Starting from the „rough” use of 1,500-1,800 subject samples in field researches with populations of the dimensions of a medium town, a capital or an entire country, which generates doubts and possible questions related to the representativeness of the investigated samples as well as to the conclusions of these researches, we developed the reconstruction of the necessary interactions in determining the sample volume, and also identified some inadvertencies. Some improvements solutions were also recommended.confidence interval, distribution laws, probable error, representativeness, sample volume.

Alexandru ISAIC-MANIU

Irina M. DRĂGAN

English

Research Papers in Economics

Management & Marketing (2010) Vol. 5, No. 1, pp. 127-134. 
 
POSSIBLE ERRORS IN ESTABLISHING THE SAMPLE 
DIMENSIONS OF SURVEY INVESTIGATIONS 
 
Alexandru ISAIC-MANIU, Irina M. DRĂGAN 
Academy of Economic Studies, Bucharest, Romania 
 
 
Abstract. Starting from the „rough” use of 1,500-1,800 subject samples in field researches 
with populations of the dimensions of a medium town, a capital or an entire country, which 
generates doubts and possible questions related to the representativeness of the investigated 
samples as well as to the conclusions of these researches, we developed the reconstruction of 
the necessary interactions in determining the sample volume, and also identified some 
inadvertencies. Some improvements solutions were also recommended. 
 
Keywords: confidence interval, distribution laws, probable error, representativeness, sample 
volume. 
 
1. Sample volume estimation 
 
In organizing a survey research, one of the main problems to solve is its 
rational dimension. It is true that the size of volume sample n – by virtue of the law of 
large numbers – increases the precision of the results and reduces the probable average 
error. Taking into consideration economical criteria, it is necessary that this volume be 
as small as possible. Considering both aspects, we determine the minimum number of 
observation units that satisfies precision and safety requirements formulated in relation 
to the research. 
In the theory and practice of surveys, „large” and „reduced volume” samples 
are used, depending on the homogeneity degree and on the volume of the origin 
general population. The calculus and interpretation of the representativeness error are 
determined differently: for large volume samples, the Laplace normal distribution, 
while for reduced volume samples, Student distribution.  
The sample volume calculus is achieved starting from the permitted maximum 
limit error, which in case of the repeated survey, is: 
n
z e
x
σ
α =          ( 1 )  
from where we can highlight n: 
n
z e
x
2
2 2 σ
α =          ( 2 )  
and thus: 
22
2
x
z
n
e
α σ ⋅
=          ( 3 )  
 Management & Marketing 
 
128
In the case of the repeated survey, the relation becomes: 
22
22
2
x
z
n
z
e
N
α
α
σ
σ
⋅
=
⋅
+
        ( 4 )  
In practice, the values resulted from this relation are majored in order to 
obtain an integer number, as well as to not fail the research, taking into account the 
fact that a number of questionnaires are usually rejected. 
So, in order to rationally dimension the sample volume n, the following 
elements are necessary: 
  admissible limit error 
x e  which is established depending on the concrete 
features of the problem to be solved, and on the required precision; 
  confidence probability 1 α − , practically close enough to certainty; 
  variance, the 
2 σ  characteristic in the general population, or its estimator, 
established on the base of a survey; 
  in the case of unrepeated sampling, the general population volume N, is 
also necessary.  
In these conditions, for the most usual probability values, probability 
coefficients equivalences, and admissible maximum errors, we have the values of the 
sample volume, in Table 1. 
 
Table 1 
Values for volume n sample, for  x e (%)=1-5% and P= 90-99,9% 
 
P(%) 
(%) x e  
P=99,9% 
Z0,001=3,29 
P=99% 
Z0,01=2,57 
P=97,5% 
Z0,025=2,24 
P=95% 
Z0,05=1,96 
P=90% 
Z0,10=1,645 
1 27.050  16.520  12.550  9.600  6.800 
   2,5  4.328  2.643  2.008  1.536  1.088 
3  3.006 1.836  1.394 1.067  756 
5  1.082 661  502 384  272 
 
The data confirm the fact that in the most usual conditions imagined by 
research and survey companies, the numbers regarding the investigated survey volume 
are placed around 1,500-1,800 units. A few observations regarding the calculus and 
results are required. 
 
1.1. Observation I 
 
•  In relations (3) and (4), variance is considered unknown and is estimated 
through the maximum variance of the binary characteristic: 
2 (1 ) p qp p σ =⋅= −   Possible errors in establishing the sample dimensions of survey investigations 
 
129
•  As known, the mean of the binary characteristic belongs to the [0; 1] 
interval. For the most probable value of the mean equal to 0.5, we have a value of the 
variance equal to 0.25. 
•  If this method is justifiable in the calculus of sample volume in the absence 
if previous information regarding the population under research, the same value (0.5) 
should be used for the mean in determining the probable error; the values of 1%-5% of 
this error indicate admissible deviations of the estimated parameters from the true but 
unknown values of the general population. In such a situation, the sample volume 
results in a range of 5,000-10,000 units, different from the ones usually used by survey 
or marketing institutes.  
Details on input indicators and results achieved are found in Table 2. 
 
