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Abstract

Obtaining a representative sample size remains critical to survey researchers because of its implication for cost, time and precision of the sample estimate. However, the difficulty of obtaining a good estimate of population variance coupled with insufficient skills in sampling theory impede the researchers' ability to obtain an optimum sample in survey research. This paper proposes an adjustment to the margin of error in Yamane's (1967) formula to make it applicable for use in determining optimum sample size for both continuous and categorical variables at all levels of confidence. A minimum sample size determination table is developed for use by researchers based on the adjusted formula developed in this paper.
_____________________________________________________________________________________________________
*Corresponding author: E-mail: aadam@ucc.edu.gh;
Journal of Scientific Research & Reports
26(5): 90-97, 2020; Article no.JSRR.58400
ISSN: 2320-0227
Sample Size Determination in Survey Research
Anokye M. Adam
1*
1
Department of Finance, School of Business, University of Cape Coast, Ghana.
Author’s contribution
The sole author designed, analyzed and interpreted and prepared the manuscript.
Article Information
DOI: 10.9734/JSRR/2020/v26i530263
Editor(s):
(1) Dr. Suleyman Korkut, Duzce University, Turkey.
Reviewers:
(1)
Sonal Kapoor, India.
(2)
Tanmoy Dasgupta, The University of Burdwan, India.
Complete Peer review History:
http://www.sdiarticle4.com/review-history/58400
Received 18 April 2020
Accepted 22 June 2020
Published 25 June 2020
ABSTRACT
Obtaining a representative sample size remains critical to survey researchers because of its
implication for cost, time and precision of the sample estimate. However, the difficulty of obtaining a
good estimate of population variance coupled with insufficient skills in sampling theory impede the
researchers’ ability to obtain an optimum sample in survey research. This paper proposes an
adjustment to the margin of error in Yamane’s (1967) formula to make it applicable for use in
determining optimum sample size for both continuous and categorical variables at all levels of
confidence. A minimum sample size determination table is developed for use by researchers based
on the adjusted formula developed in this paper.
Keywords: Sample size determination; Yamane formula; survey research; Likert scale.
1. INTRODUCTION
One of the key challenges that social science
researchers face in survey research is the
determination of appropriate sample size which
is representative of the population under study.
This is to ensure that findings generalized from
the sample drawn back to the population are with
limits of random error [1]. It is impossible to make
accurate inferences about the population when a
test sample does not truly represent the
population from which it is drawn due to sample
bias [2]. This makes the appropriate sample size
important in survey research. However,
researchers continue to incorrectly estimate
sample size due to misuse or inappropriate use
Short Research Article
Adam; JSRR, 26(5): 90-97, 2020; Article no.JSRR.58400
91
of sample size determination tables and formulas
[3]. Wunsch [4] identified two most consistent
flaws when determining sample size as disregard
for sampling error and nonresponse bias. In
addition, disregard for sample variance and
treatment of all estimand as dichotomous
(population proportion) is a common flaw in
survey research. As pointed out by Bartlett, et al.
[1], a simple survey reveals numerous errors and
questionable approaches to sampling size
selection in published manuscript surveyed.
These errors emanate from an insufficient
statistical understanding of these sample size
selection methods and quest to use the simplest
method in survey research [3]. As noted by Israel
[5], the difficulty of obtaining a good estimate of
population variance has increased the popularity
of sample size based on proportion. Taro
Yamane (1967) formula which is a simplified
formula for proportion has become popular with
researchers for these reasons. Denoting by n
the sample size, Taro Yamane formula is given
by  =

