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A New Sampling Design for Systematic SamplingZaheen Khan a , Javid Shabbir b & Sat Gupta ca Department of Mathematics and Statistics , Federal Urdu University of Arts, Science andTechnology , Islamabad , Pakistanb Department of Statistics , Quaid-i-Azam University , Islamabad , Pakistanc Department of Mathematics and Statistics , The University of North Carolina atGreensboro , Greensboro , North Carolina , USAAccepted author version posted online: 25 Apr 2013.Published online: 16 Jul 2013.

To cite this article: Zaheen Khan , Javid Shabbir & Sat Gupta (2013) A New Sampling Design for Systematic Sampling,Communications in Statistics - Theory and Methods, 42:18, 3359-3370, DOI: 10.1080/03610926.2011.628771

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Communications in Statistics—Theory and Methods, 42: 3359–3370, 2013Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610926.2011.628771

ANew Sampling Design for Systematic Sampling

ZAHEEN KHAN1, JAVID SHABBIR2, AND SAT GUPTA3

1Department of Mathematics and Statistics, Federal Urdu Universityof Arts, Science and Technology, Islamabad, Pakistan2Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan3Department of Mathematics and Statistics, The University of NorthCarolina at Greensboro, Greensboro, North Carolina, USA

A sampling design called “Modified Systematic Sampling (MSS)” is proposed. Inthis design each unit has an equal probability of selection. Moreover, it works forboth situations: N = nk or N �= nk. Consequently, the Linear Systematic Sampling(LSS) and Circular Systematic Sampling (CSS) become special cases of the proposedMSS design.

Keywords Circular systematic sampling; Modified systematic sampling;Systematic sampling.

Mathematics Subject Classification 62D05.

1. Introduction

In systematic sampling when population size N is a multiple of the sample size n,i.e., �N = nk�, where k is the sampling interval, a unit is selected at random fromthe first k units and then every kth unit is selected to get a sample of size n. Underthis sampling scheme there are k possible samples of size n and the sample mean isan unbiased estimator of the population mean. This sampling scheme is known asLinear Systematic Sampling (LSS). Linear systematic sampling can not be used whenpopulation size is not a multiple of the sample size, i.e., �N �= nk�. To overcomethis problem various authors have used alternative sampling designs like CircularSystematic Sampling (CSS), Balanced Circular Systematic Sampling (BCSS) andDiagonal Systematic Sampling (DSS) etc.

According to CSS, N units of the population are arranged around a circle. Aunit is selected at random from the entire set of N units followed by selecting everykth unit thereafter around the circle to get a sample of size n. Here, we have N

possible samples of size n.

Received February 14, 2011; Accepted September 27, 2011Address correspondence to Javid Shabbir, Department of Statistics, Quaid-i-Azam

University, Islamabad 45320, Pakistan; E-mail: javidshabbir@gmail.com

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3360 Khan et al.

Uthayakumaran (1998) suggested balanced circular systematic sampling whenN �= nk. But this method has the restriction of assuming that both the populationand sample sizes should be even numbers. Subramani (2000) introduced diagonalsystematic sampling design as an alternative to the linear systematic samplingunder the conditions N = nk and n ≤ k. Sampath and Varalakshmi (2008) modifiedthe diagonal systematic sampling and proposed diagonal circular systematicsampling to deal with the cases where N �= nk. Subramani (2010) has suggestedthe generalization of diagonal systematic sampling for finite population wherethe condition of n ≤ k is relaxed.

Under CSS scheme the possibility of coincidence of units in a sample is a majorproblem. Some authors have suggested modifications of this scheme to overcomethis problem. As quoted by Sengupta and Chattopadhyay (1987), “Bellhouse (1984)observes that for k = �N/n�+ 1 the first and the last units of the sample will coincideiff N = �n− 1�k, therefore to avoid this coincidence an alternative choice of k is�N/n+ 1/2� if N �= �n− 1�k and is �N/n� if N = �n− 1�k”. Here, ��� denotes theintegral part of the number.

The above remedy, however, seems to be inadequate because coincidence ofmore units is also possible when k = �N/n�+ 1. For example, if N = 30, n = 12, andk = 3, then the first and the second units will coincide with the �n− 1�th and nthunits, respectively.

