upward and moving t o t h e 2nd

i t e r a t i o n (I = 21, we s e t n2 = 18 wi th the assoc iated t v a l u e o f t17 =

2.110. The c a l c u l a t e d n value f rom (11) i s 17.805 which rounds up t o n3 =

18. Since n3 = n t h e i t e r a t i o n p r o c e s s h a s s t a b i l i z e d t o t h e s o l u t i o n .

Hence, i n t h e f i r s t example, t h e c l a s s i c a l i t e r a t i o n p r o c e s s f o r CV = 20% and

PE = 10% converges t o n = 18. I n the second example, c l a s s i c a l i t e r a t i o n f o r

CV = 14% and PE = 3% converges t o n = 87.

1 =

2 ’

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A CORRECTED ITERATION SCHEME

Unfortunately, convergence does not always occur with t h i s c l a s s i c a l

i t e r a t i v e method. Table 2 i l l u s t r a t e s some examples where convergence does

not occur (95% conf idence leve l ) . I n t h e f i r s t example, i t e r a t i o n f o r CV = 8%

and PE = 10% d i v e r g e s , o s c i l l a t i n g between nK = 4 and nK+l = 7. I n t h e

second example, i t e r a t i o n f o r CV = 3% and PE = 5% a g a i n d i v e r g e s , o s c i l l a t i n g

between nK = 2 and nK+l = 59. Divergent examples l i k e t h e s e a r e common

fo r low va lues o f Q ( i . e . , Q 1). 5

A mathemat ica l l y cor rec t (convergent ) i te ra t . ion scheme c o n s i s t s o f

r e w r i t i n g f o r m u l a (10) as

= Q n t

Note t ha t Q (= CV /PE i s f i x e d . S i n c e t i s a d e c r e a s i n g f u n c t i o n o f n,

n / t i s a n i n c r e a s i n g f u n c t i o n o f n. The c o r r e c t e d j - t e r a t i o n scheme then

c o n s i s t s o f c h o o s i n g t h e f i r s t n, n = 2, 3 , ... , such that n / t > 4.

2 2

2

2 6

Table 3 l i s t s some n i t values (95% Confidence level). Given an 2

e s t i m a t e d c o e f f i c i e n t o f v a r i a t i o n (CV) and prescr ibed a l lowable percentage

sampling error (PE), Q can be calculated and compared with Table 3 v a l u e s t o

determine sample s i z e ( n ) . F o r example, i f CV == 5% and PE = 8%, t h e n Q = ,391

and n = 4 i s t h e f i r s t n such t ha t n / t2 Q (95% con f idence l eve l ) .

5Apparent ly , s ince var ia t ion i s u s u a l l y f a i r l y l a r g e i n f o r e s t r y app l i ca t i ons ( i . e . , CV > PE), t h i s p rob lem with divergence has not been genera l l y recogn ized (bu t c f . Grosenbaugh 1947, Freese 1956).

6Choosing t h e f i r s t n w h i c h s a t i s f i e s n / t2 2 Q ensures tha t i t i s t h e c loses t i n tege r va lue wh ich conse rva t i ve l y app rox ima tes t he so lu t i on t o (12).

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Table 3 reso lves the ques t ions o f sample s i z e f o r t h e examples o f Table 2

(95% conf idence leve l ) . I n t h e f i r s t example, CV = 8% and PE = lo%, so

Q = .640 and n = 5 i s t h e f i r s t n such t h a t n / tL 2 Q . I n t h e second example,

CV = 3% and PE = 5%, so 4 = .360 and n = 4 i s t h e f i r s t n such t ha t n / t 2 2 Q .

SAMPLE SIZE TABLES

FOR SIMPLE RANDOM SAMPLING

U s i n g t h i s c o r r e c t e d i t e r a t i o n scheme, f o u r sample s i z e t a b l e s have heen

cons t ruc ted f o r s imp le random ~ a m p l i n g . ~ Tables 4-7 express sample sizes

( n ) r e q u i r e d f o r e s t i m a t e d c o e f f i c i e n t s o f v a r i a t i o n ( C V ) t o ensure prescr ibed

a l lowable percentage sampl ing er rors (PE) a t v a r y i n g l e v e l s o f confidence.

Four levels of confidence have been chosen: 9% confidence (Table 41, 95%

conf idence (Table 51, 90% conf idence (Table 61, and 80% conf idence (Table 7).

The c o e f f i c i e n t s o f va r ia t i on range f rom 1% t o 150% (except fo r the 99%

c o n f i d e n c e l e v e l t a b l e where t h e sample s i z e s become so l a r g e with h i g h C V ' s

as t o become meaningless; hence we on ly cons ider C V ' s up t o 100%). The

percentage sampl ing errors are: 1%, 3%, 5%, lo%, 15%, 20%, 25%, 50% and 100%.

To i l l u s t r a t e t h e use o f t h e s e t a b l e s , suppose a conf idence leve l o f 95%

i s r e q u i r e d (see Table 5). I f t h e c o e f f i c i e n t o f v a r i a t i o n i s e s t i m a t e d t o be

20% ( t h i s v a l u e c o u l d be est imated f rom prev ious exper ience wi th " s i m i l a r 1 1

popu la t i ons or from a p i l o t s u r v e y ) , sample s i z e s o f 174, 64, and 18 would be

requ i red to ensure p rescr ibed a l lowab le percentage sampl ing e r ro rs o f 3%,

'These t a b l e s have been generated using computer programs which have been

developed t o c o n s t r u c t sample s i z e t a b l e s f o r s p e c i f i e d l e v e l s o f c o n f i d e n c e

and ranges o f PE's and C V ' s . The c o n s t r u c t i o n o f a d d i t i o n a l t a b l e s o f t h i s

t y p e f o r o t h e r l e v e l s o f c o n f i d e n c e and ranges o f PE's and C V ' s can he

arranged by con tac t i ng t he au tho r .

- 9 -

5%, and lo%, r e s p e c t i v e l y . I f , however, t h e e s t i m a t e d c o e f f i c i e n t o f

v a r i a t i o n i s 30%, sample s i z e s o f 387, 141 and 38 wou ld be requ i red to

m a i n t a i n t h e same percentage sampl ing errors. Note how t h e r e q u i r e d sample

s izes inc rease as the C V ' s increase and the PE's decrease.

It may sometimes be a p p r o p r i a t e t o u s e t h e maximum o f a range o f p o s s i b l e

C V ' s i n t h e t a b l e s t o d e t e r m i n e sample s i z e when t h e p r e c i s i o n o f t h e e s t i m a t e

f o r t h e CV i s i n doubt.8 For ins tance, i f t h e C V ' s a re es t ima ted t o he

between 30% and 40%, a sample s i z e o f n = 64 will ensure an al lowable

percentage sampl ing error of PE = 10% (n = 64 i s t h e sample s i z e r e q u i r e d t o

ensure a l lowable percentage sampl ing er ror PE = 10% f o r t h e maximum CV =

40%). More r i g o r o u s l y , u p p e r l i m i t s o f o n e - s i d e d c o n f i d e n c e i n t e r v a l s f o r CV

est imates may be used i n t h e t a b l e s t o d e t e r m i n e sample size.

L i n e a r i n t e r p o l a t i o n nay be used t o g i v e a rough approx imat ion o f sample

s i z e s for in termediate percentage sampl ing er ror va lues not inc luded i n t h e

tab le.9 For ins tance, a CV = 15% and a l l owab le PE = 12% requ i res an

approximate sample s ize of n = 10 ( i n t e r p o l a t i n g between PE = 10% and PE = 15%

us ing n = 12 and n = 7, r e s p e c t i v e l y ) .

8As i n d i c a t e d e a r l i e r , t h e a l g o r i t h m used t o c o n s t r u c t t h e s e t a b l e s assumes p r e c i s e e s t i m a t e s f o r t h e c o e f f i c i e n t s o f v a r i a t i o n ( i . e . , a n e s t i m a t e t h a t i s c l o s e t o t h e t r u e v a l u e ) .

%ut, note, sample s i z e i s a d e c i d e d l y n o n l i n e a r f u n c t i o n o f percentage sampl ing er ror . I n fac t , sample s i z e i s a n i n v e r s e s q u a r e f u n c t i o n o f percentage sampling error ( c f . f o rmu la (10) ) .

- 10 -

For a f i x e d c o e f f i c i e n t o f v a r i a t i o n , t h e p e r c e n t a g e s a m p l i n g e r r o r s

decrease asymptot ical ly as the sample s izes increase. Thus, a law o f

d im in i sh ing re tu rns app l i es : con t i nued i nc reases i n sample s i z e r e s u l t i n

diminishing improvements i n prec is ion ( i .e . , decreases i n sampl ing e r ro r ) .

Hence sampl ing in tens i ty leve ls can he chosen which o f f e r a n "optimum1t

marginal improvement i n p r e c i s i o n p e r i n c r e a s e i n sample size. For example,

i f CV = 25%, doubl ing the sample s ize f rom n = 14 t o n = 28 decreases the

percentage sampl ing error f rom PE = 15% t o approximately PE = 10%. Adding

another 14 samples, however, on ly reduces the percentage sampl ing e r ro r to

approximately PE = 9%. Cost and t ime cons t ra in ts on sampl ing in tens i ty m igh t

then suggest using a sample s i z e of n = 28 t o ensure a percentage sampling

e r r o r o f PE = lo%, thus employing an "opt imal" design i n t h e sense o f r e a l i z i n g the "best" possible improvement i n p r e c i s i o n per inc rease i n sample

s ize.

SAMPLE SIZE TABLES FOR DENSITY SAMPLING

USING FIXED-AREA PLOTS

Consider now the case where f i xed-area sample p lo ts (quadra ts ) a re be ing

randomly pos i t i oned t o p rov ide es t ima tes o f dens i t y (e .g . , t r ee dens i t y ) . The

n sample measurements xl, x2, ... t h e n a r e t h e c o u n t s o f t h e number

o f p o p u l a t i o n members i n s i d e e a c h o f t h e sampled quadrats. 9 'n

An es t ima te o f dens i t y (pe r quadra t ) i s g i ven by t he mean (T) o f these

sampled quadrat counts. The (1- a ) . loo% con f idence i n te rva l es t ima te i s g i ven

by [T-E,TT+El where E=tn - 1, a,2 s/Jn i s the samp l ing e r ro r . Po in t and

c o n f i d e n c e i n t e r v a l e s t i m a t e s f o r t he dens i t y pe r hec ta re and popu la t i on

t o t a l s a r e t h e n g i v e n by ax and [a?-aE,aF+aEl, r e s p e c t i v e l y , where a i s t h e

approp r ia te sca l i ng f ac to r (a = (a rea pe r hec ta re ) / (a rea pe r Quadra t ) f o r dens i ty per hec tare ; a = ( t o t a l a r e a ) / ( a r e a p e r q u a d r a t ) f o r popu la t i on

t o t a l s ) .

- 11 -

An es t ima te o f spac ing can a l so be ob ta ined f rom th i s t ype o f samp l ing by

us ing t he sampled variancelmean r a t i o ( ~ ~ 1 x 1 . Phe u s e o f t h i s r a t i o a s an

i n d i c a t o r o f s p a t i a l d i s t r i b u t i o n ( i . e . , i n d e x o f nonrandomness) is well-known

i n t h e q u a n t i t a t i v e e c o l o g y l i t e r a t u r e ( D a v i d and Moore 1954, L loyd 1967,

M o r i s i t a 1959, P i e l o u 1969) . I f the spac ing i s random, t h e s p a t i a l

d i s t r i b u t i o n i s d e s c r i b e d by a Po isson d i s t r i bu t i on wh ich has a var iance equal

t o i t s mean; hence t h e v a r i a n c e h e a n r a t i o e q u a l s 1. I f the spac ing i s

regu la r , t he va r iance i s l ess t han t he mean and the var iance lmean ra t io i s

l e s s t h a n 1. The b i n o m i a l d i s t r i b u t i o n i s sometimes used t o d e s c r i b e r e g u l a r

spacing. Conversely, i f the spac ing i s aggregated (clumped), the variance i s

g rea te r t han t he mean and the var iance lmean ra t io i s g rea te r t han 1. The

n e g a t i v e b i n o m i a l d i s t r i b u t i o n i s sometimes used to descr ibe aggregated

spacing.

When dens i t y i s being est imated using quadrat sampl ing, (10) can be

ad jus ted by express ing the squared coef f i c ien t of v a r i a t i o n a s t h e p r o d u c t o f

the variance/mean r a t i o and t h e r e c i p r o c a l o f t h e mean.

Thus t h e c o e f f i c i e n t o f v a r i a t i o n c a n b e e x p r e s s e d a s a f u n c t i o n o f two

components, t he va r iance lmean ra t i o ( s 2 / g ) and the mean ( i i ) . Sample s i z e

tab les f o r dens i t y es t ima t ion can hence be cons t ruc ted us ing es t ima tes f o r

spac ing and dens i t y ra the r t han coe f f i c i en ts o f va r ia t i on . 10

1OThis may seem l i k e a c i r c u l a r argument t o u s e e s t i m a t e s o f d e n s i t y t o prov ide sampl ing gu ide l ines for est imat ing densi ty . However, i t i s r e a l l y j u s t a minimum o f t h e e s t i m a t e o f d e n s i t y w h i c h i s r e q u i r e d i n o r d e r t o u s e t h e t a b l e . Hence the minimum o f a range o f densi t ies o r a lower limit o f a one-s ided con f idence i n te rva l f o r a p r o j e c t e d e s t i m a t e o f d e n s i t y will s u f f i c e .

- 12 -

Tables 8-11 l i s t sample s i ze t ab les f o r dens i t y samp l ing . These t a b l e s

express sample size as a f u n c t i o n o f e s t i m a t e s o f d e n s i t y ( 2 ) and spacing

(s21X) and p resc r ibed a l l owab le samp l ing e r ro r (PE) a t v a r y i n g l e v e l s o f

conf idence: 99% (Table 81, 95% (Table 91, 90% (Table 10) and 80% (Table 11). Three types of spacing have been included: regular spacing (s21X = .042),

random spacing ( s = 1.01, and aggregated spacing (s21X = 3.0). The

regu la r spac ing var iance lmean ra t io va lue cor responds to tha t p roduced by

regular (square) 2.5 meter spacing sampled with 50 square meter quadrats

( rad ius = 3.99 meters). The aggregated spacing variancelmean r a t i o v a l u e

corresponds t o t h a t produced by a randomly spaced stand treated with s t r i p or cross-ha tch th inn ing (2 .5 m e t e r s t r i p s ) and sampled with 50 square meter

quadrats. The d e n s i t y l e v e l s c o r r e s p o n d t o t h e f o l l o w i n g t y p e s o f r e g u l a r

(square 1 spacing :

2

( i ) 625 stemslhectare = 4.0 meter (square spacing);

(ii) 816 stemslhectare = 3.5 meter (square spacing);

( i i i ) 1111 stemslhectare = 3.0 meter (square spacing) ; ( i v ) 1600 stemslhectare = 2.5 meter (square spacing);

( v ) 2500 stemslhectare = 2.0 meter (square spacing);

( v i ) 4444 stemslhectare = 1.5 meter (square spacing);

( v i i ) 10000 stemslhectare = 1.0 meter (square spacing).

