A Review of the Literature of Systematic Sampling

A Review of the Literature of Systematic SamplingAuthor(s): William R. BucklandSource: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 13, No. 2(1951), pp. 208-215Published by: Wiley for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2984061 .

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208 [No. 2,

A REVIEW OF THE LITERATURE OF SYSTEMATIC SAMPLING

By WILLIAM R. BUCKLAND

[Received September, 1950]

SUMMARY

THE literature relating to systematic sampling shows the interplay between practical needs and the provision of statistical techniques for their satisfaction. Problems of systematic sampling occur more frequently than is generally realized and, since many of the techniques are still far from satisfactory, the situation offers great incentive to further development.

Initroduction

THIS review attempts to place in perspective the main papers on systematic sampling. In view of the almost contemporary nature of. much of the development in this 'branch of sampling the review has been arranged in a nearly chronological order. In this way it is possible to examine the interplay of demand for a technique with its supply for the purpose of helping to solve the practical problems at issue. From this point of view it is interesting to note that, just as the methods of Factor Analysis grew up in the atmosphere of the psychologist, so has Systematic Sampling grown up alongside problems of forestry and land use. Since the paper by Stephan (1948) is, of itself, a review paper, it will be convenient to take this first.

Two quotations from the second part of this paper are worth setting down:

1. "Much of modern sampling practice rests on the processes of selecting individuals at random or according to certain systematic procedures".

2. ". . . systematic sampling employs a simple rule of counting cases in some convenient order and selecting for the sample every nth case or of using some similar pattern at measured intervals". In some of the physical and engineering sciences forms of systematic sampling have long been in use: other applications have been made in forestry, agriculture, animal husbandry and meteorology. Butin the case of some studies in the economic and social field-for example, by Bowley, Caradog-Jones, Hilton and Kaier-the underlying feature was that the lists used for the selection of the sample were already in an order which was largely random, at least within certain divisions of the whole. On the other hand, the material in most other fields is usually found to be arranged in an order which is certainly not random. Therefore we have the two kinds of systematic sampling which have both emerged as a result of a simnilar process of sample selection but whose interpretation is very different. The method as used by Bowley and others in the economic field is frequently referred to as quasi-random or even stratified random sampling. This general point of terminology is one to which we can briefly return at the end of this review.

Hasel (1938) gave a general review of the pre-war literature of sampling for forestry purposes, and stated the usual conditions whereby the sample can provide the information necessary for estimating the sampling error. But since growing timber was usually inspected by "cruising" and sampling done at the same time, the sample was usually obtained by some form of systematic selection which violated the conditions for the valid estimation of sampling error. As a way round the difficulty the author suggested taking an auxiliary samnile selected at random; but he qualified this solution as 'being an expensive practice. In spite of its difficulties he suggested retaining the systematic sample owing to its great advantages in administration and mapping, but pointed out that it was of considerable practical importance to evolve methods for dealing with the errors of systematic samples.

The following two references are investigations in the related field of land-use. The paper by Proudfoot (1942) applies the technique of traverse surveying to the problem of estimating the



1951] BUCKLAND-Review of the Literature of Systematic Sampling 209

adequacy of sample data on the distribution of types of land-use. He established that the greater the total traverse intercepts, the lower the root-mean-square error as a result of narrower spacing of the traverse lines. This is formally equivalent to increasing the size of a sample from a finite population by raising the sampling fraction. Various indices were calculated, based on the correlations betxween the estimates at different spacings of the traverse lines as checked by field survey data: they provide "a beginning of a traverse precision chart for forecasting the accuracy of traverse sampling".

