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INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN APPLIED SCIENCE (IJRIAS)
ISSN No. 2454-6194 | DOI: 10.51584/IJRIAS |Volume X Issue X October 2025
Comparative Analysis on Probability Proportional to Size Sampling
Scheme in Estimating Population Total of Student Enrolment in Ekiti
State University
Faweya O., Babarinde A., T Odukoya E. A
Department of Statistics, Faculty of Science, Ekiti State University, Ado-Ekiti, Ekiti State, Nigeria
DOI:
https://doi.org/10.51584/IJRIAS.2025.10100000145
Received: 26 August 2025; Accepted: 03 September 2025; Published: 18 November 2025
ABSTRACT
This study focuses on a comparative analysis of probability proportional to size (PPS) sampling schemes in
estimating the population total of student enrollment at Ekiti State University (EKSU), Ado-Ekiti. The study
population consists of all ten faculties in EKSU, with data on student enrollment for five academic years (2017–
2022) obtained from the Directorate of Academic Planning. Secondary data were utilized, and five faculties were
sampled using the recommended sampling techniques for each method. The results revealed that all three
methods provided reliable estimates for the total population, but there were notable differences in efficiency.
PPS sampling with replacement was found to be relatively simple and robust for ensuring representation from
unequal population units. The Horvitz-Thompson method produced unbiased estimates but with higher variance
compared to PPS. The Rao-Hartley-Cochran scheme was less efficient, making it less suitable for such analyses.
Keywords: Probability proportional to size, Horvitz-Thompson, Rao-Hartley-Cochran, population total, student
enrollment.
INTRODUCTION
Probability proportional to size is an important sampling method in survey research as it addresses the issues
and problems associated with traditional probability sampling methods, most especially those which are in the
category of equal probability sampling. One drawback of the traditional sampling procedures is that none of
these sampling procedures consider the size of the population units, in the process of selecting the units from the
population (Ila, Raj & Joshi, 2020). If the size of the population units varies significantly, then it may not be
appropriate to select the population units with equal probabilities, as in the population larger units may have
some important information and this kind of selection ignores the significance of the larger units. This problem
can be solved by assigning different selection probabilities to different units of the population (Pamplona, 2019).
Thus, when the size of population units varies considerably and the variance is highly correlated with the size of
the unit, then the selection probabilities can be assigned in proportion to the size of the population units. The
essence of probability proportional to size is claimed to be superior amidst unbiased sampling procedure mainly
due to involvement of auxiliary information. Developments in sampling theory with the introduction of
proportional to size, were brought by the emphasis on the need and use of auxiliary information in improving
precision of estimates (Homa, Maurya, and Singh, 2013). Sampling scheme is an important aspect most
considered in statistics, most especially in survey research, given that it is possible to get a sufficiently good
estimate of the parameter of interest at a reasonably low cost (Grafstrom, 2010). Sampling is defined as a
procedure to select a sample from individual or from a large group of population for certain kind of research
purpose (Shardwaj, 2019). Abdullah, et al (2014) examined the selection of samples in probability proportional
to size sampling using cumulative relative frequency method. They used data of a village with 10 holdings
applied to probability proportional to size under cumulative relative frequency method, cumulative total method,
and Lahiri’s Method, result showed that relative frequency to select samples in probability proportional to size
takes less time and easy to apply than Cumulative Total Method and Lahiri’s Method. Hence, the study
recommended engaging the method of selecting samples in probability proportional to size which use relative
frequency among others.
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Maleka, Si and Gelman (2017) in their study checked Bayesian inference under cluster sampling with probability
proportional to size. They revealed that challenges arise when the sizes of nonsampled cluster are unknown, and
that integrated Bayesian approach outperforms classical methods with efficiency gains. They use Stan for
computing and apply the proposal to the Fragile Families and Child Wellbeing study as an illustration of complex
survey inference in health surveys. esponse probabilities are non-uniform and a sampling fraction can be both
negligible and not negligible where both circumstances are more realistic in practice under the reverse
framework using simulation and real data sets. Result revealed that consider under less restricted situations where
response probabilities are non-uniform and a sampling fraction can be both negligible and not negligible where
both circumstances are more realistic in practice under the reverse framework.