Sengupta and Chattopadhyay (1987) proposed the following theorem to detectthose situations where one can get a sample with or without coincidence.

Theorem 1.1. A necessary and sufficient condition for a CSS of size n, drawn froma population of N units with sampling interval k, to contain all distinct units is that�N� k�/k ≥ n or equivalently, N/�N� k� ≥ n, where �N� k� and �N� k� respectively denotethe least common multiple and the greatest common divisor of N and k.

Applying the condition given in Theorem 1.1 to a situation like N = 30, n = 12,and therefore k = 3, where k = �N/n�+ 1, the least common multiple of N and k

is �N� k� = 30. Hence, �N� k�/k = 10. As the condition �N� k�/k ≥ n does not hold,obviously the coincidence is necessary in CSS. The question is: how many units willbe coincide? The answer is: n− �N� k�/k = 2 units will be coincide. To explain this,we take a sample using CSS as given below.

Suppose 30 units are arranged around a circle and labeled 1–30. Select the firstunit at random from these 30 units. Suppose the first unit selected is Unit 1. Nowtake every kth unit around the circle to get a sample of size n = 12, so we get thefollowing labeled units: 1, 4, 7, 10, 13, 16, 19, 22, 24, 27, 1, 4.

One can see that 1st selected unit coincides with the 11th unit and the2nd selected unit coincides with the 12th unit, giving us two coincidencesas expected.

This type of coincidence is very common in CSS. To reduce such frequentcases of coincidence, Sengupta and Chattopadhyay (1987) suggested usingk = �N/n+ 1/2� when N �= jk for all integers j�≤ �n− 1��. Otherwise, k should bechosen equal to �N/n�.

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A New Sampling Design for Systematic Sampling 3361

In practice, the choice of k = �N/n+ 1/2� when N �= jk for all integersj�≤ �n− 1�� is the best choice, but there are many situations where N = jk, forj�≤ �n− 1��. So k = �N/n� will be end up being used according to the Sengupta andChattopadhyay (1987). However, the drawback of this choice �k = �N/n�� is that thesample is not evenly distributed over the whole range of population units. It maybe possible that a considerable portion of Nunits is totally ignored in the sampleselection. In the example above, the coincidence of units is necessary because N =�n− 2�k if k = �N/n+ 1/2� = 3, so we use k = �N/n� = 2 to avoid this coincidence.Therefore the resulting sample will be 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23.

One can see that the last seven units of the population are totally ignored.The existence of such a drawback in Circular Systematic Sampling is our mainmotivation to introduce the following sampling design which we call “ModifiedSystematic Sampling (MSS).” In MSS, there is no restriction on the size of N andn and the problem of coincidence of units in a sample is very rare as comparedto CSS.

2. Modified Systematic Sampling (MSS)

Consider a finite population U = �U1� U2� � � � � UN � consisting of N units. Let�y1� y2� � � � � yN � be the respective values of the study variable y. Here, the interest isin estimating the population mean Y = ∑N

i=1 yi/N .In order to obtain a Modified Systematic Sample of size n�1 ≤ n ≤ N� from a

population of N units, one can proceed as follows.

Step 1. Find least common multiple of N and n, i.e., L, then compute m, k∗,s and k, where m = L

N, k∗ = L

n, s = N

k∗ and k = �k∗/m� or k = �N/n� is a round offinteger.

Consequently we have ms = n, which means that we select m sets of s units eachto get a sample of size n.

Step 2. Assume that the population units are arranged around a circle.

Step 3. Select one unit at random from the first k∗ units, say the rth unit �1 ≤r ≤ k∗�.

Step 4. Keeping �r + jk�th unit with j = 0� 1� 2� 3� � � � � �m− 1�, as first unitof each set of s units, determine the remaining �s − 1� units by picking everyk∗th unit thereafter around the circle. So the study variable values for the�j + 1�th set of “s” selected units of the sample of size n (i.e., n = ms) areyr+jk� yr+jk+k∗� yr+jk+2k∗� � � � � yr+jk+�s−1�k∗ .

