53
WAYS OF TAKING
RANDOM SAMPLES FROM A POPULATION FOR THE NEEDS
OF AN ECONOMIC INDICATORS ANALYSIS
Ka eřina Gu ino á
*Vladimí a Ho o ko á Valen o á
Technical Uni e si y o Libe ec
Facul y o Economics
S uden ská 2, 461 17, Libe ec 1, Czech Republic
[email p o ec ed]
* Technical Uni e si y o Libe ec
Facul y o Economics
S uden ská 2, 461 17, Libe ec 1, Czech Republic
ladimi a. alen o [email protected]
Abs ac
The aim o his a icle is o p o ide a lis o possibili ies how o ake andom samples om he
Czech Republic’s popula ion and, consequen ly, hei compa ison. The compa ison o he
p esen ed me hods was ca ied ou wi h he help o selec ed s a is ic, which enables us o
choose a me hod ha b ings in he mos accu a e es ima es. Howe e , we mus no o ge he
ac ha when aking andom samples in p ac ise, we a e limi ed by inancial means, wo king
o ce ha pa icipa es in he su ey, and o he ac o s. We a e, he e o e, p esen ing an ideal
solu ion in he i s example and an op imal one in he second example while ha ing aken
in o accoun all in luencing ac o s.
In oduc ion
This a icle was elabo a ed wi h he inancial help o he p ojec no. 1101 o he Fund o he
De elopmen o Highe Educa ion Ins i u ions called C ea ion o a new module “S a is ical
Da a Analysis o Ques ionnai es” and in connec ion wi h a p ojec egis e ed as WD-30-07-1
in he esea ch p og amme o he Minis y o Regional De elopmen . This p ojec called
“Inno a ion App oach o Analysis o Dispa i ies on Regional Le el” has been ca ied ou a
he Facul y o Economics, Technical Uni e si y o Libe ec du ing he yea s 2007- 2011. The
ques ion how o ake he bes andom samples om a popula ion a ose while sol ing asks o
he abo e men ioned p ojec , and du ing consul a ions wi h s uden s s udying a he Facul y o
Economics, Technical Uni e si y o Libe ec (EF TUL). We ha e been ecen ly wo king,
wi hin he ame o he p ojec WD-30-07-01, wi h he popula ion o he Czech Republic’s
municipali ies, whe e alues o ce ain economic indica o s we e being elici ed. The aim o
his a icle is o p o ide o he possible ways o aking andom samples o a ce ain numbe o
municipali ies in he Czech Republic, which will be ins umen al in explo ing selec ed
economic indica o s. This a icle is a ollow-up o he esea ch esul s published in [1]. New
sugges ions o u he esea ch we e ecei ed a e publishing he a icle [1], and hese a e
p esen ed in his a icle.
1 Theo e ical look a a andom sample
Le us quickly ecapi ula e how a andom sample is de ined, and wha kinds o p obabili y
samples we ha e. A sample is called andom when da a a e ob ained by andom sampling.
54
Randomness ensu es he ep esen a i ness o a sample and s a is ics ob ained by such a
sample can be gene alized o a popula ion by he me hods o ma hema ical s a is ics. F om
he p obabili y poin o iew, andom sampling can be implemen ed by equal o unequal
p obabili ies.
The simples kind o a andom sample is a simple andom sample - SRS. I is a di ec
selec ion o elemen s om an unso ed popula ion. Each elemen , which is in he popula ion
du ing his d aw, has, du ing each d aw, he same p obabili y o being d awn. We dis inguish
be ween wo simple andom samples, namely a simple andom sample wi h eplacemen and a
simple andom sample wi hou eplacemen . A sample wi h eplacemen has a pa e n o
independen ials. The p obabili y ha each elemen will be chosen is he same o all d aws
(1/N), and he size o he popula ion does no a y du ing he d aw. A sample wi hou
eplacemen has a pa e n o dependen ials, he p obabili y ha each elemen will be chosen
ises wi h each d aw. The size o he popula ion dec eases wi h each ollowing d aw.
Howe e , SRS is o en unsui able due o i s simplici y. The e o e, o he mo e complex kinds
o andom samples, which enable us o see he complica ed eali y be e , a e used.
One o he mo e complica ed kinds o sampling is s a i ied sampling. The popula ion mus be
i s ly subdi ided in o g oups o by o he name s a a. A co ec alloca ion o he popula ion