METHODS OF ESTIMATION
This study used three different method of estimating total population and variance under probability proportional
to size sampling scheme, as identified in the objective of the study which are probability proportional size with
replacement, Horvitz-Thompson sampling scheme and Rao-Hartley-Cochran’s sampling scheme. The
procedures for the three methods are discussed extensively below:
Probability Proportional to Size with Replacement
Let units
have sizes
respectively; where
i=1, 2,…., N is an
integer. If
’s are not integers they should be multipled by appropriate power of 10 to make them integers. If a
sample of size n units is to be selected from a population of N units
by probability
proportional to size with replacement (ppswr), we proceed as follows:
Step 1: Form cumulative totals of the sizes; assign ranges to all the population units using the cumulative totals
as in Table 1
Table 1: Probability Proportional to Size selection
Units
Sizes
Cumulative Totals
Ranges
:
:
:
:
:
:
:
:
Step 2: Using random table select a number d between 1 and
inclusive. If the number d falls in the
range of
, say, then it is selected in the sample. Another random number is drawn between 1 and X inclusive,
and if the number selected falls this time in the range of
, unit
is selected. In other words the unit selected
in the sample is the unit in whose range the selected random number falls. The process of drawing a random
number is repeated independently (i.e. the selected random number is returned into the pool before the next
selection) until n units are selected in the sample. With this selection procedure the n units selected with ppswr,
and the probability of selecting the i
th
unit in the population is
, and gives
.
Step 3: Estimation of Population Total
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An unbiased estimator of the population total under probability proportional to size with replacement, Y is given
by
Where
is the probability of selecting the i
th
unit in the sample;
is the measure of size of the i
th
sample unit.
Step 4: Estimation of Variance
Since sampling is with replacement the values
, i= 1,2…,n are independent and the covariance between any
pair of values (
) is zero.
Its sample estimator
Horvitz-Thompson Sampling Scheme
Horvitz-Thompson Sampling Scheme is a general theory of sampling scheme without replacement under
unequal probability sampling. Under this form of estimator, there are steps to be considered to ensure proper
estimation as recommended by these scholars which are as follows:
Step 1: This involves the selection of samples. To achieve this, there is need to first create a table for cumulative
total and range of the units (departments) in each of the grouped population (faculties), Afterward, select a
sample of size n without replacement, a random start between 1 and k inclusive (k= x\n is the S.I.) is selected
using a table of random numbers. If the number selected is r then the units in the sample are those in whose
ranges the numbers r, r + k, r + 2k,…,r + (n-1)k fall. The probability of selecting the unit
in the sample of size
n is
If the unit in the population has its size greater than or equal to k such unit is removed before sampling, and then
taken into the sample with probability one. The probability that any pair of units (
) is together in the sample
is
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=
=
i
is the number of the random number between 1 and k inclusive, which will select
and
simultaneously
in the sample. Given the r, r + k, r + 2k,…,r + (n-1)k, for the first selected number and second selected number,
without replacement,
has the range of
and the process of getting
generally involves
calculating the range of
, based on inequality as expressed thus
, where
are the minimum and maximum limit of the range of cumulative total for
. If the minimum
values and maximum values of the
range falls within the range of
, we can use these values and subtract
maximum from minimum to get the
. But in cases where any of minimum or maximum values or both for
does not falls within the range of
, the minimum or maximum values or both of the range
is used for
respectively.
Step 2: Estimation of Population total
An unbiased estimator of the population total for ppswor sampling as given by Horvitz-Thompson is
Step3: Estimation of variance is
Provided that
Rao-Hartley-Cochran’s Sampling Scheme
Rao-Hartley-Cochran’s Sampling Scheme in relation to probability proportional to size is proposed as a unequal
probability sampling scheme estimator that entails selection without replacement with special cases. The main
processes of this estimator are as follows:
Step 1: divide a population of N units into n groups at random with group g containing
units (g = 1,2,…,n)
such that
Step 2: select one unit with ppswor independently from each group. This gives a total of n units selected in the
sample ppswor. The probability of selecting
in the sample in
group is
Where
is the total measure of size in
group;
is the sum of the initial probabilities in
group.