The resulting modified systematic sample of size n�=ms� will be:

yr yr+k∗ yr+2k∗ · · · yr+�s−1�k∗

yr+k yr+k+k∗ yr+k+2k∗ · · · yr+k+�s−1�k∗

yr+2k yr+2k+k∗ yr+2k+2k∗ · · · yr+2k+�s−1�k∗

· · · · · · ·yr+�m−1�k yr+�m−1�k+k∗ yr+�m−1�k+2k∗ · · · yr+�m−1�k+�s−1�k∗

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3362 Khan et al.

Similarly, all possible k∗ modified systematic samples of size n are as under:

y1 y2 · yk∗y1+k∗ y2+k∗ · yk∗+k∗· · · ·y1+�s−1�k∗ y2+�s−1�k∗ · yk∗+�s−1�k∗

y1+k y2+k · yk∗+k

y1+k+k∗ y2+k+k∗ · yk∗+k+k∗· · · ·y1+k+�s−1�k∗ y2+k+�s−1�k∗ · yk∗+k+�s−1�k∗

· · · ·· · · ·y1+�m−1�k y2+�m−1�k · yk∗+�m−1�k

y1+�m−1�k+k∗ y2+�m−1�k+k∗ · yk∗+�m−1�k+k∗

· · · ·y1+�m−1�k+�s−1�k∗ y2+�m−1�k+�s−1�k∗ · yk∗+�m−1�k+�s−1�k∗

A theorem on Modified Systematic Sampling is proposed as Theorem 2.1.

Theorem 2.1. A necessary and sufficient condition for a MSS of size ms, drawn from apopulation of N units, to contain all distinct units is that �k∗� k�/k ≥ m, where k∗ = L

n,

L is the least common multiple of N and n, and �k∗� k� is the least common multiple ofk∗ and k.

Proof. Sufficiency Part. If r is the random starting point and if any numberof sets of “s” units in the sample are to be repeated, then for some integers iand j�0 ≤ i < j ≤ �m− 1��, r + ik = �r + jk��mod k∗� => �j − i�k is a multiple ofk∗ => �k∗� k�/k ≤ �j − i� ≤ �m− 1�. Thus, if �k∗� k�/k ≥ m, the sample contains alldistinct units.

Necessary Part. Suppose all units in the sample are distinct but �k∗� k�/k = jfor some integer j�j ≤ m− 1�. Then �k∗� k� = jk implying that jk is a multiple of k∗.Hence, r = �r + jk��mod k∗� for any integer r so that the first sets of s units and thelast sets of s units of the sample coincide. This completes the proof.

Corollary 2.1. A MSS of size ms from a population of N units will have somecoincidence of units iff k∗ = jk for some integer j �j ≤ m− 1�.

Proof. According to the above theorem we have, for some integer j�j ≤ m− 1�,�k∗� k� = jk and hence jk is a multiple of k∗ since jk ≤ �m− 1�k < 2k∗. This impliesthat jk = k∗.

Thus, no matter whatever the value of k is ��N/n�+ 1/2� or �N/n�+ 1modified systematic samples can be obtained without coincidence iff jk �= k∗, where�j ≤ m− 1�. If jk = k∗ provided that �j ≤ m− 1�, the only situation to get a samplewithout coincidence is to take k = �N/n�.

Example 2.1. In the example mentioned in Sec. 1, a sample with all distinct unitscannot be obtained according to the Sengupta and Chattopadhyay (1987) theorem.However, this is possible according to the proposed theorem as shown below.

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A New Sampling Design for Systematic Sampling 3363

Let L = 60, k∗ = 5�m = 2, k = 3, and �k∗� k� = 15. Consequently, we can getk∗ = 5.

Modified Systematic Samples without coincidence as given below:

1 2 3 4 56 7 8 9 1011 12 13 14 1516 17 18 19 2021 22 23 24 2526 27 28 29 30

4 5 6 7 89 10 11 12 1314 15 16 17 1819 20 21 22 2324 25 26 27 2829 30 1 2 3

3. Comparative Study

Bellhouse (1984) suggested using k = �N/n+ 1/2� instead of k = �N/n�+ 1 to avoidthe coincidence of units in a sample. We present below an empirical study using bothchoices of k. Table 1 shows the comparison of MSS and CSS for various choices ofN and n using k = �N/n�+ 1. Table 2 shows the comparison of MSS and CSS forvarious choices of N and n using k = �N/n+ 1/2�.