in o s a a has a g ea impac on he sample quali y. S a a a e de ined as g oups o elemen s,
which a e somehow simila ; i means ha he s a a a e mo e homogenous inside han he
sample as a whole. As S. L. Loh says in [5], s a i ica ion is he mos e ec i e when means
in s a a a y a lo . These g oups o elemen s can be bo h na u al and a i icial. A andom
sample o a gi en numbe o elemen s is aken in each s a um. The mos common sample is
p opo ional alloca ion, whe e sample sizes in each s a um a e in due p opo ion o he sizes
o he s a a. Ye , a di e en app oach can be aken, such as aking he same, p e iously
s a ed, numbe o elemen s om each s a um. In his case we call i uni o m sampling.
Ano he op ion is op imal alloca ion o sampling in o s a a, i means ha sample sizes a e no
only p opo ional o he sizes o s a a, bu u he mo e, hei a iabili y is also aken in o
accoun . The d awback o his p ocedu e is i s ela i e complexness. S a i ied sampling is
complex as i equi es ce ain p elimina y in o ma ion necessa y o assigning elemen s in o
s a a. The nex d awback o s a i ied sampling is he ac ha is leads o a ela i ely la ge
space a iance. All in all, i is mo e demanding su ey o ganisa ion and da a p ocessing wise,
which o cou se inc eases he su ey expenses. The ad an age, in compa ison o SRS, is he
ac ha s a i ied sampling inc eases e iciency o es ima o s.
Clus e sampling is conside ed as a mo e complica ed kind o a andom sample. I s simples
ype is wo-s age clus e sampling, ye , he p ocedu e can be gene alized in o mo e s ages. A
popula ion mus be di ided in o g oups. G oups o uni s, called p ima y uni s, a e andomly
aken om he popula ion du ing he i s s age. Then, du ing he second s age, s a is ic uni s,
called seconda y uni s, a e andomly selec ed om he p ima y uni s. The ad an age o such a
kind o sampling, compa ed o s a i ied sampling, is he ac ha space a iance o he
selec ed uni s is signi ican ly smalle , which leads o a educ ion o he su ey cos s. The
disad an age is ha i b ings in less eliable easons wi hin he same sample size han he
simple andom sampling o s a i ied sampling. I is due o he ac ha some p ima y uni s
a e en i ely le ou du ing his p ocedu e; he e o e, he e is no in o ma ion abou hem
a ailable. Clus e sampling equi es e y p ecise p epa a ion, and i s p ocessing by
ma hema ical-s a is ical me hods is mo e complex.
55
2 Summa y o ob ained esul s
The aim o he p e ious ac i i ies was o apply di e en kinds o andom samples o speci ic
da a and o ca y ou a compa ison o he ob ained samples in e ms o hei ep esen a i ness.
The popula ion we wo ked wi h consis ed o 6,248 Czech municipali ies. The esea ched
economic indica o was unemploymen a e in %, in he yea 2006. The Czech S a is ical
O ice (CSO) supplied he da a abou his indica o in all Czech municipali ies. The sample
size was es ima ed as 520 uni s. Such a size is big enough o allow us o gene alize he esul s,
and, in addi ion, i allowed us o ca y ou sys ema ic sampling. We ook 30 andom samples
om he gi en popula ion. They ep esen ed 10 SRSs, sys ema ic sampling was used in
5 cases, and o he 5 samples we e ob ained by a andom numbe gene a o , which was un in
he s a is ic so wa e STATGRAPHICS CENTURION XVI. O he 15 samples we e ob ained
by s a i ied sampling (uni o m, p opo ional, and op imal alloca ion); he las 5 samples we e
aken by wo-s age sampling. We used se e al c i e ia o compa e he quali y o ou es ima es
ob ained by di e en kinds o andom samples:
 S anda d e o o he mean
 Mean de ia ion
 Rela i e gains om s a i ica ion
I is well known ha when a eal alue is eplaced by an es ima e ob ained by sampling, so
called sampling e o occu s . I is impossible o de ine i in a eal si ua ion; we can only
specula e abou some o i s alloca ion cha ac e is ics. The e o e, measu es o s a is ic
a ia ion a e used o measu e he quali y o an es ima e, he mos common is he mean
squa ed e o - page 28, in [4]:






.
2
2
b D E (1)
This measu emen measu es he e o o poin es ima ion. In he case ha he sample
cha ac e is ic is an unbiased es ima o o he popula ion cha ac e is ic, he mean squa e e o
equals a iance. A s anda d de ia ion, hus a posi i e squa e oo o , is some imes called a
s anda d e o o he mean, and i enables us o examine he accu acy o an unbiased
es ima o .
Based on he in o ma ion s a ed abo e, a sample a e age s anda d de ia ion can be de ined as
ollows:

.
n
yD

 (2)
This cha ac e is ic canno usually be de ined p ecisely in p ac ise as we do no know he
popula ion a iance. The e o e, i is impo an o eplace he unknown alue by i s poin
es ima ion , and we ge :

.
n
s
yDodh y
 (3)
Since we we e wo king wi h a popula ion, and he cha ac e is ics o he popula ion we e
a ailable, we could calcula e he s anda d e o o he mean di ec ly. Table 1 shows selec ed
cha ac e is ics o he popula ion, which we e calcula ed wi hin he ame o he p e ious
ac i i ies desc ibed in [1].
56
Tab. 1 Selec ed cha ac e is ics o he popula ion (own calcula ions)
Popula ion



yD
9.20197 5.574498 0.244458
Tha , apa om o he hings, allowed us o compa e his measu emen wi h i s es ima o s
ob ained by pa icula samples. Tables 2, 3, 4 show an o e iew o he ob ained esul s and
selec ed cha ac e is ics o he sample desc ibed in [1].
Tab. 2 Selec ed cha ac e is ics o he simple andom sample (own calcula ions)
Sample SRS – sys ema ic SRS – wi h help o andom numbe s
i
y i
s


yDodh i
y i
s


yDodh
Sample 1 9.15269 5.43582 0.238376 9.22019 5.70719 0.250277
Sample 2 9.06769 5.50411 0.241371 9.10981 5.54754 0.243276
Sample 3 9.13635 4.96938 0.217922 9.12923 5.19733 0.227918
Sample 4 9.14038 5.73322 0.251418 9.17962 5.73837 0.251644
Sample 5 9.78346 6.42324 0.281678 9.24788 5.51413 0.241810
Tab. 3 Selec ed cha ac e is ics o uni o m and p opo ional alloca ion (own calcula ions)
Sample Uni o m alloca ion P opo ional alloca ion
i
y i
s


yDodh i
y i
s


yDodh
Sample 1 9.79962 5.75534 0.220511 8.88192 5.10039 0.192955
Sample 2 9.52788 5.40658 0.229614 9.53596 6.30391 0.235050
Sample 3 9.93827 6.02183 0.235961 9.22327 5.17975 0.198103
Sample 4 9.93871 5.98349 0.248117 9.12923 5.32163 0.202882
Sample 5 9.67692 5.64627 0.220972 8. 69404 4.90830 0.186032
Tab. 4 Selec ed cha ac e is ics o op imal alloca ion and wo-s age clus e sampling (own
calcula ions)
Sample Op imal alloca ion Two- s age clus e sampling
i
y i
s


yDodh i
y i
s


yDodh
Sample 1 9.65962 5.29517 0.186848 9.40365 5.85226 1.040275
Sample 2 9.49462 6.26057 0.206682 9.46577 5.74643 0.937844
Sample 3 9.46154 5.71500 0.204395 10.5004 6.18511 1.496538
Sample 4 9.24250 5.29480 0.198942 9.40192 5.70257 1.048770
Sample 5 9.73115 5.79131 0.206195 8.52788 4.65322 1.346991
As Tables 2, 3, and 4 show, he esul s ob ained by he simple andom sample wi h he help o
andom numbe s come he closes o he ac ual s anda d e o o he mean. The second closes
esul is om he sys ema ic sampling, ollowed by uni o m alloca ion, hen p opo ional
alloca ion, and hen he op imal alloca ion. The wo s age clus e sampling di e ed
signi ican ly.
Ano he c i e ion used o compa e he aken samples was he mean de ia ion o each sample
mean om he ac ual mean o he popula ion. The mean de ia ion is calcula ed as ollows:
57
Mean de ia ion