, where is the population size and
is the margin of error. Strictly speaking,
Yamane formula is an approximation of known
sample size formulas such as Krejcie and
Morgan [6] and Cochran [7] formulas for
proportion at 95% confidence level and
population proportion of 0.5. Yamane formula in
its present state is, therefore, best suited for
categorical variables and only applicable when
the confidence coefficient is 95% with a
population proportion of 0.5.
Bartlett, Kotrlik and Higgins [1] argued for the
different sample size for dichotomous
(categorical) variables and continuous variables.
Though sample based on proportion is
conservative, it has a cost implication for data
collection and processing.
This paper proposes an adjustment to the margin
of error in Yamane to allow it to be applicable for
use in determining sample size for both
continuous and categorical variables at all levels
of confidence. Besides, a minimum sample size
determination table is developed for use by
researchers based on the adjusted formula
developed in this paper. The paper contributes to
the existing literature by removing the restriction
of the use of Yamane formula.
The paper is structured as follows: Section 2
looks at the mathematical derivation of the
proposed adjusted formula from Krejcie and
Morgan [6] and Cochran [7] formulae. Section 3
presents the estimation of variance for both
categorical and continuous variables.
2. MATHEMATICAL DERIVATION
We begin by considering the formula used by
Krejcie and Morgan in their 1970 article
“Determining Sample Size for Research
Activities”
s =
()
()
()
(1)
s= required sample size
= the table value of chi-square for 1 degree of
freedom at the desired confidence level.
N= the population size
P= the population proportion
d= the degree of accuracy expressed as a
proportion
From equation (1), we can write that
 =
 
()
()

(2)
S
d
→ 0 =  =

 
()
(3)
Krejcie and Morgan [6] recommended the use of
.50 as an estimate of the population proportion to
maximize variance, which will also produce the
maximum sample size. So at 95% confidence
level, P = 0.5,
(1 − )≈ 1
 =
 
which is Slovin or Yamane formula.
Again, given Cochran’s [7] formula  =
()
and finite correction factor =

, Tejada and
Punzalan [2] had proved that for P=0.5 and at
95% confidence level,  =
and  =
 
. This
implies that Yamane formula is a special case of
Krejcie and Morgan [6] formula or Cochran’s [7]
formula. Hence, Krejcie and Morgan’s, Cochran’s
and Yamane’s formulas coincide when
estimating sample size using a 95% confidence
coefficient and P = 0.5.
In effect, when
(1 − )=
σ
, the general
formula for determining sample size becomes
 =
(
σ
)
(4)
This allows the adjusted Yamane’s formula
applicable at different population proportion
levels and confidence levels.
Adam; JSRR, 26(5): 90-97, 2020; Article no.JSRR.58400
92
3. ESTIMATION OF VARIANCE
Cochran [7] listed four ways of estimating
population variances for sample size
determinations: (1) take the sample in two steps,
and use the results of the first step to determine
how many additional responses are needed to
attain an appropriate sample size based on the
variance observed in the first step data.; (2) use
pilot study results: (3) use data from previous
studies of the same or a similar population; or (4)
estimate or guess the structure of the population
assisted by some logical mathematical results.
Bartlett, et al. [1] observed that the first three
ways are logical and produce valid estimates of
variance; but not feasible to use them because of
technical difficulty to implement. The fourth
option is rather likely to be used by survey
researchers due to its flexibility.
Bartlett, et al. [1] showed that the standard
deviation of survey research using Likert-type
items is estimated as the ratio on the inclusive
range of the scale to the number of standard
deviations that would include all positive values
in the range. Let, λ= the inclusive range, ρ = the
number of standard deviations that would include