Percentage sampl ing errors of 1%, 3%, 5%, lo%, 25%, and 50% have been

inc luded i n t h e t a b l e s . Quadrat sizes of 10, 20, 30, 40, 50 , 100, 150, 200,

and 300 square meters have also been included.

To i l l u s t r a t e t h e use o f t h e s e t a b l e s , r e f e r t o T a b l e 9e w h i c h l i s t s

sample s i z e s f o r 50 square meter quadrats at a 95% c o n f i d e n c e l e v e l . I n t h i s

s i tuat ion, sampl ing i n a n a t u r a l s t a n d (assumed randomly spaced) with

est imated densi ty of 1600 stems per hectare requi res 536, 195 and 51 samples

- 13 -

t o ensure p rescr ibed percentage sampl ing exrors o f 3%, 5%, and lo%, r e s p e c t i v e l y . I f , however, the s tand i s a p l a n t a t i o n (assumed r e g u l a r l y

spaced), only 25, 11, and 5 samples are required t o ensure the same percentage

sampling errors. Note how t h e r e q u i r e d sample s i zes i nc rease as t he dens i t y

decreases or as the spacing changes from regular t o random t o aggregated.

As b e f o r e , l i n e a r i n t e r p o l a t i o n c a n be used t o approximate sample sizes

fo r i n te rmed ia te samp l ing e r ro rs . Samp l ing e r ro r can a l so be es t ima ted us ing

these tab les by r e f e r r i n g t o t h e a p p r o p r i a t e c o n f i d e n c e l e v e l , a u a d r a t s i z e ,

densi ty est ima.te, spacing est imate, and sample s ize.

It should be no ted tha t these tab les can a lso be used to p rov ide

gu ide l ines fo r choos ing an t top t ima l " quadra t s ize fo r dens i ty sampl ing . T h i s

can be determined by comparing the sampling intensit ies ( i .e., percentage area

sampled) r e q u i r e d f o r d i f f e r e n t q u a d r a t s i z e s t o e n s u r e a p resc r ibed

pe rcen tage samp l ing e r ro r f o r g i ven es t ima tes o f dens i t y and spacing. Note

tha t t he es t ima tes o f spac ing , g i ven by t h e v a r i a n c e h e a n r a t i o , will vary

with t h e s i z e o f t h e q u a d r a t and should be examined using e i t h e r computer

s i m u l a t i o n or p i l o t sampling i n t h e f i e l d . 11

llA computer program, SAMPLE, has been developed which simulates quadrat sampling on a p r e s c r i b e d s p a t i a l p a t t e r n o f t r e e s . For more d e t a i l s about th is s imu la to r o r about quadra t sampl ing i n general , see: Stauffer, H.B. 1982. The use o f quadrat sampling t o estimate tree d e n s i t y and spacing (unpublished notes). B.C. M i n i s t r y o f F o r e s t s , V i c t o r i a , B.C.

- 1 4 -

ACKNOWLEDGEMENTS

I wish t o thank Jul ien Demaerschalk, Lee Wensel, Albert Stage, Char les

Hatch, Wi l l iam Reed, Ken M i t c h e l l , James Goudie, Mik Kovats, Mike Wyeth, Tony

Kozak, F ranc i s Yeh, Carole Leadem, and Jim D a n g e r f i e l d f o r t h e i r many he lp fu l .

comments which have i n f l u e n c e d t h e f i n a l v e r s i o n o f t h i s paper.

- 15 -

REFE:RENCES

Avery, T.E. _ . 1975. Natural resources measurements, 2nd ed. McGraw-Hi New York. 339 p.

11, Inc . ,

Cochran, W.G. 1977. Sampling techniques, 3rd ed. John Wiley and Sons, New York. 428 p.

Cox, D.R. 1952. Estimation by double sampling. Biometrika 39:217-227.

David, F.N. and P.G. Moore. 1954. Notes on c o n t a g i o u s d i s t r i b u t i o n s i n p l a n t populat ions. Ann. Bot. Lond. N.S. 18:47-53.

Freese, F. 1956. Guidebook f o r s t a t i s t i c a l t r a n s i e n t s . U.S. Dept. o f Agriculture, Forest Service, Southern Forest Experiment Stat ion. 77 p.

Freese, F. 1962. Elementary forest sampl ing. Agr icu l tura l Handbook No. 232. U.S. Dept. o f Agr icu l tu re , Fores t Serv ice . 91 p.

Freund, J.E. and R.E. Walpole. 1980. Mathemat ica l s ta t i s t i cs , 3rd ed. Prent ice-Hal l , Inc. , Englewood C l i f f s , New Jersey. 548 p.

Grosenbaugh, L.R. 1947. Elementary design and a n a l y s i s i n f o r e s t r e s e a r c h . Colo. State Univ. , Fort Col l ins, Colo. 133 p.

L i , J.C.R. 1964. S t a t i s t i c a l i n f e r e n c e I. Edwards Brothers, Ann Arbor, Mich. 658 p.

Lloyd, M. 1967. Mean crowding. J. Anim. Ecol. 36:l-30.

Morista, M. 1959. Measuring the d ispers ion o f i n d i v i d u a l s and t h e ana lys is of the d i s t r i b u t i o n a l p a t t e r n s . Mem. Fac. Sci. Kyushu U. Ser ies E (B io l . ) 2:215-235.

Pielou, E.C. 1969. An in t roduc t ion to mathemat ica l eco logy . John Wiley and Sons, New York. 286 p.

Schaeffer, R.L., W. Mendenhall and L. O t t . 1979. Elementary survey sampling, 2nd ed. Duxburg Press, North Scituate, Mass. 278 p.

Seelbinder, B.M. 1953. On Stein's two-stage sampling scheme. Ann. Math. S ta t . 24:640-649.

Snedecor, G.W. and A.M. Mood. 1946. Query and answer. B iomet r i cs Bu l l . 2 : 120-122.

- 16 -

S ta r r , N. 1966. The performance o f a sequent ia l procedure for the f ixed-width i n t e r v a l e s t i m a t i o n o f t h e mean. Ann. Math. S ta t . 37(1) :36-50.

Star r , N. and M.B. Woodroofe. 1968. Remarks on a stopping t ime. Proc. N.A.S. 61(4) ~1215-1218.

Stauf fe r , H.B. 1982. A sample s ize tab le for forest sampl ing. Forest Sc ience 28 ( 4 ) : 777-784.

Stein, C. 1945. A two-sample t e s t f o r a l i nea r hypo thes i s whose power i s independent o f the var iance. Ann. Math. S ta t . 16:243-258.

U.B.C. Forest Club. 1971. Fores t ry handbook f o r B r i t j - s h Columbia, 3rd ed. Univ. o f B.C., Vancouver, R.C. 815 p.

Wensel, L.C. 1977. Wild l i fe resource sampl ing (draf t copy) . Univ . Cal i f . , Berkeley. 273 p.

- 17 - .#I,@

Table 1. Examples o f convergence, us ing the c lass ica l i t e ra t ive a lgor i thm to ca lcu la te sample s i z e (95% confidence level) . Note:

CV = estimated coefficient of variation,

PE = allowable percentage sampling error,

Q = CV2lPE2,

I = i t e r a t i o n number,

n I = sample s ize ca lcu la ted a t the I t h i t e r a t i o n ,

t, = t value w i t h m degrees of freedom (2.5% probabi l i ty ) ,

n = sample s i ze .

d

a ) Convergence: CV = 20%, PE = lo%, Q = 4.000, n = 18

I n 1 tn1 -1 n1+1 "

0 30 2.045 16.731 1 17 2.120 17.976 2 18 2.110 17.805 3 18

b ) Convergence: CV = 145'6, PE = 3%, Q = 21.778, n = 87

I "I tq-1 n1+1

0 30 2 045 91.094 1 92 1.986 85.927 2 86 1.988 86.090 3 87 1.988 86.061 4 87

- 18 -

Table 2. Examples o f divergence, u s i n g t h e c l a s s i c a l i t e r a t i v e a l g o r i t h m t o ca lcu la te sample s i z e (95% confidence level) . Note:

CV = estimated coefficient o f variat ion,

PE = allowable percentage sampling error,

Q = CV2/PE2,

I = i t e r a t i o n number,

n I = sample s i ze ca l cu la t ed a t t he I t h i t e r a t i o n ,

t m = t value w i t h m degrees of freedom (2.5% probabi l i ty ) ,

n = sample s i z e .

a > Divergence: CV = 8%, PE = lo%, Q = 0.640

I "I t n 1 - 1 n1+1

30 2.045 2.677 3 4.303 11.850

12 2.201 3.100 4 3.182 6.480 7 2.447 3 . a31 4

b) Divergence: CV = 3%, PE = 5%, Q = 0.360

I "I t n 1-1 "1+1 - 0 30 2.045 1.506 1 2 12.706 58.119 2 59 2.002 1.442 3 2

- 19 - ,I,##

Table 3. Examples from Table 2, u s i n g t h e c o r r e c t e d i t e r a t i o n scheme t o c a l c u l a t e sample s i z e (95% conf idence leve l ) . Note:

CV = e s t i m a t e d c o e f f i c i e n t o f v a r i a t i o n ,

PE = al lowable percentage sampl ing error,

Q = CV2IPE2,

tm = t value with m degrees o f freedom (2.5% p r o b a b i l i t y ) ,

n = sample size.

n

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

n/ t& l

0.012 0 J.62 0.395 0.649 0.908 1.169 1.431 1.693 1.954 2.216 2.477 2.739 3.000 3.261 3.522

4.044 4.305 4.566

3:7a3

a) CV = 8%, PE = 1096, Q = 0.640: n = 5

b) CV = 3%, PE = 5%, Q = 0.360: n = 4

- 20 - Table 4. Sample size table (99% confidence): sample sizes (n)

required for estimated coefficients of variation ( C V ) to ensure prescribed allowabie percentage sampling errors (PE).

cv n: n: n: n: n: n: n: n: n: PE=l% PE=3% PE=5% PE=lO% PE=15% PE=20% PE=25% PE=50% PE=100% **** ***** ***** * x * * * X * * * * * * x * * * * * * * * x * **I*** ****** * * * * x * *

1% 1 1 2% 31 3% 64 4% 1 1 1 5% 170 6% 243 7% 330 8% 429 9% 542 10% 668 1 1 % 807 12% 960 13% 1126 14% 1305 15% 1498 16% 1703 17% 1922 18% 21 55 19% 2400 20% 2659 21% 2931 22% 3217 23% 3515 24% 3%27 25% 4152 26% 4491 27% 4843 28% 5208 29% 5586 30% 5978 31% 6382 32% 6801 33% 7232 34% 7677 35% 8135 36% 8606 37% 9090 38% 9588 39% 10099 40% 10624 41% 11161 42% 11712 43% 12276 44% 12854 45% 13444 46% 14048 47% 14665 48% 15296 49% 15940 50% 16597 51% 17267 52% 17951

4 7

1 1 16 23 31 40 52 64 78 94

1 1 1 129 149 170 193 217 243 27 1 299 330 36 1 394 429 465 503 542 582 625 668 713 759 807 857 908 960 1014 1069 1126 1184 1244 1305 1368 1432 1498 1565 1633 1703 1775 1848 1922 1998

53% 18648 2076

4 5 6 8

1 1 14 17 21 26 31 36 43 49 56 64 72 81 90 100 1 1 1 121 133 145 157 170 184 198 212 228 243 259 276 293 31 1 330 348 368 388 408 429 45 1 473 495 518 542 566 59 1 61 6 642 668 695 722 750

3 4 4 5 6 6 7 8 10 1 1 12 14 15 17 19 21 23 26 28 31 34 36 39 43 46 49 53 56 60 64 68 72 77 81 86 90 95 100 105 1 1 1 116 121 127 133 139 145 151 157 164 170 177 184 191

3 3 4 4 4 5 5 6 6 7 8 8 9 10 1 1 12 13 14 15 16 17 19 20 21 23 24 26 27 29 31 33 35 36 38 40 43 45 47 49 52 54 56 59 61 64 67 69 72 75 78 81 84 87

3 3 3 4 4 4 5 5 5 6 6 6 7 7 8 8 9 10 10 1 1 12 12 13 14 15 15 16 17 18 19 20 21 22 23 25 26 27 28 30 31 32 34 35 36 38 39 41 43 44 46 47 49 51

3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 7 7 8 8 8 9 9 10 10 1 1 1 1 12 13 13 14 14 15 16 17 17 18 19 20 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 10 10 10 10 1 1 1 1 1 1 1 1 12

2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6

- Z I - Table 4 continued ...

cv n: n: n: n: n: n: n: n: n: PE=l% PE=3% PE=5% PE=10% PE=15% PE=20% PE=25% PE=50% PE=100%

* * * x x * * * * x * * * * x * * * * ****** ****** * * * * x * x * * * * * * x * * * * * * * * * x *

54% 19358 55% 20081 56% 20818 57% 21 568 58% 22331 59% 23108 60% 23897 61% 24701 62% 2551 7 63% 26347 64% 27189 65% 28046 66% 28915 67% 29798 60% 30694 69% 31603 70% 32526 71% 33461 72% 3441 0 73% 35373 74% 36348 75% 37337 76% 38340 77% 39355 78% 40384 79% 41426 80% 42481 81% 43550 82% 44631 83% 45727 84% 46835 85% 47957 86% 49092 87% 50240 88% 51401 8 9 % 5 2 5 7 6 90% 53764 91% 54965 92% 56180 93% 57408 94% 58649 95% 59903 96% 61171 97% 62452 98% 63746 99% 65053 100% 66374

21 55 2235 2317 2400 2485 257 1 2659 2748 2839 293 1 3025 31 20 321 7 3315 341 4 351 5 3618 3722 3827 3934 4043 41 52 4264 4377 4491 4607 4724 4843 4963 5085 5208 5332 5458 5586 571 5 5 8 4 6 5978 61 1 1 6246 6382 6520 6660 680 1 6943 7087 7232 7379

7 78 807 837

897 928 960 992 1025 1058 1092 1126 1161 1 1 96 1232 1268 1305 1343 1381 1419 1458 1498 1538 f 578 1619 1661 1703 1746 1789 1833 1878 1922 1968 20 14 2060 21 0 7 21 55 2203 225 1 2300 2350 2400 2451 2502 2554 2606 2659

867

1 98 205 212 220 228 235 243 25 1 259 268 276 285 293 302 31 1 320 330 339 348 3 58 3 68 378 388 3 98 408 41 9 429 440 45 1 462 473 484 495 507 518 5 3 0 542 554 566 578 59 1 603 61 6 629 642 655 668

90 94 97 100 104 107 1 1 1 114 118 121 125 129 133 137 141 145 149 153 157

166 170 175 179 184 188 193 198 203 208 212 217 222 228 233 2 3 8 2 4 3 249 254 259 265 27 1 276 282 288 293 299

162

53 55 56 58 60 62 64 66 68 70 72 74 77 79 81 83 86 88 90 93 95 98 100 103 105 108 1 1 1 113 116 119 121 124 127 130 133 1 3 6 139 142 145 148 151 154 157 160 164 167 170

35 36 38 39 40 41 43 44 45 46 48 49 51 52 53 55 56 58 59 61 62 64 66 67 69 71 72 74 76 77 79 81 83 85 87 88 90 92 94 96 98 100 102 104 1 C6 108 1 1 1

12 12 13 13 13 13 14 14 14 15 15 15 16 16 17 17 17 18 18 18 19 19 20 20 20 21 21 22 22 23 23 23 24 24 25 2 5 26 26 27 27 28 28 29 29 30 30 3 1

6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 1 1 1 1 1 1

- 2 2 - Table 5 . Sample size table ( 9 5 % confidence): sample sizes (n)

required for estimated coefficients of variation (CV) to ensure prescribed allowable percentage sampling errors ( P E ) .