Results given in Tables 1 and 2 indicate the following.

(i) According to Table 1, out of 44 different choices of N and n, 30 cases in CSSand only 9 cases in MSS will have coincidences when k = �N/n�+ 1.

(ii) Similarly in Table 2, out of 44 different choices of N and n, 16 cases in CSS andonly 3 cases in MSS will have coincidences when k = �N/n+ 1/2�.

4. Mean and Variance Formulae

An unbiased estimator of the population mean and the corresponding variance aregiven by

ymss =∑m−1

j=0

∑s−1t=0 yr+jk+tk∗

ms(1)

E�ymss� = Y

V�ymss� =S2mss

n

(L− 1L

)�1+ �n− 1��w � (2)

where

S2mss =

1L− 1

k∗∑r=1

m−1∑j=0

s−1∑t=0

�yr+jk+tk∗�2 −

(∑k∗r=1

∑m−1j=0

∑s−1t=0 �yr+jk+tk∗�

)2

L

�

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Table

1Com

parisonof

MSS

andCSS

whenk=

�N/n

�+

1

CSS

MSS

Nn

k�N

�k�

�N�k�/k

�N�k�/k≥

nPossiblesamples

L=

�N�n�

m=

L/N

k∗=

L/n

�k∗ �k�

�k∗ �k�/k

�k∗ �k�/k≥

mPossiblesamples

144

428

7yes

1428

27

287

yes

75

342

14yes

1470

514

4214

yes

146

342

14yes

1442

37

217

yes

78

214

7no

-56

47

147

yes

710

214

7no

-70

57

147

yes

712

214

7no

-84

67

147

yes

715

63

155

no-

302

515

5yes

57

315

5no

-10

57

1515

5no

-8

230

15yes

1512

08

1530

15yes

159

230

15yes

1545

35

105

yes

510

230

15yes

1530

23

63

yes

312

230

15yes

1560

45

105

yes

518

45

905

yes

1836

29

459

yes

98

318

6no

-72

49

93

no-

102

189

no-

905

918

9yes

912

218

9no

-36

23

63

yes

314

218

9no

-12

67

918

9yes

920

64

205

no-

603

1020

5yes

108

360

20yes

2040

25

155

yes

512

220

10no

-60

35

155

yes

514

220

10no

-14

07

1010

5no

-15

220

10no

-60

34

42

no-

162

2010

no-

804

510

5yes

5

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249

324

8no

-72

38

248

yes

814

224

12no

-16

87

1212

6no

-15

224

12no

-12

05

88

4no

-16

224

12no

-48

23

63

yes

618

224

12no

-72

34

42

no-

286

514

028

yes

2884

314

7014

yes

148

428

7no

-56

27

287

yes

710

384

28yes

2814

05

1442

14yes

1412

384

28yes

2884

37

217

yes

716

228

14no

-11

24

714

7yes

1418

228

14no

-25

29

1414

7no

-20

228

14no

-14

05

714

7yes

730

84

6015

yes

3012

04

1560

15yes

159

460

15yes

3090

310

205

yes

1012

330

10no

-60

25

155

yes

514

330

10no

-21

07

1515

5no

-16

230

15no

-24

08

1530

15yes

1518

230

10no

-90

35

105

yes

520

230

10no

-60

23

63

yes

322

230

10no

-33

011

1530

15yes

1524

230

10no

-12

04

510

5yes

10

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Table

2Com

parisonof

MSS

andCSS

whenk=

�N/n

+1/2�

CSS

MSS

Nn

k�N

�k�

�N�k�/k

�N�k�/k≥

nPossiblesamples

L=

�N�n�

m=

L/N

k∗=

L/n

�k∗ �k�

�k∗ �k�/k

�k∗ �k�/k≥

mPossiblesamples

144

428

7yes

1428

27

287

yes

75

342

14yes

1470

514

4214

yes

146

214

7yes

1442

37

147

yes

78

214

7no

-56

47

147

yes

710

114

14yes

1470

57

77

yes

712

114

14yes

1484

67

77

yes

715

63

155

No

-30

25

155

yes

57

230

15yes

1510

57

1530

15yes

158

230

15yes

1512

08

1530

15yes

159

230

15yes

1545

35

105