,
2
1
k
y
k
i
i





(4)
whe e i
y a e indi idual sample means and k is a numbe o samples.
Table 5, which is also a esul o he ac i i ies published in [1], shows a compa ison o h ee
kinds o andom sampling.
Tab. 5 Compa ison o di e en kinds o andom sampling wi h he help o mean de ia ion.
(own calcula ions)
Cha ac e is ic
SRS S a i ied sampling
wo-s age
clus e
sys ema ic andom
numbe s p opo ional uni o m op imal
Mean de ia ion 0.2708 0.0578 0.3091 0.6011 0.3589 0.6768
The esul s p esen ed in Table 5 show ha he simple andom sample wi h he help o andom
numbe s accoun s o he bes alues. The a e age di e ence o sample means om he ac ual
mean is only 0.0578, he second bes is he sys ema ic simple andom sample wi h i s alue
0.2708. The wo-s age clus e sampling demons a ed he bigges di e ence 0.6768. The
uni o m alloca ion o s a i ied sampling shows he second wo s esul , whe e he mean
de ia ion is 0.6011.
Ano he c i e ion used o compa e he gi en samples was hei ela i e gains om
s a i ica ion, which S.L. men ions in [page 77, 5]. I is weighing up he a iance o s a i ied
sampling and he a iance o a simple andom sample o he same size. Mo e signi ican
ela i e gains om he s a i ica ion was no iced only in wo cases, namely when compa ing
p opo ional alloca ion o s a i ied sampling wi h he sys ema ic simple andom sample
(0.9025) and he simple andom sample wi h he help o andom numbe s (0.9434).
3 A new pe spec i e on s a i ied sampling
A e publishing he abo e esul s o ou esea ch, we we e conce ned wi h co ec de ining
s a a in s a i ied sampling. As s a ed he einbe o e, he quali y o he esul s ob ained based
on he s a i ied sampling is con ingen on he co ec de ini ion o s a a. The egions o he
Czech Republic we e conside ed as s a a in ou las esea ch, i means ha we ollowed he
o mal o ganiza ion. As he esul s demons a e in [1], egions a e no exac ly ideal g oups.
The a iabili y wi hin hem is ela i ely high and hey do no di e a lo among hemsel es.
Now, we ocused on he ac how di e en ly s a a o s a i ied sampling can be de ined, so
ha he ob ained esul s would be mo e sa is ying han in he p e ious case.
We ook in conside a ion he ac ha unemploymen a e can be di e en in di e en ly-sized
municipali ies, he e o e we made a decision ha a new g ouping c i e ion will be he size o a
municipali y gi en by he numbe o inhabi an s. We bo owed he ca ego iza ion om he
Czech S a is ic O ice – see e.g. [8], whe e we di e en ia e he ollowing g oups:
 municipali ies up o 199 inhabi an s;
 municipali ies wi h 200 – 499 inhabi an s;
 municipali ies wi h 500 – 999 inhabi an s;
 municipali ies wi h 1,000 – 1,999 inhabi an s;
 municipali ies wi h 2,000 – 4,999 inhabi an s;

58
 municipali ies wi h 5,000 – 9,999 inhabi an s;
 municipali ies wi h 10,000 – 19,999 inhabi an s;
 municipali ies wi h 20,000 – 49,999 inhabi an s;
 municipali ies wi h 50,000 – 99,999 inhabi an s;
 municipali ies wi h 100,000 inhabi an s and mo e.
Table 6 shows he numbe o municipali ies in he Czech Republic, in each ca ego y.
Tab. 6 Numbe o municipali ies in each size ca ego y acco ding o he numbe o
inhabi an s ( he CZSO and own calcula ions)
Ca ego y Up
o199 200-499 500- 999 1,000 –
1,999
2,000 –
4,999
5,000 –
9,999
10,000 –
19,999
20,000 –
49,999
50,000 –
99,999
abo e
100,000
Numbe
o
municipali ies
1 608 2 012 1 304 678 376 138 69 42 16 5
We did no see uni o m sp eading o he s a i ied sampling meaning ul due o he numbe o
municipali ies in each ca ego y. The e o e, we ca ied ou only p opo ional and op imal
alloca ion o s a i ied sampling – 5 samples om each kind.
Fi s ly, we c ea ed 5 p opo ional samples. The sample size in each s a um was de ined
acco ding o (see page 17, [13]):
,
N
N
nn h
h (5)
whe e
h
n is a sample size in h s a um,
n is a o al sample size,
h
N is a h-s a um size,
N is a popula ion size.
Table 7 shows he numbe o municipali ies selec ed in each ca ego y, based on he numbe o
inhabi an s.
Tab. 7 Numbe o municipali ies in he sampling in each size ca ego y acco ding o he
numbe o inhabi an s a p opo ional alloca ion ( he CZSO and own
calcula ions)
Ca ego y Up o
199 200 -499 500 -999 1, 000 –
1, 999
2, 000 –
4, 999
5, 000 –
9, 999
10, 000 –
19, 999
20, 000 –
49, 999
50, 000 –
99, 999
abo e
100, 000
Numbe o
municipali ies 134 167 109 56 31 12 6 4 1 0
As we can see, he las ca ego y, municipali ies wi h mo e han 100,000 inhabi an s, was no
ep esen ed in he sample. We p esume ha his ac could cause less accu a e esul s han i
all he ca ego ies we e included in he sample. Table 8 shows he calcula ed selec ed
cha ac e is ics om all 5 samples.
59
Tab. 8 Selec ed cha ac e is ics o p opo ional and op imal alloca ion (own calcula ions)
Sample P opo ional alloca ion Op imal alloca ion
i
y i
s