all possible values in the range and = the
degree of accuracy expressed as a proportion,
then σ =
λ
ρ
and mean margin of error  = λ
The adjusted Yamane’s formula in equation (4)
becomes
n=
Νɛ
(5)
Where,
n= minimum returned sample size
N = the population size
ɛ = adjust margin of error [ɛ= (
ρ
)]
e = the degree of accuracy expressed as a
proportion
ρ= the number of standard deviations that
would include all possible
t= t-value for the selected alpha level of
confidence level
Park and Jung [8] argues that respondents tend
to avoid choosing extreme responses categories,
proportion choosing the middle option of Likert-
type is larger than extreme responses, so that
the coefficient of variation is smaller than 1
(about 0.3-0.5). This coefficient of variation and
its associated mean values imply that the
standard deviation of Likert-type item rounds to 1
point. Given that the standard deviation is 1
point, the number of standard deviations that
would include all possible values in the range is
one less the number of inclusive ranges for an
odd number of points and equal to the number of
inclusive ranges for an even number. For
example, the number of standard deviations that
would include all possible (ρ) for five-point Likert-
type scale is four (i.e. two to each side of the
mean) and for six-point Likert-type scale is six
(i.e three to each side of the mean). The number
of inclusive ranges for all survey research ranges
from 2 for dichotomous responses to 10 for 11-
point Likert-type scale. The minimum returned
sample size varies inversely with the number of
standard deviations that would include all
possible ( ρ). Inferences from Rasmussen [9],
Owuor [10] and Norman [11] suggest that scales
with 5 or more points can be treated as
continuous data and be treated with parametric
statistics. A 2-point and 5-point scales are
recommended as least for categorical and
continuous variables respectively. The number of
standard deviations that would include all
possible of 2-point scale, 2, and 5-point scale, 4,
respectively yield maximum sample size for
categorical and continuous variables. This is
consistent with Cochran [7] that for a range of
sample size which is relatively close, the
researcher can settle on the largest sample size
to be confident of achieving the desired
accuracy. Thus, ρ = 2 is recommended for
categorical variables and ρ =4 for continuous
variable.
The choice of either number of standard
deviations that would include all possible values
in the range for categorical or continuous
variable depends on whether a categorical
variable will play a primary role in the data
analysis or not [1]’ if categorical variable will play
a primary role in the data analysis, use number
of standard deviations that would include all
possible values in the range for categorical else
use the number of deviations that would include
all possible values in the range for continuous
estimand. Krejcie and Morgan [6] recommended
5% as an acceptable margin of error for
categorical data and 3% for continuous data.
To illustrate the use of the two examples, let us
consider the following two examples:
Example 1:
Assume a researcher wants to examine the
gender disparity in financial literacy among the
Adam; JSRR, 26(5): 90-97, 2020; Article no.JSRR.58400
93
Cape Coast Metropolis in Ghana. If the
researcher set the alpha level a priori at.05 and
the population is 1500 retirees, what minimum
returned sample size is required at 95%
confidence level and margin of error of 0.05? The
number of standard deviations that would include
all possible categorical variables should be used
because of gender.
n=
Νɛ
Where
n= minimum returned sample size
N= Population size=1500
e= the degree of accuracy expressed as a
proportion=0.05
ρ= the number of standard deviations that
would include all possible values in the
range =2
t= t-value for the selected alpha level or
confidence level at 95% =1.96
ɛ = adjust margin of error [ɛ= (
ρ
)]
ɛ= (
(.)
.
) = 0.051
 =