Step 3: Estimation of Population Total
The Rao-Hartley-Cochran estimator of the population total is
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Where
is the value of the study variate for
unit in
group
Step 4: Estimation of variance
The unbiased estimator of variance of
is
Rao-Hartley-Cochran Estimator
Rao et al. (1962) proposed a sampling strategy for use with unequal probability sampling and the estimator of
population total. The population units are divided randomly into n groups, where the group sizes are
predetermined. Then one unit is selected from each group. Their estimator is
(23)
where
is the probability of the Tth unit being selected from the ith group. Also
and
The Rao-Hartley-Cochran estimator can be used for any sample size. The variance of (23) is
(24)
RESULTS AND DISCUSSION
The population of the study consist all the faculties in Ekiti State University (EKSU) Ado-Ekiti. There are ten
faculties in EKSU, which are Sciences, Management Sciences, Social Sciences, Law, Engineering, Agricultural
Science, Art, Education, Basic Medical Science, and Medicine. The data on students’ enrolment were collected
from Directorate of Academic Planning, and data covered the students’ enrolment for 5 years spanning from
2017 to 2022 for all the ten faculties in Ekiti State University. Hence, data used for this study is gathered from
secondary source. The sample size of this study is five faculties selected from all the faculties in Ekiti State
University. The major sampling technique is the technique appropriate and recommended under each of the
probability proportional to size sampling scheme, Horvitz-Thompson sampling scheme and Rao-Hartley-
Cochran’s sampling scheme.
Table 1 Range and Selection of Samples among the Faculties
S/N
Faculty
No of
Dept.
Cum
Total
Ranges
2017
2018
2019
2021
2022
1
Agricultural
science*
6
6
1 – 6
1391
1468
1506
1526
1275
2
Art
7
13
7 – 13
2673
3053
3936
4293
4656
3
Basic Medical
Science*
3
16
14 – 16
984
1014
1150
1194
1550
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4
Education*
9
25
17 – 25
3476
3218
3065
3625
3123
5
Engineering
4
29
26 – 29
1622
1903
2341
2469
2507
6
Law
1
30
30
768
817
664
606
592
7
Management
Science
3
33
31 – 33
3335
3925
4272
5068
4367
8
Medicine
1
34
34
288
447
1098
262
292
9
Science**
12
46
35 – 46
4884
5591
7725
9205
8053
10
Social Science
6
52
47–52
3710
3894
4629
4484
4375
Note: Faculties were selected based on random sampling with replacement
Source: Author’s, (2023)
Estimation of Total Population
An unbiased estimator of the population total under probability proportional to size with replacement, Y is given
by
Table 2:Probabilities and Data Yearly data of students’ enrolment for the sampled faculties
Faculties
Agricultural
Science
Art
Science
Education
Science
1391
2673
4884
3476
4884
1468
3053
5591
3218
5591
1506
3936
7725
3065
7725
1526
4293
9205
3625
9205
1275
4656
8053
3123
8053
Source: Author’s Computation, (2023)
Total Population Estimate for 2017
Total Population Estimate for 2018
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Total Population Estimate for 2019
Total Population Estimate for 2021
Total Population Estimate for 2022
Estimation of Variance
Variance Estimate for 2017
Standard Error (SE) is
Variance Estimate for 2018
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Standard Error (SE) is
Variance Estimate for 2019
Standard Error (SE) is
Variance Estimate for 2021
Standard Error (SE) is
Variance Estimate for 2022
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Standard Error (SE) is
Horvitz-Thompson Sampling Scheme
Table 1 is also applicable for the selection of sample units under this scheme. With the intended sample of size
5, k =
= 10.4, and using table of random numbers, a number 007 is selected, which is between 1 and 10.4.
Based on this random number, the other numbers to be used in the selection are
The first number, which is random number 007 falls within the range of 7-13 which is for faculty of Art, the
second number 17.4 lies in range 17-25 which stand for faculty of education, the third number, 27.8 within 26-
29 which is for faculty of engineering, forth number and fifty number fall in the range of 35-46 and 47-52 which
are for faculty of sciences and social sciences respectively. Hence, the five elected faculties under Horvitz-
Thompson sampling scheme are Art, education, engineering, sciences and social sciences.