yDodh i
y i
s


yDodh
Sample 1 9.25510 4.98647 0.208174 9.17675 5.12996 0.205640
Sample 2 9.23478 5.30185 0.221910 9.10845 5.78815 0.230998
Sample 3 9.88580 5.88114 0.246896 9.64418 5.81105 0.244301
Sample 4 8.90485 5.55757 0.233371 9.16308 5.52874 0.225410
Sample 5 9.41249 5.66548 0.236503 9.38354 5.86010 0.232929
S anda d e o o es ima ed mean in s a um h is calcula ed acco ding o (p esen ed in e. g.
[3]):
,
1
)( 2
2
2







 L
h
hh
h
hsN
n
N
N
yDodh (6)
whe e
N is a popula ion size,
h
N is he popula ion o al in s a um h,
h
n is he sample size in s a um h,
2
h
s is an es ima o o he popula ion a iance in s a um h ( o sampling wi h and wi hou
eplacemen ).
The es ima o o he popula ion a iance in s a um h 2
h
s is de ined as:

,
2
2
h
n
k
hhk
hn
yy
s
h

 (7)
whe e hk
y is a alue o k- h uni in s a um h and h
y is an es ima o o he popula ion mean in
s a um h.
A e p opo ional alloca ion o s a i ied sampling, we ca ied ou op imal alloca ion. The
op imal sample size in s a um h ( o sampling wi hou eplacemen ) is de e mined by he
ela ion p esen ed in e. g. [3]:
,



hh
hh
hSN
SN
nn (8)
whe e
h
N is he popula ion o al in s a um h,
h
n is he sample size in s a um h,
h
S is he popula ion s anda d de ia ion in s a um h.
The s anda d de ia ion h
S is calcula ed acco ding o he o mula:
60