CV n: n: n: n: n: n: n: n: n: P E = ~ % pE=3% pE=5% PE=10% PE=15% PE=20% PE=25% PE=50% PE=100% **** ***** ***** * * * * x X * * * * * x * * * * * * * * X * * * * * x * * * x * * * * * * * x * * *

1 % 7 2% 18 3% 38 4 % 64 5 % 9 9 6% 1 4 1 7% 191 8% 249 9% 314

10% 387 1 1 % 468 12% 556 13% 652 14% 756 15% 867 16% 986 17% 1113 18% 1248 19% 1 3 9 0 20% 1539 21% 1697 22% 1862 2 3 % 2035 24% 2216 25% 2404 26% 2600 27% 2803

29% 3234 30% 3460 31% 3694 32% 3936 33% 4186 34% 4444 35% 4709 36% 4981 37% 5262 38% 5550 39% 5846 40% 6149 4 1 % 6460 42% 6779 43% 71 06 44% 7440 45% 7782 46% 8131 47% 8 4 8 9 48% 8 8 5 3 49% 9226 50% 9606 51% 9994 52% 10390

28% 301 5

3 5 7

1 0 1 4 18 24 30 38 46 5 5 64 75 87 99

112 126 1 4 1 157 174 191 210 229 249 270 2 9 1 314

362 387 41 3 440 468 496 5 2 6 556 587 61 9 652 686 720 756 792 8 2 9 867 906 946 9 8 6

1028 1070 1 1 13 1157

338

53% 10793 1202

3 3 4 5 7 9

1 1 1 3 1 5 18 22 2 5 29 3 3 38 42 47 5 3 58 64 71 77 8 4 91 99

107 115 1 2 3 132 1 4 1 151 160 170 181 191 202 21 3 2 2 5 237 249 26 1 274 287 3 0 0 314 3 2 8 342 357 372 387 403 4 1 8 435

2 3 3 3 4 4 5 5 6 7 8 9 9

1 1 12 13 14 15 1 7 18 20 22 23 25 27 29 31 33 3 5 38 40 42 45 47 50 5 3 56 58 61 64 68 71 74 77 81 84 8 8 91 9 5 99

103 107 1 1 1

2 3 3 3 3 3 4 4 4 5 5 5 6 6 7 7 8 9 9

10 1 1 1 1 12 1 3 1 4 15 1 5 16 17 18 1 9 20 22 23 24 2 5 26 28 29 30 32 33 3 5 36 38 3 9 41 42 44 46 47 49 51

2 2 3 3 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 7 7 8 8 9 9 9

10 1 1 1 1 12 12 1 3 1 3 1 4 1 5 1 5 16 17 18 18 19 20 21 22 22 23 24 2 5 26 27 28 29 30

2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 7 7 7 7 8 8 9 9 9

10 10 1 1 1 1 1 1 12 12 1 3 1 3 1 4 1 4 1 5 1 5 16 17 17 18 18 19 20 20

2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7

2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4

- 23 - Table 5 continued ...

CV n: n: n: n: n: n: n: n: n: PE=l% PE=3% PE=5% PE=10% PE=15% PE=20% PE=25% PE=50% PE=100%

x * * * ***** ***** ***** ****** ****** * * x * * * ****** ****** x * * * * * *

54% 11204 55% 11623 56% 12050 57% 12484 58% 12925 59% 13375 60% 13832 61% 14297 62% 14769 63% 15249 64% 15737 65% 16233 66% 16736 67% 17247 68% 17765 69% 18292 70% 18826 71% 19367 72% 19917 73% 20474 74% 21038 75% 2161 1 76% 22191 77% 22778 78% 23374 79% 23977 80% 24588 81% 25206 82% 25832 83% 26466 84% 27108 85% 27757 86% 28414 87% 29078 88% 29751 89% 30431 90% 31118 91% 31813 92% 32516 93% 33227 94% 33945 95% 34671 96% 35405 97% 361 46 98% 36896 99% 37652 100% 38417 101% 39189 102% 39969 103% 40756 104% 41551 105% 4 2 3 5 4 106% 43165 107% 43983

1248 1294 1341 1390 1439 1489 1539 1591 1644 1697 1751 1806 1862 1919 1977 2035 2094 21 55 221 6 2277 2340 2404 2468 2534 2600 2667 2735 2803 2873 2943 301 5 3087 31 60 3234 3308 3384 3460 3537 361 6 3694 3774 3855 3936 401 9 41 02 4186 427 1 4357 4444 453 1 461 9 4709 4799 4890

45 1 468 485 502 520 538 556 57 5 594 61 3 632 652 672 693 713 734 756 777 799 822 844 867 890 91 4 938 962 98 6 101 1 1036 1061 1087 1 1 13 1139 1166 1193 1220 1248 1275 1303 1332 1361 1390 1419 1449 1479 1509 1539 1570 1602 1633 1665 1697 1729 1762

108% 44809 4981 1795

115 119 123 128 132 137 141 146 151 155 160 165 170 175 181 186 191 197 202 208 21 3 219 225 23 1 237 243 249 255 26 1 268 274 280 287 294 300 307 31 4 32 1 328 335 342 350 357 364 372 379 387 395 403 410 418 4 2 6 435 443 45 1

53 55 56 58 60 62 64 66 69 71 73 75 77 80 82 84 87 89 91 94 96 99 102 104 107 109 112 115 118 121 123 126 129 132 135 138 141 144 147 151 154 157 160 164 167 170 174 177 181 184 188 191 195 198 202

31 32 33 34 35 36 38 39 40 41 42 44 45 46 47 49 50 51 53 54 56 57 58 60 61 63 64 66 68 69 71 72 74 76 77 79 81 82 84 86 88 90 91 93 95 97 99 101 103 YO5 107 " 09 '! 1 1 113 115

21 22 22 23 24 24 25 26 27 27 28 29 30 31 31 32 33 34 35 36 37 38 38 39 40 41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 60 61 62 63 64 66 67 68 69 71 72 73 75

7 8 8 8 8 8 9 9 9 9 9 9 10 10 10 10 1 1 1 1 1 1 1 1 1 1 12 12 12 12 13 13 13 13 14 14 14 14 15 15 15 15 16 16 16 17 17 17 17 18 18 18 19 19 19 20 20 20 21 21

4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7

- 2 4 - T a b l e 5 continued ...

cv n: n: n: n: n: n: n: n: n: PE=1% PE=3% PE=5% PE=10% PE=15% PE=20% PE=25% PE=50% PE=100% **** ***** ***** * * X * * * x * * * * ****** * * * * * x * * x * * * X * * * * * * * * * x * *

109% 45642 1 1 0 % 46484 1 1 1 % 47333 112% 48189 1 1 3% 49053 114% 49925 1 1 5% 50805 116% 51693 1 1 7% 52588 1 1 8 % 53490 1 19% 54401 120% 55319 12 1 % 56245 122% 57178 123% 58119 124% 59068 125% 60024 126% 60989 127% 61961 128% 62940 129% 63927 130% 64922 13 1 % 65925 132% 66935 133% 67953 134% 68979 135% 70012 136% 71053 137% 721 02 138% 73158 139% 74222 140% 75294 141% 76373 142% 77461 143% 78555 144% 79658 145% 80768 146% 8 1 8 8 6 147% 8301 1 148% 8 4 1 4 5 149% 8 5 2 8 5 150% 86434

5074 51 67 5262 5357 5 4 5 3 5550 5648 5746 5846 5946 6047 61 49 6252 6356 6460 6566 6672 67 79 6887 6996 71 06 721 6 7328 7440 7 5 5 3 7667 7782 7897 8 0 1 4 8131 8250 8 3 6 9 8 4 8 9 8 6 0 9 8 7 3 1 8 8 5 3 8 9 7 7 91 01 9226 9352 9479 9606

1828 1862 1896 1930 1965 2000 2035 2071 21 06 21 42 21 79 221 6 2253 2290 2328 2366 2404 2442 248 1 2520 2560 2600 2640 2680 272 1 2762 2803 2845 2887 2929 2972 3015 3058 3101 31 45 31 89 3234 3278 3323 3369 341 4 3460

459 468 476 4 8 5 493 502 51 1 520 529 538 5 4 7 556 5 6 5 5 7 5 584 594 6 0 3 61 3 622 6 3 2 642 652 662 672 68 2 6 9 3 7 0 3 7 1 3 724 7 3 4 7 4 5 7 5 6 767 77 7 7 8 8 7 9 9 81 1 8 2 2 8 3 3 8 4 4 8 5 6 8 6 7

206 21 0 21 3 217 22 1 225 229 233 237 24 1 245 249 253 257 26 1 265 270 274 278 28 3 287 29 1 296 300 305 309 3 1 4 319 323 328 333 338 342 347 352 357 362 367 372 377 382 387

1 1 7 119 121 1 2 3 126 128 130 132 134 137 139 141 144 146 1 4 8 151 153 1 5 5 158 160 1 6 3 1 6 5 168 170 1 7 3 175

181 1 8 3 1 8 6 188 191 194 197 1 9 9 202 2 0 5 208 210 2 1 3 216 219

178

76 77 79 8 0 81 8 3 8 4 8 6 8 7 8 9 90 91 9 3 94 96 97 9 9

1 0 1 102 104 105 107 108 110 112 1 1 3 1 1 5 117 118 120 122 123 1 2 5 127 129 130 132 134 1 3 6 138 1 3 9 141

2 1 22 22 22 23 23 23 24 24 24 25 25 2 5 26 26 27 27 27 28 28 29 29 29 30 30 3' 1 31 31 32 32 33 33 3 3 34 34 3 5 3 5 36 36 37 37 38

8 8 8 a 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9

10 10 10 10 10 10 10 1 0 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 12

- 25 - Table 6. Sample size table (90% confidence): sample sizes (n)

required for estimated coefficients of variation (CV) to ensure prescribed allowable percentage sampling errors (PE).

u u r cv n: n: n: n: n: n: n: n: n: PE=l% PE=3% PE=5% PE=10% PE=15% PE=20% PE=25% PE=50% PE=100%

* x * * ***** ***** * * * X * * * * x * * * x * * * * x * * * * * ****** * * X * * * * * * x * * *

1% 5 2% 13 3% 27 4% 46 5% 70 6% 100 7% 135 8% 176 9% 221 10% 273 11% 330 12% 392 13% 460 14% 533 15% 61 1 16% 695 17% 784 18% 879 19% 979 20% 1085 21% 1195 22% 1312 23% 1434 24% 1561 25% 1693 26% 1831 27% 1975 28% 21 23 29% 2278 30% 2437 31% 2602 32% 2773 33% 2949 34% 3 1 3 0 35% 3317 36% 3509 37% 3706 38% 3909 39% 4117 40% 4331 41% 4550 42% 4775 43% 5005 44% 5240 45% 5481 46% 5727 47% 5979 48% 6236 49% 6498 50% 6766 51% 7039 52% 7318 53% 7602

3 4 5 7 10 13 17 22 27 32 39 46 53 61 70 79 89 100 1 1 1 123 135 148 161 176 190 206 22 1 238 255 273 29 1 31 0 330 350 37 1 392 41 4 436 460 483 508 533 558 584 61 1 638 666 695 724 754 784 81 5 847

2 3 4 4 5 6 8 9

1 1 13 16 18 21 24 27 30 34 37 41 46 50 55 60 65 70 76 81 87 93 100 106 113 120 1 2 7 135 143 151 159 167 176 184 193 202 21 2 22 1 23 1 24 1 252 262 27 3 284 295 306

2 2 3 3 3 4 4 4 5 5 6 6 7 8 9 9 10 1 1 12 13 14 16 17 18 19 21 22 24 25 27 28 30 32 34 36 37 39 41 44 46 48 50 52 55 57 60 62 65 67 70 73 76 78

2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 6 6 6 7 7 8 8 9 9 10 1 1 1 1 12 13 13 14 15 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 34 35 36

2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 6 6 6 6 7 7 7 8 8 9 9 9 10 10 1 1 1 1 12 12 13 13 14 14 15 16 16 17 17 18 19 19 20 21 21

2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 9 9 9 10 10 10 1 1 1 1 12 12 12 13 13 14 14 15

2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

- 26 - Table 6 continued ...