yes

510

230

15yes

1530

23

63

yes

312

115

15yes

1560

45

105

yes

518

45

905

yes

1836

29

459

yes

98

218

9yes

1872

49

189

yes

910

218

9no

-90

59

189

yes

912

218

9no

-36

23

63

yes

314

118

18yes

1812

67

918

9yes

920

63

6020

yes

2060

310

3010

yes

108

360

20yes

2040

25

155

yes

512

220

10no

-60

35

155

yes

514

120

20yes

2014

07

1010

10yes

1015

120

20yes

2060

34

44

yes

416

120

20yes

2080

45

55

yes

5

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249

324

8No

-72

38

248

yes

814

224

12no

-16

87

1212

6no

-15

224

12no

-12

05

88

4no

-16

224

12no

-48

23

63

yes

618

124

24yes

2472

34

44

yes

428

65

140

28yes

2884

314

7014

yes

148

428

7no

-56

27

287

yes

710

384

28yes

2814

05

1442

14yes

1412

228

28yes

2884

37

217

yes

716

228

14no

-11

24

714

7yes

1418

228

14no

-25

29

1414

7no

-20

128

28yes

2814

05

77

7yes

730

84

6015

yes

3012

04

1560

15yes

159

330

10yes

3090

310

3010

yes

1012

330

10no

-60

25

155

yes

514

230

15yes

3021

07

1530

15yes

1516

230

15no

-24

08

1530

15yes

1518

230

10no

-90

35

105

yes

520

230

10no

-60

23

63

yes

322

130

30yes

3033

011

1515

15yes

1524

130

30yes

3012

04

55

5yes

5

Note:

Symbo

l“−

”indicate

that

theun

itsin

thesesamples

will

becoinciding

.

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3368 Khan et al.

As we know, �y1� y2� � � � � yN � are replicated m times. Therefore,

S2mss =

1L− 1

{m

N∑i=1

�yi�2 −

(m∑N

i=1 �yi�)2

L

}or

S2mss =

m

mN − 1

{N∑i=1

�yi�2 −

(∑Ni=1 �yi�

)2N

}�

Now (2) becomes

V�ymss� =S2

n

(N − 1N

)�1+ �n− 1��w � (3)

where S2 = 1N−1

{∑Ni=1 �yi�

2 − �∑N

i=1 �yi��2

N

}and �w = E��yi−Y ��yj−Y �

E�yi−Y �2, �i �= j� is the intra-

class correlation coefficient. An alternative form is �w = 1− (n

n−1

) 2w2, where 2

w isthe variance within the systematic sample and 2 is the usual population variance.

5. Efficiency Comparison

We use the following data sets for efficiency comparison. The results are given inTables 3 and 4.

Data Set 1. [Source: Government of Pakistan (2007–08 & 2008–2009)].Let y be the production of wheat in different districts of Punjab (Pakistan) in

year 2008–2009.

Data Set 2. [Source: (Singh and Mangat, 1996, P. 150)].Let y be the timber volume (in cubic meters) of Dalbergia sisoo trees.

Table 3Efficiency of modified systematic sampling with respect to circular systematic

sampling for Data Set 1

Circular Modified

N Y 2 n 2w V�C� 2

w V�M� Eff

15 392.450 80576.93 9 78276.20 2300.73 79193.81 1383.117 166.3410 78081.37 2495.55 78967.13 1609.801 155.02

20 420.395 67863.01 6 64187.40 3675.61 64212.75 3650.262 100.698 65074.85 2788.16 65879.95 1983.061 140.60

30 479.223 63738.50 8 61518.14 2220.36 61848.22 1890.281 117.469 60085.04 3653.46 62280.57 1457.93 250.59

36 511.667 68512.08 8 66375.44 2136.64 66664.33 1847.755 115.6310 65851.64 2660.44 67192.29 1319.789 201.5816 67517.65 994.44 68021.03 491.0496 202.51

Here, V�C� = Variance of circular systematic sampling (CSS) and V�M� =Variance ofmodified systematic sampling (MSS).