Table 2 
Values for sample volume n and a mean m = 0.5 
 
 
 
 
 
 
 
 
 
 
1.2. Observation II 
 
 
 
The values of the normal standardized variable for the one-tailed case are 
used, although the confidence interval for the estimated parameters is two-tailed. The 
values of probabilistic coefficients are presented in Table 3, in the two testing 
alternatives. 
Table 3 
The values for Zα  variable 
 
α   12 () Z α φ = −⋅   0.5 ( ) Z α φ = −  
0,001 Z0,001=3,29 Z0,001=3,09 
0,01 Z0,01=2,57 Z0,01=2,33 
0,025 Z0,025=2,24 Z0,025=1,96 
0,05 Z0,05=1,96 Z0,05=1,645 
0,10 Z0,10=1,645 Z0,10=1,28 
 
This mistaken option also leads to distortions in the values of survey 
indicators. 
P 
x e  
 
P=99,9% 
Z0,001=3,29 
P=99% 
Z0,01=2,57 
P=97,5% 
Z0,025=2,24 
P=95% 
Z0,05=1,96 
P=90% 
Z0,1=1,645 
 
0,01 
 
 
 
 
  108.200  66.080  50.200  38.400  27.200 
 
0,025 
 
 
 
 
 
17.318  10.576  8.035  6.146  4.354 
 
0,03 
 
 
 
 
 
12.022  7.342  5.578  4.267  3.022 
 
0,05 
 
 
 
 
 
4.328  2.643  2.008  1.536  1.088 
1%
2,5%
3%
5%
0,01x0,5=0,005
0,0052 =0,000025
0,025x0,5=0,0125
0,01252 =0,0001525
0,03x0,5=0,015
0,0152 =0,000225
0,05x0,5=0,025
0,0252 =0,000625Management & Marketing 
 
130
1.3. Observation III 
 
  Another fundamental hypothesis, supposedly confirmed, but seldom tested, is 
the normality of the data (Gauss-Laplace law). The abusive use of the normal law is 
due to the fact that most of the times there is no preoccupation for preliminary testing 
in order to validate or to not validate the distribution law, and therefore final results 
may be corrupted in the un-normality case of data obtained through survey. 
The usual Gauss-Laplace model, from a mathematical point of view is a 
continuous statistic distribution defined by the expression: 
2
2
2
1( )
(; ; ) e x p
2 2
x x
Fx d x
μ
μσ
σ σπ −∞
⎡ ⎤ −
=− ⎢ ⎥
⎣ ⎦ ∫  
where: 
, 0,  R xR μ σ ∈> ∈ . 
 
Therefore, the confidence interval for the mean results: 
() 1
x Pxm e α −< = −  
/2 /2 1 Px Z m x Z
nn
αα
σσ
α
⎛⎞
−≤ ≤ + = − ⎜⎟
⎝⎠
 
If the sample volume was determined based on the hypothesis of binary 
characteristic and variance  (along with the previous observations), the confidence 
interval is usually established by introducing a new methodological discordance, for 
supposed normally distributed measurable characteristics. 
 
1.4. Observation IV 
 
In many practical situations, the t probabilistic coefficient is used, which has 
values different from the ones of Laplace variable, besides the fact that its values are 
dependent on the number of degrees of freedom, related to the volume of the origin 
population, respectively of the sample. 
The t distribution is also known as Student distribution after the name of the 
mathematician statistician William Sealy Gosset (1876-1937), who was mostly known 
under the pseudonym Student and an important representative of the Anglo-Saxon 
statistic school. He led the laboratory of beer factory (Park Royal) using statistic 
procedures in interpreting the results of the analyses. He also worked at the London 
University College and collaborated with R.A. Fisher and K. Pearson. The original 
results in the theory of probable error of the mean, in reduced volume surveys, are 
found in the „t law”. The t distribution is the probability distribution of a continuous 
random variable, with a probability density function given by:  Possible errors in establishing the sample dimensions of survey investigations 
 
131
1
2 2
1
11 2 (; )
1 2
ft
t
ν
ν
ν
ν πν
ν
+
⎛⎞
⎛⎞ + ⎡⎤ ⎜⎟ Γ ⎜⎟ ⎢⎥ ⎜⎟ ⎣⎦ ⎜⎟ = ⎜⎟
⎛⎞ ⎜⎟ ⎜⎟ Γ ⎡⎤ ⎜⎟ ⎜⎟ ⎜⎟ + ⎝⎠ ⎝⎠ ⎢⎥ ⎜⎟ ⎣⎦ ⎝⎠
,  
where  +∞ < < ∞ − t  with the parameter  ,... 2 , 1 = ν  and Γ is the Gamma function. 
If the X random variable has a Student distribution with  2 > ν degrees of 
freedom, then: 
()0; ( )
2
MX V a r X
ν
ν
==
−
 
The use of the Student variable values in calculus, without appropriate 
justification, is another source of errors that might influence the final results. 
 