CV n: n: n: n: n: n: n: n: n: ? E = l % PE=3% PE=5% PE=lO% PE=15% PE=20% PE=25% PE=50% PE=100%

* * x * X * * * * ***** * * * * x * * * * x * x * * * * * * * * x * * * * * x * * ****** ******* 54% 7e91 55% 8 1 8 6 56% 8487 57% 8792 58% 9 1 0 3 59% 9420 60% 9742 61% 10069 62% 10402 63% 10740 64% 1 1 084 65% 11433 66% 11787 67% 12147 68% 12512 69% 12883 70% 13259 71% 13641 72% 14027 73% 14420 74% 14817 75% 15221 7 6 % 15629 77% 16043 78% 16462 79% 16887 6 0 % 17317 8 1 % 17753 8 2 % 18 194 83% 18640 84% 19092 8 5 % 19549 86% 20012 8 7 % 20480 8 8 % 20953 8 9 % 21432 90% 21917 91% 22406 92% 22901 93% 23402 94% 23908 95% 2 4 4 1 9 96% 24936 97% 25458 98% 2 5 9 8 6 99% 26519

100% 27057 101% 27601 102% 28 150 103% 28704 104% 29264 105% 29830 106% 3040 1 107% 30977

8 7 9 91 2 945 979

1314 1049 1 0 8 5 1121 1158 1195 1234 1272 1312 1352 1392 1434 1475 1518 1561 1604 1648 1693 1739 1785 1831 1878 1926 1975 2024 2073 2123 2174 2226 2278 2330 2383 2437 2492 2547 2602 2659 2715 2773 283 1 2889 2949 3008 3 0 6 9 31 30 3191 3254 3317 3380 3444

318 330 342 3 54 366 379 392 405 4 1 8 432 446 460 474 488 503 518 533 548 5 6 3 579 5 9 5 61 1 627 644 66 1 678 6 9 5 71 2 730 7 46 766 784 8 0 3 8 2 1 8 4 0 8 6 0 8 7 9 8 9 9 91 8 938 959 97 9

1000 1021 1042 1063 1085 1106 1128 1150 1173 1 1 9 5 1218 1241

108% 31559 3509 1265

81 84 87 90 93 97

100 103 106 110 113 117 120 124 127 131 135 139 143 147 151 155 159 1 6 3 167 171 176 180 1 8 4 189 193 198 202 207 212 217 22 1 2 2 6 231 236 24 1 247 252 257 262 268 273 278 284 289 2 9 5 30 1 306 312 318

37 39 40 4 1 43 44 46 47 49 50 52 5 3 5 5 5 6 58 60 61 6 3 6 5 66 68 70 7 2 74 76 77 7 9 81 8 3 8 5 8 7 8 9 91 9 3 9 5 9 8

100 102 104 106 109 1 1 1 1 1 3 1 1 5 116 120 1 2 3 1 2 5 127 130 132 1 3 5 137 140 1 4 3

22 23 24 24 2 5 26 27 28 26 2 9 30 31 32 33 34 3 5 36 36 37 38 3 9 40 4 1 42 44 45 46 47 4 8 49 50 51 52 54 5 5 5 6 57 58 60 61 62 6 3 6 5 66 67 6 9 70 71 7 3 74 76 77 78 80 81

1 5 16 1 6 16 17 17 18 19 1 9 20 20 21 21 22 22 23 24 24 2 5 2 5 26 27 27 28 2 9 29 30 31 31 32 3 3 34 34 3 5 36 37 37 38 3 9 40 4 1 41 4 2 43 44 4 5 46 47 47 48 49 50 51 5 2 5 3

6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9

10 10 10 10 10 1 0 1 1 1 1 1 1 1 1 1 1 12 12 12 12 12 1 3 13 1 3 13 1 3 14 1 4 1 4 1 4 15 15 1 5

3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6

- 2 7 - Table 6 continued ...

cv n: n: n: n: n: n: n: n: n: p ~ = 1 % p ~ = 3 % PE=5% PE=10% PE=15% PE=20% PE=25% PE=50% PE=100% **** ***** * x * * * ***** * * * * * x x * * * * * * x * * * * ****** ****** *******

109% 32146 3574 1288 324 145 83 54 15 6 110% 32738 3640 1312 330 148 84 55 16 6

112% 33940 3773 1360 342 153 87 57 16 6 113% 34548 3841 1384 348 156 89 58 16 6 114% 35162 3909 1409 354 159 90 59 16 6 115% 35782 3978 1434 360 161 92 60 17 6 116% 36407 4047 1459 366 164 93 61 17 6 117% 37037 4117 1484 373 167 95 62 17 6 118% 37673 4188 1509 379 170 97 63 17 6

120% 38961 4331 1561 392 176 t o o 65 18 6 121% 39613 4404 1587 398 178 101 66 18 6 122% 40270 4477 1613 405 181 103 67 19 7 123% 40933 4550 1640 412 184 105 68 19 7 124% 41601 4624 1666 418 187 106 69 19 7 125% 42275 4699 1693 425 190 108 70 19 7 126% 42954 4775 1720 432 193 110 71 20 7 127% 43639 4851 1748 439 196 1 1 1 72 20 7 128% 44329 4927 1775 446 199 113 73 20 7 129% 45024 5005 1803 453 202 115 74 20 7 130% 45724 5083 1831 460 206 117 76 21 7 131% 46431 5161 1859 467 209 118 77 21 7 132% 47142 5240 1888 474 212 120 78 21 7 133% 47859 5320 1917 48 1 21 5 122 7s 22 7 134% 48581 5400 1946 488 21 8 124 80 22 7 135% 49309 5481 1975 495 22 1 126 81 22 7 136% 50042 5562 2004 503 225 127 82 22 7 137% 50781 5644 2033 510 228 129 84 23 8 138% 51525 5727 2063 51 8 23 1 131 85 23 8 139% 52274 5810 2093 525 235 133 86 23 8 140% 53029 5894 2123 533 238 135 87 24 8 141% 53789 5979 2154 540 24 1 137 88 24 8 142% 54555 6064 2184 548 245 139 90 24 8 143% 55326 6149 2215 556 248 141 91 25 8 1 4 4 % 5 6 1 0 3 6 2 3 6 2 2 4 6 5 6 3 2 5 2 1 4 3 9 2 2 5 a 145% 56884 6323 2278 57 1 255 145 93 25 8 146% 57672 6410 2309 579 259 147 95 25 8 147% 58464 6498 2341 587 262 149 96 26 8 148% 59263 6587 2373 595 266 151 97 26 8 149% 60066 6676 2405 603 269 153 98 26 8 150% 60875 6766 2437 61 1 273 155 100 27 9

111% 33336 3706 1336 336 151 86 56 16 6

119% 38314 4259 1535 385 173 98 64 18 6

- 28 - Table 7. Sample size table (80% confidence): sample sizes (n)

required for estimated coefficients of variation ( C V ) to ensure prescribed allowable percentage sampling errors (PE).

cv ****

1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 1 5% 16% 17% 18% 19% 20% 21% 22% 23% 24% 2 5% 26% 27% 28% 2 9% 30% 31% 32% 33% 34% 3 5% 3 6% 37% 38% 39% 40% 41% 42% 43% 44% 45% 46% 47% 48% 4 9% 50% 51% 52% 53%

n: PE= 1 %

4 9 17 28 43 61 82 107 135 166 20 1 238 279 324 37 1 422 476 534 595 659 726 797 87 1 948 1028 1 1 12 1199 1289 1383 1480 1580 1684 1790 1900 2014 21 30 2250 2373 2500 2630 2763 28 99 3039 3181 3328 3477 3630 3786 3945 41 08 4274 4443 461 5

* * * * x

n: PE= 3%

2 3 4 5 7 9

1 1 14 17 20 24 28 33 38 43 49 55 61 68 75 82 90 98 107 116 125 135 145 155 166 177 189 20 1 21 3 225 238 252 265 279 294 309 324 339 355 37 1 388 405 422 440 458 476 495 514

x * * * *

n: PE= 5%

2 2 3 3 4 4 5 6 7 9 10 1 1 13 15 17 19 21 23 2 6 28 31 34 37 40 43 46 50 53 57 61 65 69 73 78 82 87 92 97 102 107 112 118 123 129 135 141 147 153 160 166 173 179 186

* * * x *

n: n: PE=10% PE=15% x * * * * *

2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 7 7 8 9 9 10 1 1 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 26 27 28 29 31 32 34 35 37 38 40 41 43 45 46 48

* x * * * *

2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 5 5 5 6 6 6 7 7 7 8 8 9 9 9 10 10 1 1 1 1 12 12 13 14 14 15 15 16 17 17 18 19 19 20 21 22 22

n: n: PE=20% PE=25% * * x * * *

2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 1 1 1 1 1 1 12 12 13 13 13

****** 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9

n: PE=50%

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4

* * * * x *

n: PE= 100%

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2

2 2 2 2 2 3 3 3 3 3 3 3 3

x * * * * * *

- 29 - Table 7 continued ...

W'

cv n : n: n: n: n: I1 : n: n: n: PE=l% PE=3% PE=5% PE=10% PE=15% PE=20% PE=25% PE=50& PE=100% **** * * * X * * * * * x * x * * * ****** x * * * * * x * * * * * x * * * * * ****** * * * * * x *

54% 4791 55% 4970 56% 51 52 57% 5338 58% 5527 59% 5719 60% 591 4 61% 6 1 1 3 62% 6 3 1 5 63% 6520 64% 6729 65% 6941 66% 7156 67% 7374 68% 7596 69% 7821 70% 8 0 4 9 71% 8281 72% 8 5 1 6 73% 8754 74% 8 9 9 6 75% 9240 76% 9488 77% 9740 78% 9994 79% 10252 80% 10513 8 1 % 10777 8 2 % 11045 8 3 % 11316 8 4 % 11590 8 5 % 1 1 8 6 8 8 6 % 12149 8 7 % 12433 8 8 % 12720 89% 1 3 0 1 1 90% 13305 91% 13602 92% 13903 93% 14207 94% 14514 95% 14824 96% 151 38 97% 15455 98% 15775 99% 16099

100% 16426 1 0 1 % 16756 102% 17089 103% 17426 1 0 4 % 17766 105% 18109 106% 18456 107% 18805

534 554 574 5 9 5 61 6 637 659 68 1 703 726 749 773 797 8 2 1 8 4 6 87 1 8 9 6 922 948 974

1001 1028 1056 1 0 8 4 1 1 1 2 1141 1170 1199 1229 1259 1289 1320 1351 1383 1415 1 4 4 7 1480 1513 1546 1580 1614 1649 1684 1719 1754 1790 1827 1863 1900 1938 1976 201 4 2052 2091

108% 19159 2130

1 9 3 20 1 208 21 5 223 23 1 238 246 254 263 27 1 279 288 297 306 31 5 324 333 342 352 362 37 1 38 1 391 402 4 f 2 422 433 4 4 4 454 465 476 488 499 51 1 5 2 2 534 546 558 570 582 5 9 5 607 620 633 646 659 672 6 8 5 699 712 726 740 754 7 6 8

50 52 53 5 5 57 59 61 63 6 5 67 6 9 71 73 76 78 80 8 2 8 5 87 8 9 92 94 97 99

102 104 107 110 112 115 118 120 123 126 129 132 135 138 141 144 147 150 1 5 3 156 160 163 166 169 173 176 179 183 186 190 193

23 24 2 5 26 26 27 28 29 30 31 32 33 34 3 5 36 3 7 38 3 9 40 41 42 43 44 4 5 46 47 49 50 51 52 5 3 5 5 5 6 57 5 8 60 61 6 2 64 6 5 66 68 6 9 71 72 7 3 7 5 7 6 7 8 7 9 81 8 2 84 8 5 8 7

14 1 4 1 5 1 5 16 16 17 17 18 18 19 19 20 20 21 21 22 23 23 24 24 2 5 26 26 27 27 28 29 25 30 31 32 32 33 34 3 4 3 5 36 37 37 38 39 40 40 4 1 42 4 3 44 45 4 5 46 47 48 49 50

10 10 10 10 1 1 i t 1 1 12 12 12 13 13 1 3 14 14 14 1 5 15 16 16 16 17 17 17 18 18 19 19 20 20 20 2 1 2 1 22 22 23 23 24 24 2 5 2 5 26 26 27 27 28 28 29 2 9 30 30 31 31 32 33

4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9

10

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

- 30 - T a b l e 7 continued ...

cv n : n : n: n : n : n : n: n: n : PE= 1 % PE=3% PE=5% PE=10% PE= 15% PE=20% PE=25% PE=50% PE=lOO%

X * * * ***** ***** * * * x * x * * * * * ****** * * x * * * x * * * * * * * * * x * * * x * * * *

109% 110% 1 1 1 % 112% 113% 1 1 4 % 1 1 5% 116% 117% 118% 119% 120% 1 2 1 % 122% 123% 124% 125% 126% 127% 128% 129% 130% 131% 132% 133% 134% 135% 136% 137% 138% 139% 140% 1 4 1 % 142% 143% 144% 1 4 5 % 146% 147% 1 4 8 % 149% 150%

19515 2170 782 19875 2210 797 20238 2250 81 1 20604 2291 8 2 6 20973 2332 841 21 346 2373 8 5 6 21722 2415 871 221 02 2457 8 8 6 22484 2500 901 22870 2543 91 7 23260 2586 932 23652 2630 948 24048 2674 964 24447 2718 980 24849 2763 996 25255 2808 1012 25664 2853 1028 26076 2899 1 0 4 5 26492 2945 1061 26911 2992 1078 27333 3039 1 0 9 5 27758 3086 1112 28187 3133 1129 28619 3181 1146 29054 3230 1164 29492 3279 1181 29934 3328 1199 30379 3377 1217 30828 3427 1 2 3 5 31279 3477 1253 31734 3528 1271 32192 3579 1289 32654 3630 1308 3 3 1 1 9 3681 1326 33587 3733 1 3 4 5 34058 3786 1364 34533 3839 1383 3501 1 3892 1402 35492 3945 1421 35977 3999 1441 36464 4053 1460 36955 4108 1480

197 20 1 204 208 21 2 21 5 21 9 223 227 23 1 234 238 242 246 250 254 258 263 267 27 1 275 279 284 288 292 297 30 1 306 31 0 31 5 319 324 328 333 338 342 347 352 357 362 366 37 1

8 9 90 92 9 3 95 97 98

100 102 1 0 3 1 0 5 107 109 110 112 1 1 4 116 118 120 121 123 1 2 5 127 129 131 1 3 3 1 3 5 137 139 141 1 4 3 1 4 5 147 1 4 9 151 1 5 3 155 157 160 162 164 166

51 52 52 5 3 54 5 5 5 6 57 58 5 9 60 61 6 2 6 3 64 6 5 66 67 68 6 9 70 71 72 73 74 76 77 7 8 79 8 0 81 8 2 8 3 8 5 8 6 8 7 8 8 8 9 91 92 9 3 94

33 34 34 3 5 3 5 36 37 37 38 38 39 40 40 4 1 42 42 43 44 44 45 46 46 47 48 48 49 50 50 51 52 5 3 5 3 54 55 56 56 57 58 59 5 9 60 61

10 10 10 10 1 0 10 1 1 1 1 1 1 1 1 1 1 1 1 12 12 1 2 12 12 12 12 1 3 1 3 13 1 3 13 1 4 1 4 1 4 1 4 1 4 1 4 1 5 1 5 1 5 15 1 5 16 16 1 6 16 16 16 17

4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6

- 31 - Table 8 . Sample size table for density sampling ( 9 9 % confidence):

sample sizes required for estimated stand densities (stems/hectare) and tree spatial patterns (V/m = variance/mean ratios) to ensure prescribed allowable percentage sampling errors (PE).

a) 1 0 square meter quadrats ( 1 . 7 8 meter radius)

Density Spacing ( v/m )

******* X * * * * * *

625 0 .042 625 1 . o o o 6 2 5 3 . 0 0 0 8 1 6 0 . 0 4 2 8 1 6 1 . o o o 816 3 .000

1 1 1 1 0 . 0 4 2 1 1 1 1 1 . o o o 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 . o o o 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 . 0 0 0 2500 3 . 0 0 0 4444 0 .042 4444 1 . 0 0 0 4444 3 . 0 0 0

1 0 0 0 0 0 . 0 4 2 10000 1 . 0 0 0 10000 3 . 0 0 0

Sample Sample size: size:

PE = 1 % PE = 3%

4464 500 1 0 6 1 9 5 11803 3 18578 3540 1

3420 38 4 8 1339 9042

2440 10 271 16 251 3 283

59743 6642 179220 19917

1747 198 41 485 461 3

124447 1383 1 1 1 19 128

26552 2954 79648 8 8 5 4

632 74 14939 1664 44808 4983

28 3 3 5 664 1 742

19915 221 7

* * x * * * * *******

Sample size:

PE = 5 %

183 4252

12747 1 4 1

3258 9765

105 2394 71 7 3

74 1664 4982

49 1066 3190

29 60.2

1796 1 5

270 80 1

* * * * * x *

Sample size:

PE = l o %

49 1066 31 90

38 818

2444 29

602 1796

22 41 9

1 2 4 9 15

270 8 0 1

10 154 452

7 71

2 0 3

**x.****

b) 20 square meter quadrats ( 2 . 5 2 meter radius)

Density Spacing Sample Sample Sample Sample ( v/m ) size: size: size: size:

PE = 1 % PE = 3% PE = 5% PE = l o %

625 0 . 0 4 2 2234 252 9 4 27 625 1 . 0 0 0 5 3 100 5 9 0 4 21 28 5 3 5 6 2 5 3 . 0 0 0 159291 17703 6376 1597 8 1 6 0 . 0 4 2 1712 194 7 :3 21 8 1 6 1 . o o o 40672 4523 1631 4 1 1 8 1 6 3 .000 122007 13560 4884 1224

1 1 1 1 0 . 0 4 2 1259 144 54 17 1 1 1 1 1 . o o o 29874 3 3 2 3 1199 3 0 3 1 1 1 1 3 . 0 0 0 8 9 6 1 2 996 1 3589 9 0 0 1600 0 .042 8 7 5 101 39 1 3 1600 1 . o o o 20745 2309 8 3 4 21 2 1600 3 . 0 0 0 62226 691 8 2493 627 2500 0 . 0 4 2 562 66 2 ‘7 10 2500 1 . 0 0 0 13278 1 4 7 9 5 3 !5 137 2500 3 . 0 0 0 39826 4429 1 59‘7 4 0 3 4444 0 . 0 4 2 318 39 17 7 4444 1 . 0 0 0 7472 8 3 4 3 0 3 7 9 4444 3 . 0 0 0 22406 2 4 9 3 900 2 2 8

10000 0 . 0 4 2 1 4 4 20 10 5 10000 1 . 0 0 0 3 3 2 3 3 7 3 13’7 37

******* * x * * * * * * * * * * * x * * X * * * * * * X * * * * * x * * * * *

Sample size:

PE = 2 5 %

1 1 174 51 4

10 134 395

8 100 29 1

7 71

203 6

47 132

5 28 76

4 1 5 36

X * * * * * *

Sample size:

PE =25%

8 8 9

259 7

6 9 200

6 5 2

148 5

37 104

5 26 6 8

4 1 6 40

4 10

*******

Sample size:

PE = 5 0 %

6 47

132 5

37 102

5 28 76

4 21 54

4 1 5 36

4 10 22

3 7

1 2

* * X * * * *

Sample size:

PE = 5 0 %

5 26 68

4 21 5 3

4 16 40

4 13 29

4 10 20

3 7

1 3 3 5

* * * * x * *

1 0 0 0 0 3 .000 9960 1110 403 104 20 8

- 32 - Table 8 continued ...

c ) 30 square meter quadrats ( 3 . 0 9 meter radius)

Density Spacing (v/m )

X * * * * * * * x * * * * *

6 2 5 0 . 0 4 2 6 2 5 1 . 0 0 0 625 3 .000 81 6 0 .042 8 1 6 1 . 0 0 0 81 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 . o o o 1 1 1 1 3 .000 1600 0 . 0 4 2 1600 1 , 0 0 0 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 . o o o 2500 3 . 0 0 0 4444 0 . 0 4 2 4444 1 . 0 0 0 4 4 4 4 3.000

10000 0 . 0 4 2 10000 1 , 0 0 0 10000 3 . 0 0 0

Sample size:

PE = 1 %

1491 3540 1

1 0 6 1 9 5 1 1 4 3

271 16 8 1 3 3 9

8 4 1 19917 5 9 7 4 3

5 8 5 1 3 8 3 1 41 4 8 5

376 8 8 5 4

26552 21 3

4 9 8 3 1 4 9 3 9

9 7 221 7 6 6 4 1

*******

Sample size t

PE = 31k ******%

170 3937

1 1 8 0 3 131

301 7 9042

9 7 221 7 6642

6 9 1541 461 3

4 6 988

2954 2 8

557 1 6 6 4

15 250 7 4 2

Sample Sample size: size:

PE = 5% PE = l o %

6 4 1 9 1420 3 5 8 4252 1066

50 16 1 0 8 9 2 7 5 3 2 5 8 8 1 8

38 13 8 0 1 203

2394 602 28 10

5 5 7 1 4 3 1664 419

19 8 358 9 3

1066 270 1 3 6

2 0 3 5 4 6 0 2 1 5 4

8 5 93 26

270 71

* * * x * * * * x * * * * *

d ) 40 square meter quadrats ( 3 . 5 7 meter radius)

Density Spacing (v/m)

******* x * * * * * *

62 5 0 . 0 4 2 6 2 5 1 . 0 0 0 6 2 5 3 . 0 0 0 8 1 6 0 . 0 4 2 81 6 1 . 0 0 0 81 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 .000 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 . o o o 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 . o o o 2500 3 . 0 0 0 4444 0 . 0 4 2 4444 1 . o o o 4444 3 . 0 0 0

10000 0 . 0 4 2 10000 1 .000 10000 3 . 0 0 0

Sample size:

PE = 1 %

1 1 19 26552 7 9 6 4 8

858 20338 6 1 0 0 6

6 3 2 1 4 9 3 9 44808

440 1 0 3 7 5 31 115

2 8 3 6 6 4 1

1 9 9 1 5 161

3738 1 1 2 0 5

7 4 1664 4 982

*******

Sample size:

PE = 3%

1 2 8 2954 8 8 5 4

9 9 2264 6782

7 4 1664 4 9 8 3

5 3 1157 346 1

35 7 4 2

221 7 2 2

41 9 1249

12 1 8 9 5 5 7

* * x * * * *

Sample size:

PE = 5%

4 9 1066 31 90

38 818

2444 2 9

602 1796

22 41 9

1249 1 5

270 8 0 1

1 0 154 4 5 2

7 71

2 0 3

*******

Sample size:

PE = l o %

1 5 270 8 0 1

13 208 61 4

10 154 452

9 108 31 5

7 71

203 5

42 116

4 21 5 4

*******

Sample size:

PE =25%

6 61

1 7 4 6

48 1 3 4

5 3 6

100 5

2 6 71

4 18 47

4 1 2 28

3 8

15

*******

Sample size:

PE = 2 5 %

6 47

1 3 2 5

3 7 102

5 2 8 7 6

4 21 5 4

4 15 36

4 10 2 2

3 7

12

* * * * * x *

Sample size:

PE =50%

4 1 8 47

4 15 37

4 12 2 8

4 10 21

3 8

1 5 3 6

1 0 3 5 7

* t '

* * x * * * *

,,, ,

Sample size:

PE =50%

4 15 3 6

4 1 2 2 9

4 10 2 2

3 8

17 3 7

1 2 3 5 9 3 4 6

x * * * * * *

- 33 - Table 8 continued ...

e) 50 square meter quadrats ( 3 . 9 9 meter radius)

"l)rl" Density Spacing (v/m )

******* ******* 6 2 5 0 . 0 4 2 6 2 5 1 . 0 0 0 6 2 5 3 . 0 0 0 8 1 6 0 . 0 4 2 8 1 6 1 . 0 0 0 8 1 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 . 0 0 0 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 . 0 0 0 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 , 0 0 0 2500 3 . 0 0 0 4444 0 . 0 4 2 4444 1 . o o o 4444 3 . 0 0 0

10000 0 . 0 4 2 10000 1 .000 10000 3 . 0 0 0

Sample size:

PE = 1 %

8 9 6 2 1 2 4 3 637 19

688 1627 1 4 8 8 0 5

506 1 1952 35847

3 5 3 8 3 0 1

24893 227

531 4 15933

130 299 1 8 9 6 5

6 0 1332 3 9 8 6

* X * * * * *

Sample size:

FE = 3%

1 0 3 2364 7084

8 0

5427 6 0

1332 3987

4 3 9 2 6

2770 2 9

594 1774

18 336

1000 10

1 5 2 447

* * * x * * *

l a 1 2

Sample size:

PE = 5%

40 8 5 4

2 5 5 3 3 2

6 5 5 1956

2 4 482

1438 18

336 1000

13 217 6 4 1

9 1 2 4 3 6 3

6 5 7

164

* * * * * X *

Sample size:

PE = l o %

1 3 217 6 4 1

1 1 167 492

9 1 2 4 3 6 3

8 8 7

253 6

57 164

5 34 94

4 18 44

* * * * x * *

i1-J

f ) 1 0 0 square meter quadrats ( 5 . 6 4 meter radius)

Density Spacing ( v/m )

* * * * * x * ******* 6 2 5 0 . 0 4 2 6 2 5 1 . 0 0 0 6 2 5 3 . 0 0 0 81 6 0 . 0 4 2 8 1 6 1 .000 8 1 6 3 .000

1 1 1 - 1 0 .042 1 1 1 1 1 .000 1 1 1 1 3 . 0 0 0 1600 0 .042 1600 1 . o o o 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 . 0 0 0 2500 3 . 0 0 0 4444 0 . 0 4 2 4444 1 . o o o 4444 3 . 0 0 0

10000 0 . 0 4 2 1 0 0 0 0 1 . 0 0 0 10000 3 . 0 0 0

Sample size:

PE = 1 %

450 10623 3 1862

346 8 1 3 8

2 4 4 0 5 2 5 5

5978 17926

179 41 5 2

12449 1 1 6

2659 7969

6 7 1498 4485

3 2 668

1995

* * * * x * *

Sample size:

PE = 3%

5 4 1184 3544

42

271 5 3 2

6 6 8 1996

24 4 6 5

1387 1 7

2 9 9 8 8 9

1 1 1 7 0 5 0 2

7 7 8

2 2 6

x * * * * * *

9 o a

Sample size:

PE = 5%

22 429

1279 18

3 3 0 980

14 2 4 3 7 2 1

1 1 1 7 0 502

9 1 1 1 3 2 3

7 6 4

1 8 4 5

31 8 4

* x * * * * *

Sample size:

PE = l o %

9 1 1 1 3 2 3

8 8 6

248 7

64 184

6 4 6

129 5

31 84

4 19 4 9

4 1 1 24

*******

Sample size:

PE = 2 5 %

5 38

1 0 6 5

3 0 8 2

5 2 3 6 2

4 18 44

4 13 30

3 9

1 9 3 6

1 1

* * * * * X *

Sample size:

PE =25%

4 21 55

4 1 7 4 3

4 14 3 3

4 1 1 24

3 8

17 3 6

1 1 3 5 7

x * * * * * *

Sample size:

PE =50%

4 13 3 0

4 1 1 24

3 9

1 9 3 7

14 3 6

1 1 3 5 8 3 4 6

x * * * * * *

Sample size:

PE = 5 0 %

3 8

17 3 7

14 3 6

1 1 3 6 9 3 5 7 3 4 6 3 4 5

* x * * * * *

- 34 - Table 8 continued . . .

g ) 150 square meter quadrats ( 6 . 9 1 meter radius)

Density

* x * * * * *

6 2 5 6 2 5 6 2 5 8 1 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4 4 4 4

10000 7 0 0 0 0 10000

Spac ing (v/m )

******* 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 .000 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 , 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0

Sample size:

PE = 1 %

302 7084

21 243 232

5427 16271

172 3987

1 1952 120

2770 8 3 0 1

7 9 1774 531 4

46 1000 2 9 9 1

23 447

1332

* * X * * * *

Sample size:

PE = 3%

37 7 9 1

2364 30

6 0 7 1812

2 3 447

1332 17

31 2 9 2 6

1 3 2 0 1 594

9 1 1 5 336

6 53

152

* * * X * * *

Sample size:

PE = 5%

16 287 8 5 4

1 3 22 1 6 5 5

1 1 1 6 4 482

9 1 1 5 336

7 7 5

217 6

44 1 2 4

4 22 57

*******

Sample size:

PE = 10%

7 7 5

217 6

5 9 1 6 7

6 44

124 5

32 8 7

4 22 57

4 14 3 4

3 9

18

*******

h ) 200 square.meter quadrats ( 7 . 9 8 meter radius)

Density

* * * * * * * 6 2 5 6 2 5 6 2 5 8 1 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 1 0 0 0 0 10000

Spac in5 Sample (v/m 1 size:

PE = 1 %

0 . 0 4 2 227 1 . 0 0 0 531 4 3 . 0 0 0 15933 0 . 0 4 2 1 7 5 1 . 0 0 0 407 1 3 . 0 0 0 12205 0 . 0 4 2 130 1 .000 299 1 3 . 0 0 0 8 9 6 5 0 . 0 4 2 91 1 .000 2078 3 . 0 0 0 6226 0 .042 60 1 .000 1332 3 . 0 0 0 3986 0 . 0 4 2 3 6 1 .000 7 5 1 3 . 0 0 0 2245 0 . 0 4 2 1 8 1 .000 336 3 . 0 0 0 1000

* * * * * x * X * * * * * *

Sample size:

PE = 3%

2 9 5 9 4

1774 2 3

456 1360

18 3 3 6

1000 1 4

2 3 5 6 9 6

10 1 5 2 447

8 8 7

2 5 3 5

41 115

* * * x * * *

Sample size:

PE = 5%

1 3 21 7 6 4 1

1 1 1 6 7 492

9 1 2 4 3 6 3

8 8 7

2 5 3 6

57 164

5 3 4 94

4 18 44

*******

Sample size:

PE = l o %

6 5 7

164 6

4 5 126

5 3 4 9 4

5 2 5 67

4 18 44

4 12 27

3 7

14

* * * * * x *

Sample size:

PE =25%

4 16 38

4 13 30

4 1 1 2 3

3 9

18 3 7

13 3 6 9 3 4 6

* * x * * * *

Sample size:

PE' = 2 5 %

4 1 3 30

4 1 1 24

3 9

1 9 3 7

14 3 6

1 1 3 5 8 3 4 6

* * * * * x *

Sample size:

PE =50%

3 7

13 3 6

1 1 3 6 9 3 5 7 3 4 6 3 4 5 3 3 4

w

*******

Sample size:

PE = 5 0 %

3 6

1 1 3 6 9 3 5 8 3 5 7 3 4 6 3 4 5 3 3 4

m,,

*******

Table 8 continued . . . - 35 -

i) 300 square meter quadrats ( 9 . 7 7 meter radius)

Density

x * * * * * *

6 2 5 6 2 5 6 2 5 8 1 6 816 81 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spacing (V/ITI )

* * * * x * *

0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3.000 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . O O O 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 .042 1 . o o o 3 . 0 0 0

Sample size:

PE = 1 %

1 5 3 3544

10623 118

271 5 81 38

8 8 1996 5978

62 1387 41 52

4 1 8 8 9

2659 25

502 1498

1 4 226 668

* * * * x * *

Sample size:

PE = 3%

21 398

1184 17

306 908

1 4 226 668

1 1 1 58 465

8 103 299

6 60

170 5

29 78

x * * * * * *

Sample size:

PE = 5%

10 146 4 2 9

9 1 1 :3 3 3 0

7 8 4

2 4 3 6

60 170

!5 4 0

1 1 1 5

24 64

4 1 :3 31

******:k

Sample size:

PE = l o %

5 40

1 1 1 5

31 8 6

5 24 64

4 18 46

4 13 31

3 9

19 3 6

1 1

* * * * * * X

Sample size:

PE =25%

4 10 2 1

3 9

17 3 7

1 4 3 6

1 1 3 5 8 3 5 6 3 4 5

* * * * * x *

Sample size:

PE =50%

3 5 8 3 5 7 3 5 6 3 4 6 3 4 5 3 3 4 3 3 4

X * * * * * *

- Jb - Table 9. Sample size table for density sampling ( 9 5 % confidence):

sample sizes required for estimated stand densities (stems/hectare) and tree spatial patterns (V/m = variance/mean ratios) to ensure prescribed allowable percentage sampling errors (PE).

a ) 10 square meter quadrats ( 1 . 7 8 meter radius)

Density Spacing ( v/m )

* * * x * * * * * * * * * x

6 2 5 0 . 0 4 2 625 1 .000 6 2 5 3 . 0 0 0 8 1 6 0 . 0 4 2 8 1 6 1 . 0 0 0 8 1 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 . o o o 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 . o o o 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 . o o o 2 5 0 0 3 . 0 0 0 4 4 4 4 0 . 0 4 2 4444 1 .000 4 4 4 4 3 . 0 0 0

10000 0 . 0 4 2 10000 1 .000 10000 3 . 0 0 0

Sample size:

PE = 1 %

2584 6 1465

184389 1980

47079 141230

1455 34579

10373 1 101 1

2401 2 72029

648 15368 46099

366 8647

25935 164

3844 1 1 527

* * * * * * X

Sample size:

PE = 3%

290 6832

20490 223

5234 15695

164 3845

1 1 528 115

267 1 8006

7 5 1710 51 2 5

43 963

2884 21

430 1283

*******

Sample size:

PE = 5%

106 2461 7378

8 2 1886 5652

61 1386 4 1 52

43 963

2884 29

61 8 1 8 4 7

17 349

1040 9

157 464

*******

Sample size:

PE = l o %

29 61 8

1847 23

474 1415

17 349

1040 13

243 7 2 3

9 157 4 6 4

7 8 9

262 5

41 118

* x * * * * *

b) 20 square meter quadrats ( 2 . 5 2 meter radius)

Density

* * * * * * x

625 625 6 2 5 8 1 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4 4 4 4

10000 10000 10000

Spac i ng ( v/m )

******* 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . ooo 3 . 0 0 0 0 . 0 4 2 1 , 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 I . 000 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0

Sample Sample size: size:

PE = 1 % PE = 3%

1294 146 30734 341 7 92 196 10247

992 113 2354 1 261 8

729 84 1729 1 1924 51 867 5766

507 59 12007 1337 360 16 4004

326 39 7686 857

2305 1 2564 184 23

4325 483 12969 1 4 4 4

8 4 12 1924 216 5765 6 4 3

* x * * * * * * X * * * * *

7,06 16 7849

Sample size:

PE = 5%

5 5 1232 369 1

42 944

2827 32

694 2077

23 483

1443 16

31 0 9 2 5

10 176 522

6 8 0

233

* * * X * * *

Sample size:

PE = l o %

16 310 9 2 5

13 238 709

10 176 522

8 1 2 3 3 6 3

6 8 0

233 5

46 1 3 3

4 22 61

* * * * * x *

Sample size:

PE = 2 5 %

7 101 2 98

6 78

229 5

58 169

5 4 1

118 4

28 7 7

3 17 4 4

3 9

21

* * x * * * *

Sample size:

PE = 2 5 %

5 52

150 5

4 1 116

4 31 8 6

4 22 61

3 1 5 40

3 10 24

3 6

12

x * * * * * *

Sample size:

PE =50%

4 28 77

4 22 59

3 17 44

3 13 32

3 9

2 1 3 6

1 3 3 5 8 -3

x * * * * * *

Sample size:

PE = 5 0 %

3 1 5 40

3 12 31

3 10 24

3 8

17 3 6

12 3 5 8 3 4 5

* * * * X * *

-in

- 37 - Table 9 continued ...

c) 30 square meter quadrats ( 3 . 0 9 meter radius)

wr Density Spacing (v/m )

* * * x * * * * x * * * * *

6 2 5 0 . 0 4 2 625 1 . o o o 6 2 5 3 . 0 0 0 8 1 6 0 . 0 4 2 8 1 6 1 . 0 0 0 8 1 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 .000 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 . 0 0 0 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 . 0 0 0 2500 3 . 0 0 0 4444 0 . 0 4 2 4444 1 . 0 0 0 4444 3 . 0 0 0

10000 0 . 0 4 2 10000 1 , 0 0 0 10000 3 . 0 0 0

Sample size:

PE = 1 %

8 6 3 20490 61 465

662 15695 47079

40 7 1 1528 34579

339 8 0 0 6

24012 2 1 8

51 25 15368

124 2804 8 6 4 7

57 1 2 8 3 3844

*******

Sample size:

PE = 3%

99 2279 6832

76 1746 5234

57 1284 3845

40 892

267 1 27

572 1710

16 323 963

9 145 430

X * * * * * *

Sample size:

PE = 5%

37 022

2461 29

63 1 1886

22 464

1386 16

323 9 6 3

12 208 6 1 8

8 118 349

5 54

157

* * * x * * *

Sample size:

PE = l o %

12 208 61 8

10 160 474

8 118 349

6 0 3

243 5

54 157

4 32 8 9

3 16 4 1

*******

d ) 40 square meter quadrats ( 3 . 5 7 meter radius) :-v Density

X * * * * * *

625 6 2 5 6 2 5 8 1 6 816 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spacing Sample (v/m 1 size:

PE = 1 %

0 . 0 4 2 648 1 . 0 0 0 15368 3 . 0 0 0 46099 0 . 0 4 2 497 1 . 0 0 0 1 1 7 7 2 3 . 0 0 0 3531 0 0 . 0 4 2 366 1 . 0 0 0 8 6 4 7 3 . 0 0 0 25935 0 . 0 4 2 2 5 5 1 . 0 0 0 6 0 0 5 3 . 0 0 0 18009 0 . 0 4 2 164 1 .000 3044 3 . 0 0 0 1 1 527 0 . 0 4 2 94 1 . 0 0 0 2164 3 . 0 0 0 6 4 8 6 0 . 0 4 2 4 3 1 . o o o 963 3 . 0 0 0 2884

* * * * * x * *******

Sample size:

PE = 3%

7 5 1710 51 2 5

50 131 1 3926

43 9 6 3

2884 31

670 2004

21 430

1203 1 3

243 7 2 3

7 110 323

*******

Sample size:

PE = 5%

2 9 618

1847 2 3

4 7 4 1 4 1 5

17 349

1040 1 3

243 7 2,3

9 157 464

7 89

262 5

4 1 118

*******

Sample size:

PE = l o %

9 157 464

8 121 356

7 8 9

262 6

6 3 1 8 3

5 4 1

118 4

2 5 68

3 13 32

* * * * * * x

Sample size:

PE = 2 5 %

4 36

101 4

28 70

4 21 58

3 16 4 1

3 1 1 28

3 8

17 3 5 9

*******

Sample size:

PE = 2 5 %

4 28 77

4 2 2 5 9

3 17 4 4

3 1 3 32

3 9

21 3 6

1 3 3 5 8

* x * * * * *

Sample size:

PE = 5 0 %

3 1 1 28

3 9

22 3 8

17 3 6

13 3 5 9 3 4 6 3 3 5

x * * * * * *

Sample size:

PE = 5 0 %

3 9

21 3 8

17 3 6

13 3 5

10 3 5 8 3 4 6 3 3 4

*******

- 38 - Table 9 continued ...

e) 50 square meter quadrats ( 3 . 9 9 meter radius)

Density

* * * x * * *

625 625 625 816 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4 4 4 4 4444

10000 10000 10000

Spacing ( v/m )

******* 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 .042 1 .000 3 . O O G 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0

Sample size:

PE = 1 %

5 1 9 1 2 2 9 5 36880

3 9 8 9418

28248 293

691 8 20748

2 0 5 4 8 0 5

14408 132

3 0 7 6 9222

76 1732 51 8 9

3 5 77 1

2308

* * * * * x *

Sample size:

PE = 3%

60 1369 41 0 0

47 1049 3141

3 5 77 1

2308 2 5

536 1603

17 344

1027 1 1

1 9 5 5 7 9

7 88

259

*******

Sample size:

PE = 5%

24 495

1478 19

380 1133

1 5 280 8 3 3

1 1 195 579

8 126 372

6 72

210 4

34 95

*******

Sample size:

PE = l o %

8 126 372

7 97

285 6

72 21 0

5 51

147 4

34 95

4 20 5 5

3 1 1 26

* * * * * x *

f ) 1 0 0 square meter quadrats ( 5 . 6 4 meter radius)

Density

******* 625 625 625 81 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spac i ng (v/m 1

******* 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0

Sample size:

PE = 1 %

26 1 61 49

18442 20 1

4710 14126

148 3 4 6 1

10376 104

2404 7 2 0 6

67 1 5 3 9 461 3

3 9 8 6 7

2 5 9 6 19

387 1 1 5 5

* * * * * * x

Sample size:

PE = 3%

32 68 6

2052 2 5

526 1572

1 9 387

1 1 5 5 1 4

270 8 0 3

10 174 51 5

7 99

29 1 5

46 131

* * * x * * *

Sample size:

PE = 5%

13 249 740

1 1 191 568

9 1 4 1 4 1 8

7 99

29 1 6

64 187

4 38

107 4

18 49

* * * * * x *

Sample size:

PE = l o %

6 64

187 5

50 144

4 38

107 4

27 7 5

4 18 49

3 12 2 9

3 7

15

****if*

Sample size:

PE =25%

4 23 6 2

4 18 48

3 1 4 36

3 1 1 26

3 8

18 3 6

1 1 3 4 7

* x * * * * *

Sample size:

PE = 2 5 %

3 1 3 32

3 1 1 26

3 9

20 3 7

15 3 5

10 3 4 7 3 3 5

*******

Sample *: size:

PE = 5 0 %

3 8

1.8 3 7

1 4 3 6

1 1 3 5 9 3 4 7 3 4 5 3 3 4

* * * x * * *

Sample size:

PE = 5 0 %

3 5

10 3 5 9 3 4 7 3 4 6 3 3 5 3 3 4 3 3 3

VI,,

* * * * x * *

- 39 - Table 9 continued ...

g ) 150 square meter quadrats ( 6 . 9 1 meter radius)

" Density

* * X * * * *

6 2 5 625 625 81 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 1 0 0 0 0 10000

Spac i ng ( v/m 1

****x.** 0 . 0 4 2 1 ,000 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 .042 1 .000 3 . 0 0 0

Sample size:

PE = 1 %

1 7 5 4 1 0 0

12295 1 3 5

3141 9 4 1 8

100 2308 691 8

70 1 6 0 3 4 8 0 5

46 1027 3076

27 5 7 9

1732 14

259 77 1

* * * * * X *

Sample size:

PE = 3%

22 458

1369 18

352 1049

14 259 77 1

10 181 536

8 117 344

6 67

195 4

31 8 8

*******

Sample size:

PE = 5 %

1 0 167 495

8 128

7 9 5

280 6

67 1 9 5

5 44

1 2 6 4

26 72

3 1 3 3 4

*******

380

Sample size:

PE = l o %

5 44

126 4

34 97

4 26 72

4 19 51

3 13 34

3 9

20 3 6

1 1

* * * X * * *

h ) 200 square meter quadrats ( 7 . 9 8 meter radius) " I

Density

* x * * * * *

6 2 5 6 2 5 6 2 5 8 1 6 816 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

x * * * * * *

0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 .042 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 .000 3 .000 0 .042 1 . o o o 3.000 0 .042 1 .000 3 .000 0 .042 1 .000 3 . 0 0 0

Sample size:

PE = 1 %

132 3 0 7 6 9222

102 2 3 5 7 7064

7 6 1732 51 8 9

5 3 1 2 0 3 3604

3 5 77 1

2308 21

4 3 5 1300

1 1 1 9 5 5 7 9

* * X * * * *

Sample size:

PE = 3%

17 344

1027 1 4

2 6 4 788

1 1 195 579

9 136 403

7 8 8

259 5

51 147

4 24 67

* * * * * x *

Sample size:

PE = 5%

8 126 372

7 97

285 6

72 210

5 51

147 4

34 9 5

4 20 5 5

3 1 1 26

* x * * * * *

Sample size:

PE = l o %

4 34 95

4 2 6 74

4 20 5 5

3 1 5 39

3 1 1 26

3 7

16 3 5 9

*******

Sample size:

PE = 2 5 %

3 1 0 23

3 8

18 3 7

1 4 3 6

1 1 3 5 8 3 4 6 3 3 4

X * * * * * *

Sample size:

PE = 2 5 %

3 8

18 3 7

1 4 3 6

1 1 3 5 9 3 4 7 3 4 5 3 3 4

* * * * X * *

Sample size:

PE =50%

3 5 8 3 4 7 3 4 6 3 4 5 3 3 4 3 3 4 3 3 3

* X * * * * *

Sample size:

PE =50%

3 4 7 3 4 6 3 4 5 3 3 4 3 3 4 3 3 3 3 3 3

* * * * * * X

Table 9 continued ... - 40 -

i) 300 square meter quadrats ( 9 . 7 7 meter radius)

Density

* * * * X * *

625 625 625 81 6 81 6 81 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

1 0 0 0 0 10000 10000

Spacing ( v/m 1

* * * * * * x

0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0

Sample size:

PE = 1 %

8 9 2052 51 49

6 9 1572 471 0

51 1155 346 1

37 8 0 3

2404 24

51 5 1539

15 29 1 8 6 7

8 131 38 7

* x * * * * *

Sample size:

PE = 3%

1 3 2 3 1 68 6

10 177 5 2 6

8 131 38 7

7 92

270 5

60 174

4 3 5 9 9

3 17 46

* X * * * * *

Sample size:

PE = 5%

6 8 5

2 4 9 6

66 191

5 49

141 4

3 5 99

4 2 3 64

3 1 5 38

3 8

18

* x * * * * *

Sample size:

PE = l o %

4 23 64

4 19 50

3 15 38

3 1 1 27

3 8

18 3 6

12 3 4 7

*******

Sample size:

PE = 2 5 %

3 6

1 3 3 6

1 1 3 5 9 3 4 7 3 4 5 3 3 4 3 3 3

* * * * * * X

Sample size:

PE = 5 0 %

3 4 5 3 3 5 3 3 4 3 3 4 3 3 3 3 3 3 3 3 3

-11Ui1

* * * * * X *

- 41 - Table 10. Sample size table for density sampling ( 9 0 % confidence):

sample sizes required f o r estimated stand densities (stems/hectare) and tree spatial patterns (V/m = variance/mean ratios) to ensure prescribed allowable percentage sampling errors (PE).