,
1
2



h
N
k
hhk
hN
YY
S
h
(9)
whe e
hk
Y is a alue o k- h uni in he popula ion,
h
N is he popula ion o al in s a um h,
h
Y is he popula ion mean in s a um h.
We calcula ed selec ed cha ac e is ics on a base o i e andom samples – hey a e also
p esen ed in Table 8.
Focusing on he compa ison o hese samples, we ha e o supplemen commen s wi h he
calcula ions o he a e age de ia ion o sample means om he popula ion mean. Table 9
con ains ou calcula ions.
Tab. 9 Compa ison o p opo ional and op imal alloca ion o s a i ied sample wi h he
help o he a e age de ia ion (own calcula ion)
Cha ac e is ic S a i ied sampling
P opo ional alloca ion Op imal alloca ion
A e age de ia ion 0.3476 0.2190
The esul s in he p e ious ables show ha s anda d e o s a e smalle in he case o op imal
alloca ion o s a i ied sampling bu he di e ences a e no so signi ican . I we compa e he
new samples wi h he p e ious ones (p esen ed in [1]), i is e iden ha new alloca ion o
s a a does no b ing any bene i because he s anda d e o s a e g ea e han in he case when
he s a a we e de ined as egions o he Czech Republic.
Le us look a he alues o a e age de ia ions. The a e age de ia ion is conside ably smalle
in he case o op imal alloca ion o s a i ied sampling in compa ison o p opo ional
alloca ion. We can also no ice by he compa ison o he new esul s wi h he esul s om he
p e ious esea ch (see Table 5) ha he a e age de ia ion o op imal alloca ion is
signi ican ly smalle in compa ison o he s a i ied sampling when he s a a we e de ined as
egions o he Czech Republic. We can e en egis e ha i is he second bes esul ( he bes
is SRS using he andom numbe gene a o ). The alue o his cha ac e is ic is now wo se o
p opo ional alloca ion o s a i ied sampling han in he p e ious esea ch.
We omi he compa ison wi h he help o he ela i e gain om s a i ica ion because i does
no b ing signi ican bene i s which would make he decision on a kind o sampling easie .
Conclusion
The calcula ions and compa isons o he a ious kinds o sampling men ioned abo e indica e
ha nei he s a i ied sampling no wo-s age clus e sampling imp o e he quali y o
es ima es. The a e age de ia ion shows ha he es ima ions ob ained by SRS do no di e
om he popula ion cha ac e is ics as much as he es ima es ob ained om o he kinds o
samplings. We achie ed a ce ain imp o emen in he quali y o es ima ions, wi h espec o
his c i e ion o he compa ison, by changing he de ini ion o s a a om he “ egions o he
Czech Republic“ o he “municipal size ca ego ies“. Howe e , we did no ge as a signi ican
imp o emen o he es ima ions quali y by changing he s a a de ini ion as we had expec ed.
61
The eason can be unemploymen a e blindness in ela ion o he numbe o inhabi an s, o
la ge a iabili y o alues wi hin he s a a, and small a iabili y among hem. So, we assume
ha municipal size ca ego ies a e no a sui able so ing c i e ion ei he .
In conclusion, le us add ha i is necessa y o ake in o conside a ion he ac ha we a e
limi ed by many a ious ac o s when ca ying ou sampling in p ac ise. Fi s ly, i is he
a ailabili y o da a, which a e no o en in such a s uc u e ha can be subdi ided in o sui able
subg oups. Fu he mo e, he e is he means, which makes us minimize he su ey cos s. I is
necessa y o ha monize all hese equi emen s and choose a sui able comp omise. E en
hough i is e iden ha he bes way o aking andom samples om a popula ion would be
simple andom sampling in ela ion o he es ima ions quali y, i s inancial and o ganiza ional
demandingness makes us use some o he mo e complex kinds o sampling. Two-s age clus e
sampling is e y o en he mos common solu ion in p ac ise. I allows us o educe su ey
cos s bu he e iciency o es ima o s is small. We ha e he e p esen ed he cha ac e is ics and
esul s o wo-s age clus e sampling wi h equal p obabili ies only. We could ha e ob ained
mo e e icien es ima o s i we had ca ied ou sampling wi h unequal p obabili ies. This idea
can be an impulse o he nex esea ch.
Li e a u e
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ýbě ů z populace ČR za účelem zkoumání ý oje hospodářských ukaza elů. In VII.
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Labem 23. – 24. 9. 2010. Sbo ník příspě ků. Ús í nad Labem: UJEP, FSE, 2010 – no
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škola ekonomická P aze, P aha, 1999. ISBN 80-7079-191-8.
[3] ČERMÁK, V.; VRABEC, M. Teo ie ýbě o ých še ření. Čás 3. 1. ydání. Vysoká
škola ekonomická P aze, P aha 1999. ISBN 80-245-0003-5.
[4] KAHOUNOVÁ, J. P ak ikum k ýuce ma ema ické s a is iky I. Odhady. 1. ydání.
Vysoká škola ekonomická P aze, P aha, 2000. ISBN 80-245-0070-1.
[5] LOHR, L. S. Sampling: Desing and Analysis. 2. ydání. B ooks/Cole, Bos on (USA)
2010. ISBN 978-0-495-11084-2.
[6] PACÁKOVÁ, V. a kol. Š a is ické me ódy p e ekonómo . P é ydanie. Iu a Edi ion,
B a isla a, 2009. ISBN 978-80-8078-284-9.
[7] PECÁKOVÁ, I; NOVÁK, I.; HERZMANN, J. Pořizo ání a yhodnoco ání da e
ýzkumech eřejného mínění. Vysoká škola ekonomická P aze, P aha, 2000. ISBN 80-
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[8] Český s a is ický úřad: Tab. 2.3.1 P ůmě ná oby ná plocha / m2/ na 1 by podle poč u
oby ných mís nos í a podle elikos ní ka ego ie obce k 1. 3. 2001 [online]. P aha, Český
s a is ický úřad, [ci . 2010-10-14]. A ailable om WWW:
<www.czso.cz/csu/2005edicniplan.ns /p/4131-05>
__________________________________________________________________________
Ing. Ka eřina Gu ino á, Ph.D.
Ing. Vladimí a Ho o ko á Valen o á, Ph.D.