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A New Sampling Design for Systematic Sampling 3369

Table 4Efficiency of modified systematic sampling with respect to circular systematic

sampling for Data Set 2

Circular Modified

N Y 2 n 2w V�C� 2

w V�M� Eff

15 1.807 0.1172 9 0.114091 0.003105 0.115411 0.001785 173.9510 0.113529 0.003666 0.115502 0.001694 216.47

20 1.870 0.1753 6 62085.77 0.006069 0.170917 0.004121 147.278 63106.41 0.005575 0.172127 0.002971 187.61

30 1.889 0.1806 8 59712.20 0.003586 0.177682 0.002951 121.529 60472.09 0.010121 0.178203 0.002430 416.57

36 1.826 0.1872 8 64835.73 0.005210 0.183718 0.003522 147.9210 63142.96 0.004466 0.184503 0.002737 163.1616 64322.16 0.002935 0.186384 0.000856 342.75

We use the following expression for efficiency comparison, i.e., Efficiency =V�C�

V�M�× 100.In Tables 3 and 4, we observed that the efficiency of MSS is more as compared

to CSS in both data sets.

6. Conclusion and Discussion

One can see that the Modified Systematic Sampling (MSS) is superior to the LinearSystematic Sampling (LSS) and Circular Systematic Sampling (CSS) because of thefollowing reasons.

(i) In LSS there are k possible samples of size n, and in CSS the number ofpossible samples increases to N samples each with n units. However, in MSSthe number of samples is k∗ (where k ≤ k∗ ≤ N�. Consequently, LSS and CSSbecome special cases of the MSS as given below.

(a) If L = N then k∗ = k, m = n and s = 1, so it becomes LSS.(b) If L = N × n then k∗ = N , m = 1 and s = n, so it becomes CSS.

(ii) The MSS design is a more flexible design than the CSS design for differentchoices of N and n, because when k = �N/n�+ 1 or k = �N/n+ 1/2� often asample cannot be obtained without coincidence of the units in CSS, but thisproblem occur very rarely in MSS. In Tables 1 and 2, the symbol (-) indicatesthat the units in the samples coincide. The reason of coincidence in these casesis that in CSS, we have N = jk, for some integer j�≤ �n− 1�� in CSS whereasin MSS we have k∗ = jk for some integer j�j ≤ m− 1�. One can see that thenumber of samples with coincidence is less frequent in Table 2 as compared toTable 1. Consequently, k = �N/n+ 1/2� is a better choice than k = �N/n�+ 1.If coincidence still occurs then we can take k = �N/n� to get a sample withoutcoincidence.

(iii) The efficiency comparison in Tables 3 and 4 show that MSS is more efficientthan CSS.

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3370 Khan et al.

(iv) In most of the cases MSS provide a sample without coincidence even whencoincidence is necessary according to Sengupta and Chattopadhyay (1987).

(v) In MSS each unit has equal probability of selection. So the modified systematicsample mean ymss becomes an unbiased estimator of population mean Y .

References

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Sampath, S., Varalakshmi, V. (2008). Diagonal circular systematic sampling. Model Assis.Statist. Applic. 3:345–352.

Sengupta, S., Chattopadhyay, S. (1987). A note on circular systematic sampling. Sankhya B49(2):186–187.

Singh, R., Mangat, N. S. (1996). Elements of Survey Sampling. London: Kluwer AcademicPublisher.

Subramani, J. (2000). Diagonal systematic sampling scheme for finite population. J. Ind. Soc.Agricult. Statist. 53(2):187–195.

Subramani, J. (2010). Generalization of diagonal systematic sampling scheme for finitepopulation. Model Assist. Statist. Applic. 5:117–128.

Uthayakumaran, N. (1998). Additional circular systematic sampling methods. Biometr. J.40(4):467–474.

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