ɛ
n=

(.)
= 306
Example 2:
A researcher examines how financial literacy,
financial behaviour, and retirement planning to
influence on the financial well-being of retirees in
Cape Coast Metropolis of Ghana. A cross-
sectional survey strategy was employed and a
seven-point Likert-type scale was employed to
measure financial literacy, financial behaviour,
and retirement planning and financial well-being
of retirees. If the researcher set the alpha level a
priori at.05 and the population is 1500 retirees,
what minimum returned sample size is required
at 95% confidence level and margin of error of
0.05?
Unlike example 1, the number of standard
deviations that would include all possible
continuous variables should be used (i.e. 4).
n=
Νɛ
Where
n= minimum returned sample size
N= Population size=1500
e= the degree of accuracy expressed as a
proportion=0.05
ρ= the number of standard deviations that
would include all possible values in the
range =2
t= t-value for the selected alpha level or
confidence level at 95% =1.96
ɛ = adjust margin of error [ɛ= (
ρ
)]
ɛ= (
(.)
.
) = 0.06218
⟹  =

(.)
= 226
The advantage of this adjustment hinges not only
on its simplicity but its ability to determine the
sample size of both continuous and categorical
survey variables.
4. SAMPLE SIZE DETERMINATION
TABLE
We present in Table 1 the minimum sample size
values for many common sampling problems
based on our adjusted formula for both
continuous and categorical data. The table
assumed one of the three commonly used
confidence levels in survey research: 90%, 95%
or 99% and used a margin of error of 3% for
continuous data and 5% for categorical data. The
Table is recommended for use by researchers if
the indicated margin of error is appropriate for
their study.
To validate the ability of the adjustment formula
in estimating the required minimum sample size,
we compared the sample obtained to sample
size obtained from the frequently used sample
size determination approaches such as Krejcie
and Morgan [6] and Bartlett, Kortlik and Huggins
[1]. Fig. 1 shows the plot of sample size obtained
from the proposed adjusted margin of error, SS1,
and Sample size obtained from Krejcie and
Morgan [6], SS2, for categorical estimand versus
Population at 5% significance level. The plot
shows that sample sizes obtained from the two
approaches are virtually the same with
correlation = 0.9992,  < 0.001. Similarly, a plot
of sample size obtained from the proposed
adjusted margin of error, CSS1, and Sample size
obtained from Bartlett, Kortlik and Huggins [1],
SS2, for continuous estimand versus Population
is shown by Fig. 2. The plot shows that the new
approach provides a more conservative sample
size with = 0.97,  < 0.001.
Adam; JSRR, 26(5): 90-97, 2020; Article no.JSRR.58400
94
Table 1. Table for determining minimum returned sample size for a given population size for continuous and categorical data
Popula-
tion size
Sample size
Categorical data (margin of error=.05),
=2 Continuous data (margin of error=.03),
=4
90% confidence
Level
= . 
95% confidence
Level
= . 
99% confidence
Level
= . 
90% confidence
Level
= . 
95% confidence
Level
= . 
99% confidence Level
= . 
10 10 10 10 10 10 10
15 15 15 15 14 15 15
20 19 20 20 19 19 20
25 23 24 25 23 23 24
30 28 28 29 26 27 29
35 31 33 34 30 31 33
40 35 37 38 33 35 37
50 43 45 47 40 43 46
60 50 52 56 46 49 54
70 56 60 64 52 56 61
80 62 67 72 57 62 69
90 68 73 80 61 68 76
100 74 80 87 66 73 83
110 79 86 95 70 78 89
120 84 92 102 74 83 96
130 88 98 109 77 88 102
140 93 103 116 81 92 108
150 97 108 123 84 97 114
160 101 113 129 87 101 119
170 105 118 136 90 104 125
180 109 123 142 92 108 130
190 112 128 148 95 111 135
200 116 132 154 97 115 140
220 122 140 166 102 121 150
250 130 152 182 108 130 163
300 143 169 207 116 142 182
350 153 184 230 123 152 200
400 162 196 250 128 161 215
Adam; JSRR, 26(5): 90-97, 2020; Article no.JSRR.58400
95
Popula-
tion size
Sample size
Continuous data (margin of error=.03),=4
90% confidence
Level
= . 
95% confidence
Level
= . 
99% confidence
Level
= . 
90% confidence
Level
= . 
95% confidence
Level
= . 
99% confidence Level
= . 
450 169 208 269 133 168 229
500 176 218 286 137 174 241
600 187 235 316 144 185 262
700 196 249 342 149 194 279
800 203 260 364 153 201 293
900 209 270 383 156 206 306
1000 213 278 400 159 211 317
1200 221 292 429 163 219 334
1500 230 306 462 167 227 354
2000 239 323 500 172 236 376
3000 249 341 545 177 245 401
5000 257 357 588 182 254 424
8000 262 367 615 184 259 437
10000 264 370 625 185 260 442
20000 267 377 645 187 264 452
50000 270 382 657 188 266 459
100000 270 383 662 188 267 461
150000 271 384 663 188 267 461
200000 271 384 664 188 267 462
>1000000 271 385 666 188 267 463
Adam; JSRR, 26(5): 90-97, 2020; Article no.JSRR.58400
96
Fig. 1. Sample size of categorical estimand Vs population
Note: CSS1 is sample size obtained from the proposed approach and CSS2 is sample size obtained from the
approach proposed by Krejcie and Morgan [6]
Fig. 2. Sample size of continuous estimand vs population
Note: CSS1 is sample size obtained from the proposed approach and CSS2 is sample size obtained from the
approach proposed by Bartlett, Kortlik and Huggins [1]
0
50
100
150
200
250
300
350
400
450
Minimum Required Sample Size
Population Size
Sample Size of Categorical Estimand Vs Population Size
SS1
SS2
0
50
100
150
200
250
300
Minimum Required Sample Size
Population Size
Sample Size of Continuous Estimand Vs Population Size
CSS1
CSS2
Adam; JSRR, 26(5): 90-97, 2020; Article no.JSRR.58400
97
5. CONCLUSION
In this paper, we propose an adjustment to the
margin of error in Yamane (1967) formula to
make it applicable for use in determining
optimum sample size for both continuous and
categorical variables at all levels of confidence. It
has been shown that the degree of accuracy
expressed as a proportion (margin of error in