2. A few solutions 
 
Performing survey investigations in conditions of statistic fidelity able to 
assure high representativeness and accuracy to the final results must correspond to 
some rules such as: 
2.1. The dimensioning of sample volume, in the case of binary characteristics, 
is achieved by using the binomial or hypergeometric model, depending on the 
extraction type (with or without replacement), respectively the dimension of the 
reference population (reduced volume or high volume). 
a)  Binomial distribution (formulated by J. Bernoulli) 
When the values of a random variable X are the proportion of A or non A 
elements (for example the proportion of citizens who intend to vote respectively who 
do not intend to), the typical values are: 
-  mean:  () E Xn p = ; 
-  variance:  () Var X npq = ; 
-  mode: np q Mo np p −≤ ≤ +. 
where: p and q are the elements of A and non A, and n – the sample volume. 
 
The theoretical model considers there is the same probability for the occurrence 
of the A1, A2, …, An events, and therefore the probability for the occurrence of x out of 
n events is: 
!
()
!( )!
x xn x xn x
n
n
PX x Cpq pq
xn x
−− == =
−
, 
where x is the frequency of the event under observation, with values x = 0,1, ... ,n. 
 Management & Marketing 
 
132
b) The hypergeometric distribution 
It is used in investigations as model especially for the description of reduced 
volume samples operations, with replacement sampling. 
The probability that k favorable results be obtained out of n units, is determined 
as: 
() () ;, ; 0 , 1 , 2 , , ,
xn k
kN k
n
N
CC
P Nnx x nx
C
−
− == …  
The typical values of this distribution are: 
-  mean  [ ] M Xn p = ; 
-  variance  [] ()
2 1
1
Nn
DX n p p
N
−
=−
−
. 
when N is large enough, practically when  /1 / 1 0 nN < , the hypergeometric law can 
be approximated through the binomial law. 
2.2. Establishing the main characteristic for which the research is designed; 
the calculations are performed for this characteristic which assures the 
representativeness of the entire investigation, the rest of the characteristics becoming 
subordinated.  
2.3. In the case of binary characteristics, that is in the case of alternative 
questions, designing the survey parameters must be accomplished by using the 
binomial law model, if the survey is unrepeated, or if the population under 
investigation has a high volume, while in the case of comeback sampling or reduced 
volume populations, the hypergeometric law model is recommended. 
2.4. If designing the sample volume was based on binary characteristic, then 
the confidence interval as well as the generalization of the results, must be 
accomplished based on the same hypotheses; if the essential characteristic is 
measurable and continuous, as are most in sociologic questionnaires, then the sample 
design as well as expanding the results will be performed in the hypothesis that must 
be tested in regard of validating the normality. 
2.5. Estimating variance 
a.  For binary characteristics, the use of the maximum variance (pq = 0.25) 
can be chosen only as a temporary solution, and only in the case of a first research 
over a population. Later on, based on supplementary information, the survey estimator 
can be used, which gets closer and closer to the unknown variance of the general 
population with each research.  
b.  A specific survey can also be organized, with no restrictions regarding 
representativeness, on a reduced number of subjects, and on the basis of the values 
registered for characteristic X: 
12 ,,, n Xx x x =…  
   Possible errors in establishing the sample dimensions of survey investigations 
 
133
The corrected survey variance can be estimated: 
( ) ˆ
1
i x x
S
n
−
=
−
∑  
which can be used as reference element in the calculus of the sample volume 
for the first research, whose quality shall be improved after further information.  
c.  If establishing the sample is compulsory, the variance can be estimated 
through the maximum variance of the measurable characteristic: 
( ) ()
22
min max 2
max 2
x xxx
σ
−+ −
=  
d.  This issue becomes more complicated in the case of heterogeneous 
structured populations (e.g. the populations of house-holds structured by the 
occupational status of the head of the house-hold, by income class, by number of 
children, etc., the populations of firms structured by number of employees, by activity, 
by profit volume etc.) its solving consisting in applying survey schemes with unequal 
probabilities. 
 
 
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Abordarea şase sixgma.

Ancheta sociologică şi sondajul de opinie,

Cunoaşterea vieţii sociale. Fundamente metodologice,

Dicţionar de statistică generală,

Metode de anchetă în investigarea fenomenelor sociale,

Mică enciclopedie de statistică,

Organizational research: Determining appropriate sample size for survey research”,

Possible errors in establishing the sample dimensions of survey investigations

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