Probability of Inclusion
Given the number of unit under each of the selected faculties to be 7, 9, 4, 12, and 6, based on the presentation
in table 4.1, as well as k which is 10.4, the probability for each of the faculties selected is given in Table 3 below;
Table 3: Probability of the selected samples among all the faculties
Faculties
Art
Education
Engineering
Sciences
Social Science
Source: Author’s Computation, (2023)
Computation of probability of selecting two faculties (
)
For Faculty of Art and Education (Art=i; Education=j)
Where
Since 6.6 and 14.6 are not within
For Faculty of Art and Engineering (Art = i; Engineering = j)
Where
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Since 5.2 is not within , while 8.2 is within the range
For Faculty of Art and Science (Art = i; Science = j)
Where
Since 3.8 and 14.8 are not within
For Faculty of Art and Social Science (Art=i; Social Science=j)
Where
Since 5.4 is not within the range, , while 10.4 is within the range
For Faculty of Education and Engineering (Education=i; Engineering=j)
Where
Since 15.6 is not within the range, , while 18.6 is within the range
For Faculty of Education and Science (Education=i; Science=j)
Where
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Since 14.2 and 25.2 are not within
For Faculty of Education and Social Science (Education=i; Social Science=j)
Where
Since 15.8 is not within the range , while 20.6 is within the range
4.6
For Faculty of Engineering and Science (Engineering=i; Science=j)
Where
Since 24.6 and 35.6 are not within the range
For Faculty of Engineering and Social Science (Engineering=i; Social Science=j)
Where
Since 26.2 is within the range while 31.2 is not within the range
For Faculty of Science and Social Science (Science=i; Social Science=j)
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Where
Since 36.6 and 41.6 are within the range
Estimation of Total Population
An unbiased estimator of the population total for ppswor sampling as given by Horvitz-Thompson is
Table 4: Probability of the selected samples among all the faculties with annual enrolment data
Sample
faculties
Serial no
2017
2018
2019
2021
2022
Art
1
2673
3053
3936
4293
4656
Education
2
3476
3218
3065
3625
3123
Engineering
3
1622
1903
2341
2469
2507
Sciences
4
4884
5591
7725
9205
8053
Social
Science
5
3710
3894
4629
4484
4375
Source: Author’s Computation, (2023)
Based on the Horvitz-Thompson sampling scheme criteria, Table 4 reveals the probability of selecting each of
the faculties which are sampled, art, education, engineering, science and social sciences based on the number
007 selected from the random. In the Table 4.4 probabilities stood at
,
,
,
and
for art, education, engineering, sciences and social sciences respectively, in regards to k and the number
of units (departments) in each of the faculties.
Total Population Estimate for 2017
Total Population Estimate for 2018
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Total Population Estimate for 2019
Total Population Estimate for 2021
Total Population Estimate for 2022
4.2.2. Estimation of Variance
Variance Estimate for 2017
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Standard error (SE) is
SE
SE
3631.64
Variance Estimate for 2018
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Standard error (SE) is
SE
SE
4137.73
Variance Estimate for 2019
Standard error (SE) is
SE
SE
Variance Estimate for 2021
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Standard error (SE) is
SE
SE
Variance Estimate for 2022
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Standard error (SE) is
SE
SE
15384.42
Rao-Hartley-Cochran’s Sampling Scheme
Faculties was selected under Rao-Hartley-Cochran’s sampling scheme
Table 5: Categories of faculties under Rao-Hartley-Cochran’s sampling scheme
First Random Group
Second Random Group
Third Random Group
Faculties
Cum
Range
Faculties
Cum
Range
Faculties
Cum
Range
Agricultural
science
6
6
1-6
Education
9
9
1-9
Medicine
1
1
1
Law*
1
7
7
Engineering
4
13
10-13
Management
science*
3
4
2-4
Social
Science
6
13
8-13
Art*
7
20
14-20
Basic
Medical
Science
3
7
5-7
Science
12
11
8-11
Note: * indicates selected faculties based on random sampling
Source: Author’s Computation (2023)
Result presented in Table 5 reveals that there are three random groups for the faculties covered in the study. For
the first random group, there are faculties of agricultural science, law and social sciences, in the second random
group we have faculties of education, engineering and art, while the third random group consist faculties of
medicine, management science, basic medical science and science. In line with selection of one faculty from
each of the groups, based on random sampling, the faculties selected were law, art and management science for
first, second and third group respectively.