a ) 10 square meter quadrats ( 1 . 7 8 meter

Density Spacing Sample Sample Sample ( v/m 1 size: size: size:

PE = 1 % PE = 3% PE = 5%

6 2 5 0 . 0 4 2 1820 204 7 5 625 1 , 0 0 0 43289 4812 1734 625 3 . 0 0 0 129864 14431 51 97 8 1 6 0 . 0 4 2 1395 157 58 8 1 6 1 . o o o 33 157 3686 1329 8 1 6 3 . 0 0 0 99467 1 1 054 398 1

1 1 1 1 0 . 0 4 2 1025 116 43 1 1 1 1 1 . o o o 24354 2708 976 1 1 1 1 3 . 0 0 0 73057 81 20 2925 1600 0 . 0 4 2 7 1 3 81 31 1600 1 . 0 0 0 1691 1 1881 679 1600 3 . 0 0 0 50730 5639 2 0 3 1 2500 0 . 0 4 2 457 5 3 21 2500 1 . 0 0 0 10824 1 2 0 5 435 2500 3 .000 32468 3610 t 301 4444 0 . 0 4 2 258 31 13 4444 1 . 0 0 0 6090 679 246 4444 3 . 0 0 0 18266 2032 7 3 3

10000 0 . 0 4 2 116 15 7 10000 1 . o o o 2708 3 0 3 1 1 1 10000 3 . 0 0 0 81 19 904 327

b ) 20 square meter quadrats ( 2 . 5 2 meter

Density Spacing Sample Sample Sample

PE = 1 % PE = 3% PE = 5%

625 0 .042 91 1 103 39 6 2 5 1 , 0 0 0 2 1 6 4 6 2 4 0 7 8 6 8 625 3 . 0 0 0 64933 7217 2600 8 1 6 0 . 0 4 2 699 8 0 30 8 1 6 1 . 0 0 0 16580 1844 6 6 5 8 1 6 3 . 0 0 0 49735 5528 1992

1 1 1 1 0 . 0 4 2 514 5 9 23 1 1 1 1 1 .000 12178 1355 489 1 1 1 1 3 . 0 0 0 36530 4061 1463 1600 0 . 0 4 2 357 42 17 1600 1 . o o o 8457 942 34 1 1600 3 . 0 0 0 25366 282 1 1017 2500 0 . 0 4 2 230 28 12 2500 1 . o o o 541 3 604 219 2500 3 . 0 0 0 16235 1806 652 4 4 4 4 0 . 0 4 2 130 17 8 4444 1 . o o o 3046 34 1 124 4444 3 . 0 0 0 91 3 4 1017 3 68

1 0 0 0 0 0 . 0 4 2 59 9 5 10000 1 . o o o 1355 153 56

* * X * * * * * X * * * * * ******* * x * * * * * * * x * * * *

( v/m ) size: size: size:

******* * * * X * * * ******* ******* *******

radius 1

Sample size:

PE = l o %

21 435

1301 16

334 997

1 3 246 7 3 3

10 171 51 0

7 1 1 1 327

5 6 3

185 4

29 84

*******

radius 1

Sample size:

PE = l o %

12 21 9 652

9 168 500

8 124 368

6 8 7

256 5

56 1 6 5

4 3 3 94

3 16

* * * * x * *

Sample size:

PS = 2 5 %

5 72

21 0 5

5 5 162

4 4 1

119 4

29 8 4

3 20 54

3 12 32

3 7

1 5

* * x * * * *

Sample size:

PE = 2 5 %

4 3 7

106 4

29 8 2

3 22 61

3 16 43

3 1 1 28

3 7

17 3 5

* X * * * * *

Sample size:

PE =50%

3 20 54

3 16 42

3 12 32

3 9

2 3 3 7

15 3 5

1 0 3 4 6

* * X * * * *

Sample size:

PE = 5 0 %

3 1 1 28

3 9

22 3 7

17 3 6

1 3 3 5 9 3 4 6 3 3

* X * * * * *

1 0 0 0 0 3 .000 406 1 453 165 43 9 4

Table 10 continued ... - 42 -

c ) 30 square meter quadrats ( 3 . 0 9 meter radius)

Density

* * * x * * *

625 625 6 2 5 816 81 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spacing Sample ( v/m 1 size:

PE = 1 %

0 . 0 4 2 608 1 . 0 0 0 1443 1 3 . 0 0 0 43289 0 . 0 4 2 467 1 . 0 0 0 1 1054 3 .000 33 157 0 . 0 4 2 343 1 , 0 0 0 81 20 3 . 0 0 0 24354 0 . 0 4 2 239 1 . 0 0 0 5639 3 . 0 0 0 1691 1 0 . 0 4 2 154 1 . 0 0 0 361 0 3 . 0 0 0 10824 0 . 0 4 2 8 8 1 . o o o 2032 3 . 0 0 0 6090 0 . 0 4 2 4 0 1 . 0 0 0 904 3 . 0 0 0 2708

* * * * * * x *******

Sample Sample size: size:

PE = 3% PE = 5%

70 27 1606 580 4812 1734

54 21 1230 444 3686 1329

40 16 904 327

2708 976 29 12

629 228 1881 679

19 9 403 147

1205 435 12 6

228 8 4 679 246

7 4 103 38 3 0 3 1 1 1

* * x * * * * * * x * * * *

Sample size:

PE = l o %

9 147 43 5

7 113 334

6 84

246 5

5 9 171

4 38

1 1 1 3

23 6 3

3 1 1 29

* * X * * * *

d ) 40 square meter quadrats ( 3 . 5 7 meter radius)

Density

X * * * * * *

625 625 625 8 1 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spacing Sample Sample (v/m) size: size:

PE = 1 % PE = 3%

0 . 0 4 2 457 5 3 1 . 0 0 0 10824 1205 3 . 0 0 0 32468 361 0 0 . 0 4 2 3 50 41 1 . o o o 8 2 9 1 9 2 3 3 . 0 0 0 24869 2765 0 . 0 4 2 258 31 1 . o o o 6090 679 3 . 0 0 0 18266 2032 0 . 0 4 2 180 22 1 . 0 0 0 4230 472 3 . 0 0 0 12684 1 4 1 1 0 . 0 4 2 116 15 1 . 0 0 0 2708 303 3 . 0 0 0 81 19 904 0 . 0 4 2 66 10 1 . 0 0 0 1524 171 3 . 0 0 0 4568 51 0 0 . 0 4 2 31 6 1 . o o o 679 78 3 . 0 0 0 203 1 228

x * * * * * * ******* x * * * * * *

Sample size:

PE = 5%

21 435

1301 16

334 997

1 3 246 7 3 3

10 171 51 0

7 1 1 1 327

5 6 3

185 4

2 9 8 4

x * * * * * *

Sample size:

PE = l o $

7 1 1 1 327

6 8 5

2 5 1 5

63 1 8 5

4 45

129 4

29 8 4

3 18 48

3 9

23

* * * * * X *

Sample size:

PE = 2 5 %

4 2 5 72

3 20 5 5

3 15 41

3 1 1 2 9

3 8

20 3 6

12 3 4 7

*******

Sample size:

PE = 2 5 %

3 20 54

3 16 42

3 12 32

3 9

23 3 7

15 3 5

10 3 4 6

X * * * * * *

Sample size: ‘,a -

PE =50%

3 8

20 3 7

16 3 6

12 3 5 9 3 4 7 3 3 5 3 3 4

x * * * * * *

Sample %I @

size: PE = 5 0 %

3 7

1 5 3 6

12 3 5

10 3 4 8 3 4 6 3 3 4 3 3 3

* x * * * * *

- 43 - Table 10 continued ...

e) 5 0 square meter quadrats ( 3 . 9 9 meter radius)

" Density Spacing (v/m 1

* * * * * x * * * * * * x *

6 2 5 0 . 0 4 2 6 2 5 1 . o o o 6 2 5 3 . 0 0 0 81 6 0 . 0 4 2 8 1 6 1 . o o o 8 1 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 .000 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 .000 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 . o o o 2500 3 . 0 0 0 4444 0 . 0 4 2 4444 1 . o o o 4444 3 . 0 0 0

10000 0 . 0 4 2 10000 1 . 0 0 0 1 0 0 0 0 3 . 0 0 0

Sample size:

PE = 1 %

366 8660

25975 28 1

6 6 3 3 19895

207 4873

14613 144

3384 10148

93 21 67 6495

54 1220 3 6 5 5

2 5 543

1626

*******

Sample size:

PE = 3%

43 964

2888 33

739 221 3

25 543

1626 18

378 1130

13 243 724

8 138 408

5 62

183

* x * * * * *

Sample size:

PE = 5%

17 349

1041 1 4

268 798

1 1 197 587

8 138 408

6 8 9

262 5

51 1 4 8

3 24 6 7

* * * x * * *

Sample size:

PE = l o %

6 8 9

262 5

6 9 20 1

5 51

148 4

36 104

3 24 6 7

3 15 39

3 8

19

* * * * * * x

f ) 100 square meter quadrats ( 5 . 6 4 meter radius)

%aPf Dens i ty

******* 6 2 5 6 2 5 6 2 5 8 1 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1 6 0 0 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spac i ng ( v/m )

* X * * * * *

0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 , 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0

Sample size:

PE = 1 %

184 433 1

12989 142

3318 9949

105 2437 7308

7 3 1693 5075

48 1085 3249

28 61 1

1829 1 4

2 7 3 8 1 4

*******

Sample size:

PE = 3%

23 483

1445 18

37 1 1108

1 4 273 814

10 190 566

8 1 2 3 363

5 70

205 4

32 93

* * x * * * *

Sample size:

PE = 5%

10 176 522

8 1 3 5 400

7 100 295

5 70

205 4

46 132

4 27 7 5

3 1 3 35

* X * * * * *

Sample size:

PE = l o %

4 46

132 4

36 102

4 27 7 5

3 19 5 3

3 1 3 3 5

3 9

21 3 5

1 1

x * * * * * *

Sample size:

PE = 2 5 %

3 16 4 4

3 13 34

3 10 26

3 8

19 3 6

13 3 4 8 3 3 5

* * * * * * X

Sample size:

PE =25%

3 9

23 3 8

18 3 6

1 4 3 5

1 1 3 4 8 3 4 5 3 3 4

* * * * * x *

Sample size:

PE -50%

3 6

1 3 3 5

1 0 3 4 8 3 4 7 3 3 5 3 3 4 3 3 3

*******

Sample size:

PE =50%

3 4 8 3 4 6 3 4 5 3 3 5 3 3 4 3 3 3 3 3 3

* * * * * x *

Table 10 continued ... - 44 - g) 150 square meter quadrats ( 6 . 9 1 meter radius)

Density Spacing (v/m 1

* * * * * x * ******* 6 2 5 0 . 0 4 2 625 1 .000 625 3 .000 8 1 6 0 . 0 4 2 816 1 .000 8 1 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 .000 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 . o o o 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 .ooo 2500 3 . 0 0 0 4444 0 . 0 4 2 4444 1 .000 4444 3 . 0 0 0

10000 0 . 0 4 2 10000 1 . o o o

Sample size:

PE = 1 %

124 2888 8 6 6 0

95 221 3 6633

71 1 6 2 6 4873

50 1130 3384

33 724

2167 19

408 1 2 2 0

10 183

*******

Sample size:

PE = 3%

16 323 964

1 3 248 739

10 1 8 3 543

8 128 378

6 83

243 4

47 1 3 8

3 22

* * * * X * *

Sample size:

PE = 5%

7 118 349

6 91

268 5

67 197

4 47

138 4

31 8 9

3 19 51

3 10

* * x * * * *

10000 3 .000 543 62 24

h ) 200 square meter quadrats ( 7 . 9 8 meter c

Density Spacing ( v/m )

* * * x * * * ******* 6 2 5 0 . 0 4 2 6 2 5 1 , 0 0 0 6 2 5 3 . 0 0 0 8 1 6 0 . 0 4 2 8 1 6 1 . 0 0 0 8 1 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 . 0 0 0 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 , 0 0 0 1600 3 .000 2500 0 . 0 4 2 2500 1 . 0 0 0 2500 3 . 0 0 0 4444 0 . 0 4 2 4444 1 . 0 0 0 4444 3 . 0 0 0

10000 0 . 0 4 2 10000 1 . 0 0 0 10000 3 . 0 0 0

Sample size:

PE = 1 %

9 3 21 6 7 6 4 9 5

72 1660 4976

54 1220 3 6 5 5

38 8 4 8

2539 2 5

543 1626

1 5 307 91 6

8 138 408

* * x * * * *

Sample size:

PE = 3%

1 3 2 4 3 724

10 187 5 5 5

8 138 408

6 96

284 5

62 183

4 36

104 3

17 47

*******

Sample size:

PE = 5%

6 8 9

262 5

6 9 20 1

5 51

148 4

36 104

3 24 67

3 15 39

3 8

19

* * * x * * *

Sample size:

PE = l o %

4 31 8 9

3 24 6 9

3 19 51

3 1 4 36

3 10 24

3 7

1 5 3 4 8

* * x * * * *

radius)

Sample size:

PE = l o %

3 24 67

3 19 52

3 1 5 39

3 1 1 28

3 8

19 3 6

12 3 4 7

* * * * X * *

Sample size:

PE = 2 5 %

3 7

16 3 6

1 3 3 5

10 3 4 8 3 4 6 3 3 4 3 3 3

x * * * * * *

Sample size:

!?E = 2 5 %

3 6

13 3 5

1 0 3 4 8 3 4 7 3 3 5 3 3 4 3 3 3

*******

Sample size:

PE =50%

3 4 6 3 3 5 3 3 4 3 3 4 3 3 3 3 3 3 3 3 3

%*I

*******

Sample size:

PE =50%

3 3 5 3 3 5 3 3 4 3 3 4 3 3 3 3 3 3 3 3 3

."I

*******

Table 10 continued . . . - 45 -

i ) 300 square meter quadrats ( 9 . 7 7 meter radius)

'W ' Density

x * * * * * *

625 625 625 81 6 8 1 6 81 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4 4 4 4

1 0 0 0 0 10000 10000

Spac i ng ( v/m )

******* 0 .042 1 , 0 0 0 3 .000 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 .000 0 . 0 4 2 1 .000 3 .000 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0

Sample size:

PE = 1 %

6 3 1 4 4 5 433 1

49 1108 331 8

36 8 1 4

2437 26

566 1693

18 3 6 3

1 0 8 5 1 1

2 0 5 61 1

6 9 3

2 7 3

* * x * * * *

Sample size:

PE = 3rk * * * * * X *

9 1 6 3 483

8 125 37 1

6 93

273 5

6 5 190

4 42

1 2 3 3

2 5 70

3 12 32

Sample size:

?E = 5% * * * x * * *

5 60

176 4

47 1 3 5

4 3 5

100 3

25 7 0

3 17 46

3 l i 27

3 6

1 3

Sample size:

PE = l o %

3 17 46

3 13 36

3 1 1 27

3 8

19 3 6

1 3 3 5 9 3 3 5

* * * * x * *

Sample size:

PE = 2 5 %

3 5 9 3 4 8 3 4 6 3 3 5 3 3 4 3 3 4 3 3 3

*******

Sample size:

PE = 5 0 %

3 3 4 3 3 4 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3

* * X * * * *

- 46 - Table 1 1 . Sample size table for density sampling ( 8 0 % confidence):

sample sizes required for estimated stand densities (stems/hectare) and tree spatial patterns (V/m = variance/mean ratios) to ensure prescribed allowable percentage sampling errors ( P E ) . " 1 ,

a ) 10 square meter quadrats ( 1 . 7 8 meter radius)

Density

* * * * X * *

625 625 6 2 5 81 6 8 1 6 81 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4 4 4 4 4444

10000 10000 10000

Spacing Sample ( v/m ) size:

PE = 1 %

0 . 0 4 2 1105 1 .000 26280 3 . 0 0 0 7 8 8 3 5 0 . 0 4 2 847 1 . o o o 20129 3 . 0 0 0 6 0 3 8 3 0 . 0 4 2 6 2 3 1 . 0 0 0 14785 3 .000 44350 0 . 0 4 2 433 1 . 0 0 0 10267 3 . 0 0 0 3 0 7 9 6 0 . 0 4 2 278 1 . o o o 657 1 3 . 0 0 0 19710 0 . 0 4 2 157 1 .000 3697 3 . 0 0 0 1 1 089 0 . 0 4 2 71 1 . 0 0 0 1644 3 . 0 0 0 4929

* * * * * * x * x * * * * *

Sample size:

PE = 3%

124 2922 8761

96 2238 671 1

71 1644 4929

50 1142 3423

33 732

2 1 92 19

4 1 2 1234

10 1 8 4 5 4 9

*******

Sample size:

PE = 5%

46 1053 31 55

36 8 0 7

241 7 27

5 9 3 1776

19 412

1234 13

2 6 5 790

8 150 4 4 5

5 68

1 9 9

* * * X * * *

Sample size:

PE = l o %

13 265 790

10 203 606

8 150 445

6 104 310

5 68

199 4

39 113

3 18 51

*******

b ) 20 square meter quadrats ( 2 . 5 2 meter radius)

Density

******* 6 2 5 625 625 81 6 8 1 6 81 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spacing Sample ( v/m 1 size:

PE = 1 %

0 . 0 4 2 554 1 . ooo 13141 3 . 0 0 0 394 18 0 . 0 4 2 424 1 .000 1 0 0 6 5 3 . 0 0 0 30192 0 . 0 4 2 312 1 . 0 0 0 7 3 9 3 3 . 0 0 0 22176 0 . 0 4 2 21 7 1 . 0 0 0 51 34 3 . 0 0 0 15399 0 . 0 4 2 140 1 . 0 0 0 3287 3 . 0 0 0 9856 0 . 0 4 2 7 9 1 . ooo 1850 3 . 0 0 0 5545 0 . 0 4 2 36 1 , 0 0 0 8 2 3 3 . 0 0 0 2 4 6 5

X * * * * * * *******

Sample Sample size: size:

PE = 3% PE = 5%

6 3 2 4 1462 527 438 1 1578

49 19 1120 404 3356 1209

36 1 4 8 2 3 2 9 7

2466 a w 26 1 1

572 207 1713 618

17 7 367 133

1097 396 1 1 5

207 76 618 224

6 3 9 3 3 5

276 100

* x * * * * * *******

Sample size:

PE = l o %

7 133 396

6 102 304

5 76

224 4

5 3 156

3 3 5

100 3

20 57

3 10 26

* * * * * X *

Sample size:

PE = 2 5 %

4 44

128 3

34 98

3 26 7 3

3 18 51

3 12 33

3 8

20 3 5

10

x * * * * * *

Sample size:

PE = 2 5 %

3 23 6 5

3 18 50

3 1 4 37

3 10 26

3 7

18 3 5

1 1 3 3 6

* * X * * * *

Sample size:

PE = 5 0 %

3 12 3 3

3 10 26

3 8

20 3 6

1 4 3 5

10 3 4 6 3 3 4

* * X * * * *

VI,

Sample size:

PE = 5 0 %

3 7

18 3 6

1 4 3 5

1 1 3 4 8 3 3 6 3 3 4 3 3 3

x * * * * * *

*ld'

Table 1 1 continued ... - 47 -

c ) 30 square meter quadrats ( 3 . 0 9 neter radius)

WW' Density Spacing ( v/m )

* * * X * * * * * * x * * *

6 2 5 0 . 0 4 2 6 2 5 1 . o o o 6 2 5 3 . 0 0 0 8 1 6 0 . 0 4 2 8 1 6 1 . 0 0 0 8 1 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 . o o o 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 .000 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 1 .000 2500 3 .000 4444 0 . 0 4 2 4444 1 . 0 0 0 4444 3 . 0 0 0

10000 0 . 0 4 2 10000 1 . 0 0 0 10000 3 . 0 0 0

Sample size:

PE = 1 %

370 8 7 6 1

26280 284

671 1 20 129

209 4929

14785 146

3423 10267

94 21 92 657 1

54 1234 3697

25 549

1644

X * * * * * *

Sample size:

PE = 3%

43 975

2922 33

747 2238

25 549

1644 18

382 1142

12 245 732

139 412

5 63

1 8 4

*******

a

Sample size:

PE = 5%

17 352

1053 1 3

270 8 0 7

1 0 195, 5 9 3

8 139 412

6 8 9

2 6 5 4

51 150

3 24 6 8

x * * * * * *

Sample size:

PE = l o %

6 8 9

265 5

69 203

4 51

150 4

36 104

3 24 68

3 14 39

3 7

18

*******

d ) 4 0 square meter quadrats ( 3 . 5 7 meter radius) r

Density

* * * x * * *

6 2 5 6 2 5 6 2 5 81 6 816 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spac i ng ( v/m )

* * * X * * *

0 .042 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 .042 1 . o o o 3 . 0 0 0 0 .042 1 . o o o 3 .000 0 . 0 4 2 1 . o o o 3 .000 0 . 0 4 2 1 . o o o 3 .000 0 . 0 4 2 1 . o o o 3 . 0 0 0

Sample size:

PE = 1 %

278 657 1

19710 21 3

5 0 3 4 15097

157 3697

1 1 089 110

7700 71

1644 4929

41 926

2774 19

412 1 2 3 4

x * * * * * *

2568

Sample size:

PE = 3%

33 732

21 92 25

56 1 1679

19 412

1234 1 4

287 8 57

10 184 5 4 9

6 104 310

4 47

139

* * * * * * x

Sample size:

PE = 5%

1 3 2 6 5 7 90

10 2 0 3 6 0 6

a 150 4 4 5

6 104 310

5 68

1 9 9 4

3 9 113

3 18 51

* * x * * * *

Sample size:

PE = l o %

5 68

199 4

52 153

4 39

113 3

28 79

3 18 51

3 1 1 30

3 6

1 4

* * x * * * *

Sample size:

PE = 2 5 %

3 16 44

3 13 34

3 10 26

3 7

18 3 6

12 3 4 8 3 3 5

* * * * x * *

Sample size:

PE =25%

3 12 33

3 1 0 26

3 8

20 3 6

14 3 5

10 3 4 6 3 3 4

* x * * * * *

Sample size:

PE = 5 0 %

3 6

12 3 5

10 3 4 8 3 3 6 3 3 5 3 3 4 3 3 3

* * x * * * *

Sample size:

PE =SO%

3 5

10 3 4 8 3 4 6 3 3 5 3 3 4 3 3 3 3 3 3

**I****

Table 1 1 continued . . . - 48 - e) 50 square meter quadrats ( 3 . 9 9 meter radius)

Densi ty

* * * * x * *

6 2 5 6 2 5 6 2 5 81 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4 4 4 4

10000 10000 10000

*I*****

0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 .042 1 .000 3 .000 0 . 0 4 2 1 . 0 0 0 3 .000 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 .000 0 .042 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0

Sample size:

?E = 1 %

223 5257

15768 171

4027 12078

126 2958 887 1

8 8 2055 6161

57 1316 3943

33 74 1

2 2 1 9 16

330 987

*******

Sample size:

PE = 3%

26 586

1754 21

449 1344

16 330 987

1 1 230 6 8 6

8 148 440

5 8 4

2 48 4

38 1 1 1

*******

Sample size:

PE = 5%

i t 212 632

9 1 6 3 485

7 120 3 57

5 8 4

2 4 8 4

54 159

3 31 91

3 15 4 1

*******

Sample size:

PE = l o %

4 54

159 4

42 123

3 31 91

3 22 63

3 15 41

3 9

24 3 5

12

*******

f) 100 square meter quadrats ( 5 . 6 4 meter radius)

Density

******* 6 2 5 6 2 5 6 2 5 81 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spacing ( v/m )

* * ***** 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 .000 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0

Sample size:

PE = 1 %

1 1 2 2630 7885

8 6 201 5 6040

64 1480 4437

45 1028 308 1

29 659

1973 17

37 1 1 1 1 1

9 166 455

*******

Sample size:

PE = 3%

1 4 2 94 8 7 8

1 1 225 673

9 166 495

7 116 344

5 75

22 1 4

43 1 2 5

3 20 57

* * * * * x *

Sample size:

PE = 5%

6 107 317

5 8 2

243 5

61 179

4 43

125 3

28 81

3 17 46

3 9

22

* x * * * * *

Sample size:

PE = l o %

3 28 81

3 22 62

3 17 46

3 12 33

3 9

22 3 6

13 3 4 7

* * * x * * *

Sample size:

PE = 2 5 %

3 10 27

3 8

21 3 7

16 3 5

12 3 4 8 3 3 6 3 3 4

* * * * * X *

Sample size:

PE = 2 5 %

3 6

15 3 5

12 3 4 9 3 4 7 3 3 5 3 3 4 3 3 3

*******

Sample size:

?E =50%

3 4 8 3 4 7 3 3 6 3 3 5 3 3 4 3 3 3 3 3 3

.<,.*,

*******

Sample size:

PE =50%

3 3 5 3 3 4 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3

v,.,

* * * * x * *

Table 1 1 continued ... - 49 -

g ) 150 square

Density Spacing ."

( v/m 1

******* x * * * * * *

6 2 5 0 . 0 4 2 625 1 .000 625 3 .000 8 1 6 0 . 0 4 2 8 1 6 1 . o o o 8 1 6 3 . 0 0 0

1 1 1 1 0 . 0 4 2 1 1 1 1 1 . 0 0 0 1 1 1 1 3 . 0 0 0 1600 0 . 0 4 2 1600 1 . o o o 1600 3 . 0 0 0 2500 0 . 0 4 2 2500 ? . o o o 2500 3 . 0 0 0 4444 0 .042 4444 1 . 0 0 0 4444 3 . 0 0 0

10000 0 . 0 4 2 10000 1 . o o o 10000 3 .000

h ) 200 square

* x * * * * *

6 2 5 625 6 2 5 8 1 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1600 1600 1600 2500 2500 2500 4444 4444 4444

10000 10000 10000

Spac i ng ( v/m )

******* 0 . 0 4 2 1 .000 3 .000 0 .042 1 .000 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 .000 0 . 0 4 2 1 . o o o 3 .000 0 .042 1 , 0 0 0 3 .000 0 . 0 4 2 1 .000 3 .000 0 .042 1 . o o o 3 .000

meter quadrats ( 6 . 9 1 meter radius)

Sample Sample Sample Sample size: size: size: size:

PE = 1 % PE = 3% PE = 5% PE = l o %

75 10 5 3 1754 196 72 19 5257 586 212 54

58 8 4 3 1344 151 56 15 4027 449 163 42

43 7 4 3 987 1 1 1 41 12

2958 330 120 31 31 5 3 3

686 78 2 9 9 2055 230 8 4 2 2

20 4 3 3 440 51 19 6

1316 1 4 8 5 4 1 5 12 3 3 3

248 29 12 5 74 1 84 31 9

7 3 3 3 1 1 1 1 4 6 3 330 38 1 5 5

******* * * * x * * * ******* *******

meter quadrats ( 7 . 9 8 meter radius)

Sample size:

PE = 1 %

57 1316 3 9 4 3

44 1008 302 1

33 74 1

221 9 23

51 5 1542

16 330 98 7

10 187 556

5 84

248

* * * * x * *

Sample size:

PE = 3%

8 148 440

7 1 1 4 337

5 8 4

4 59

1 7 3 4

38 1 1 1

3 22 6 3

3 1 1 29

* * * * * x *

248

Sample size:

PE = 5%

4 5 4

159 4

42 123

3 31 91

3 22 6 3

1 5 41

3 9

2 4. 3 5

1 2

*******

7

Sample size:

PE = l o %

3 1 5 4 1

3 12 32

3 9

24 3 7

17 3 5

12 3 4 7 3 3 5

* x * * * * *

Sample size:

PE = 2 5 %

3 5

10 3 4 8 3 4 7 3 3 5 3 3 4 3 3 3 3 3 3

*******

Sample size:

PE = 2 5 %

3 4 8 3 4 7 3 3 6 3 3 5 3 3 4 3 3 3 3 3 3

* * * x * * *

Sample size:

PE =50%

3 3 4 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

* * * x * * *

Sample size:

PE =50%

3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

* * * X * * *

Table 1 1 continued . . . - 50 - i) 3 0 0 square meter quadrats ( 9 . 7 7 meter r a d i u s )

Density

* * * * * x *

6 2 5 6 2 5 6 2 5 8 1 6 8 1 6 8 1 6

1 1 1 1 1 1 1 1 1 1 1 1 1 6 0 0 1 6 0 0 1 6 0 0 2 5 0 0 2 5 0 0 2 5 0 0 4 4 4 4

4 4 4 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0

4 4 4 4

Spac ing ( v/m )

******* 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 ? . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . 0 0 0 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0 0 . 0 4 2 1 . o o o 3 . 0 0 0

Sample size:

PE = 1 %

39 8 7 8

2 6 3 0 30

6 7 3 2 0 1 5

2 3 4 9 5

1 4 8 0 1 6

3 4 4 1 0 2 8

1 1 2 2 1 6 5 9

7 1 2 5 3 7 1

4 5 7

1 6 6

* * * * * * x

Sample size:

PE = 3%

6 99

2 9 4 5

7 6 2 2 5

4 57

1 6 6 4

40 1 1 6

3 2 6 7 5

3 1 6 43 3 8

2 0

* * * * * x *

Sample size:

PE = 5%

4 3 7

1 0 7 3

2 9 8 2

3 2 2 61

3 1 6 4 3

3 1 1 2 8

3 7

1 7 3 4 9

*******

Sample size:

PE = l o %

3 1 1 28

3 9

2 2 3 7

1 7 3 5

1 2 3 4 9 3 3 6 3 3 4

*******

Sample size:

PE = 2 5 %

3 4 6 3 3 5 3 3 4 3 3 4 3 3 3 3 3 3 3 3 3

* * * * x * *

Sample size:

PE = 5 0 %

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

x * * * * * *

Queen's Printer for British Columbia 0 Vlctona. 1983

MSO-181