Yamane formula), , be adjusted by a factor of
the ratio of the number of standard deviations
that would include all possible values in the
range to the t-value for the selected alpha level
or confidence level,
ρ
,. Accordingly, ρ = 2 was
recommended for categorical variables and ρ=4
for continuous variable. A minimum sample size
determination table is developed for use by
researchers based on the adjusted formula when
certain assumptions are met.
COMPETING INTERESTS
Author has declared that no competing interests
exist.
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Organizational research: Determining
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2. Taherdoost H. Determining
sample size; how to calculate survey
sample size. International Journal of
Economics and Management Systems.
2017;2:237–239.
3. Tejada JJ, Punzalan JRB. On the misuse
of Slovin's formula. The Philippine
Statistician. 2012;61(1):129–136.
4. Wunsch D. Survey research: Determining
sample size and representative response.
Business Education Forum. 1986;40(5):
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5. Israel GD. Determining sample size,
University of Florida Cooperative
Extension Service, Institute of Food and
Agriculture Sciences, EDIS; 1992.
6. Krejcie RV, Morgan DW. Determining
sample size for research activities.
Educational and Psychological
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7. Cochran WG. Sampling techniques (3
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ed.). New York: John Wiley & Sons; 1977.
8. Park J, Jung M. A note on determination of
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Communication of the Korean Statistical
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______________________________________________________________________________
© 2020 Adam; This is an Open Access article distributed under the terms of the Creative Commons Attribution License
(http://creativecommons.org/licenses /by/4.0), which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
Peer-review history:
The peer review history for this paper can be accessed here:
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... The research population was whittled down to 139 officials, who served as the sample population (Wisdom & Creswell, 2013;Radhakrishnan, 2014;Umsl.edu, 2021), using the Taro Yamane formula (Adam, 2020;Taherdoost, 2017). ...
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... This formula was used to calculate the sample sizes in tables 2 and 3 and is shown in Equation 1. A 95% confidence level and a sampling of error e = 0.05 are assumed for that formula [10]. ...
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Reviewers of research reports frequently criticize the choice of statistical methods. While some of these criticisms are well-founded, frequently the use of various parametric methods such as analysis of variance, regression, correlation are faulted because: (a) the sample size is too small, (b) the data may not be normally distributed, or (c) The data are from Likert scales, which are ordinal, so parametric statistics cannot be used. In this paper, I dissect these arguments, and show that many studies, dating back to the 1930s consistently show that parametric statistics are robust with respect to violations of these assumptions. Hence, challenges like those above are unfounded, and parametric methods can be utilized without concern for "getting the wrong answer".
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When a social scientist prepares to conduct a survey, he/she faces the problem of deciding an appropriate sample size. Sample size is closely connected with cost, time, and the precision of the sample estimate. It is thus important to choose a size appropriate for the survey, but this may be difficult for survey researchers not skilled in a sampling theory. In this study we propose a method to determine a sample size under certain assumptions when the quantity of interest is measured by a Likert scale.
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Research has shown that when an appropriate data transformation is known a priori, then it can lead to a substantial increase in power of analysis of variance while maintaining an appropriate Type I error rate. It was unknown, however, whether data transformation selected on sample characteristics would yield accurate Type I error rates and increased power. The present Monte Carlo study demonstrates that correct data transformation values could be selected on samples as small as four per group, that legitimate approaches do not inflate the nominal significance levels and that power could be increased by sample-based transformations.
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The determination of sample size is a common task for many organizational researchers. Inappropriate, inadequate, or excessive sample sizes continue to influence the quality and accuracy of research. The procedures for determining sample size for continuous and categorical variables using Cochran's (1977) formulas are described. A discussion and illustration of sample size formulas, including the formula for adjusting the sample size for smaller populations, is included
On the misuse of Slovin's formula. The Philippine Statistician
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Tejada JJ, Punzalan JRB. On the misuse of Slovin's formula. The Philippine Statistician. 2012;61(1):129-136.
Determining sample size, University of Florida Cooperative Extension Service, Institute of Food and Agriculture Sciences, EDIS
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Implications of using Likert data in multiple regression analysis. University of British Columbia: Doctoral dissertation
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