Probability of Selection under Rao-Hartley-Cochran’s sampling scheme
Table 6: Sampled Faculties and Selection Probabilities
Sample
Faculty
*
2017
2018
2019
2021
2022
Law
768
817
664
606
592
Art
2673
3053
3936
4293
4656
Management
Science
3335
3925
4212
5068
4367
Source: Author’s Computation (2023)
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Estimation of Total Population
The Rao-Hartley-Cochran’s estimator of the population total is
Total Population Estimate for 2017
Total Population Estimate for 2018
Total Population Estimate for 2019
Total Population Estimate for 2021
Total Population Estimate for 2022
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Estimation of Variance
The unbiased estimator of variance of
is
Variance Estimate for 2017
Standard error (SE) is
SE
SE
Variance Estimate for 2018
Standard error (SE) is
SE
SE
Variance Estimate for 2019
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Standard error (SE) is
SE
SE
Variance Estimate for 2021
Standard error (SE) is
SE
SE
Variance Estimate for 2022
Standard error (SE) is
SE
SE
Comparison of Estimated Population and Variance for the three PPS estimators
Table 7: Comparison of Estimated Population for PPS, HT and RHC
Estimated Population
YEAR
PPS
HT
RHC
2017
18864.66
22868.69
38743.00
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2018
20490.18
24797.40
44202.19
2019
25390.04
30194.76
46933.71
2021
29167.42
32736.40
52241.04
2022
26694.72
31607.09
48656.53
Note: PPS= Probability proportional to size; HT= Horvitz-Thompson; RHC= Rao-Hartley-Cochran’s
Figure 1: Estimated Population Total for PPS, HT and RHC
Result in Table 7 and Figure 1 reveals the estimated population total of student’s enrolment in Ekiti State
University under probability proportional to size sampling scheme, Horvitz-Thompson sampling scheme and
Rao-Hartley-Cochran’s sampling scheme for the year 2017, 2018, 2019, 2021 and 2022. It is revealed that
population proportional to size sampling scheme with replacement has the lowest estimation population while
Rao-Hartley-Cochran’s sampling scheme without replacement and sample selection has the highest estimated
population total of students’ enrolment in Ekiti State University over the period covered.
Table 8: Comparison of Estimated Variance for PPS, HT and RHC
Estimated Variance
YEAR
PPS
HT
RHC
2017
2970827.17
13188794.22
94617751.48
2018
4832052.04
17120788.35
135244730.4
2019
17837550.85
22385929.7
149634501.7
2021
27953583.24
6762123.26
255461577.0
2022
25822303.28
19606328.23
148086909.0
0
10000
20000
30000
40000
50000
60000
2017 2018 2019 2021 2022
ESTIMATED POPULATION
PPS HT RHC
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Figure 2: Estimated Variance for PPS, HT and RHC
Table 8 and Figure 2 shows estimated variance of the student’s enrolment in Ekiti State University under
probability proportional to size sampling scheme, Horvitz-Thompson sampling scheme without replacement and
Rao-Hartley-Cochran’s sampling scheme without replacement and special selection for the year 2017, 2018,
2019, 2021 and 2022. It is revealed that population proportional to size sampling scheme with replacement has
the lowest variance in 2017 to 2019, but Hovitz-Thompson has the lowest variance in 2021 and 2022, while Rao-
Hartley Cochran’s sampling scheme has the highest variance of students’ enrolment in Ekiti State University in
all the period covered.
DISCUSSION OF FINDING
The study revealed that the probability proportional to size (PPS) sampling scheme is more efficient than the
Rao-Hartley-Cochran sampling scheme for estimating the total student enrolment at Ekiti State University.
However, PPS showed inconsistencies in efficiency when compared to the Horvitz-Thompson sampling scheme.
The results indicated that PPS produced the lowest estimated population total among the three estimators.
Regarding variance, PPS had the minimum variance between 2017 and 2019, but the Horvitz-Thompson scheme
outperformed it in 2021 and 2022.
A key finding is the difficulty in determining the more efficient estimator between PPS without replacement and
Horvitz-Thompson when analyzing continuous (time series) data instead of discrete (point-in-time) data.
Nonetheless, PPS remains more efficient than the Rao-Hartley-Cochran sampling scheme.
CONCLUSION
This study concludes that there is a close competition between the probability proportional to size without
replacement (PPSWOR) estimator and the Horvitz-Thompson estimator. Both estimators produce population
estimates within the total student enrolment, though the PPSWOR estimator generates a lower estimate than the
Horvitz-Thompson, but both outperform the Rao-Hartley-Cochran estimator, which overestimates the
population. In terms of efficiency, the study finds no clear distinction between PPSWOR and Horvitz-Thompson
due to the irregular student enrolment flow in 2020. However, PPSWOR is deemed more efficient than the Rao-
Hartley-Cochran estimator.
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VARIANCE
PPS HT RHC
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