Osborne's paper (1942) appears to be the first attempt to face up to the problems of systematic sampling in the more rigorous interpretation of that term. Against a background of systematic line sampling for the estimation of the area covered by different kinds of forest, this paper reported on the accuracy of sample estimates ffom both random and systematically selected samples and gave a procedure for estimating the sampling error of the latter. The author concluded that if data taken systematically are evaluated with random sample formulae, then biased estimates of the sampling errors must result. It was also no solution to subdivide the area into blocks, since this did not produce a set of homogeneous strata. The variate was, in fact, undergoing a change which could be considered as a continuous function of position. The problem of sampling then became associated with that of curve fitting, and regard must be paid to the possibility that internal correlations might exist between the observed values.

Thus, at this early stage, it was brought out clearly that the serial correlation occupies a central position in this problem which is closely bound up with the general analysis of series ordered in space or time. It is this aspect of the problem which distinguishes the modern interpretation of systematic sampling from the older and less rigorous interpretation.

Development of the Theory In 1944 there was published the first paper (Madow and Madow, 1944) dealing with the theory

of systematic sampling: since then the names of the authors, William and Lilian-Madow, have figured throughout the subsequent stages of development of the theory. It is interesting to note that the only reference to previous literature given in.this paper was to the work by Hansen and Hurwitz (1943) on sampling from a finite population.

The paper coVered systematic sampling from stratified and unstratified populations of single elements: a later paper (Madow, 1949) deals with the extension to clusters of equal and unequal size. In this first paper it was considered that. "The major omission of sampling theory was possibly the absence of any statistical method for reaching a decision on whether or not to take a completely random sample or one selected systematically". A systematic sampling procedure was defined as a random sampling procedure with many of the possible selections specifically excluded. Looked at in another way, the systematic design is a form of cluster sampling, with the difference that a knowledge of the order of the elements in the population is used to obtain values of the intra-class correlation: hence the reference to the work of Hansen and Hurwitz. Thus from a knowledge of the population variance and certain serial correlations, the sampling error of the systematic sample can be assessed. Three basic results were stated:

1. If the serial correlations have a positive sum then systematic sampling is worse than random sampling, in the sense of having a higher sampling variance.

2. If the serial correlations have a negative sum then systematic sampling is better than random sampling.

3. If the serial correlations have a sum which is approximately zero then there is no great difference between the two sampling procedures.

Since the number of systematic samples from a finite population is small it is difficult to assume a normal distribution of sample means. We are led to regard the elements of the population as single observations on random variables: it should be noted that it was not until Cochran's paper (1946) that this concept appeared to be fully incorporated into the development.

For the case of unstratified systematic sampling of single units it was shown that the mean of the systematic sample (xc) is an unbiased estimate of the population mean with a variance of

Va 2 (2 Var(x n { ?(n - 1) Tkl n{ +2 Pk.m},

m



210 BUCKLAN D-Review of the Literature of Systematic Sampling [No. 2

where k is the number of classes with n in each class and km is the lag in the serial correlation, a2 is the population variance and Pk iS the intraclass correlation coefficient. The authors make the comment that the biased estimates of population variance and serial correlations which are obtained from the single sample can only be corrected by using more than one sample: a course, it will be remembered, was also suggested by Hasel (1938), and deemed to be an expensive one. Indicating that the estimate rkm (of P km) is likely to produce only negligible bias, some alternative methods are given for overcoming the estimation of a2, which is rather more complex.

In the case of stratified systematic sampling two basic forms were considered: (i) Where the sampling ratio (or interval) is constant for all strata. (ii) Where a variable sampling ratio (or interval)is used.

The sample mean is the. weighted average of the strata means and the following practical points emerge from a consideration of the variance:

1. That, if corresponding items in the different strata are positively correlated, it is not advisable to use a constant sampling interval unless considerations other than sampling efficiency are dominant (see also Madow, W. G., 1949).

2. That, if the corresponding items are negatively correlated, then sampling with a constant interval will yield a smaller sampling variance.

3. Where the variable sampling ratio is used, then the independent selection as between strata eliminates the covariance between pairs of strata, and the expression for the variance of the mean reduces to the known case of stratified sampling.

An expression is given for the ratio of the systematic sampling variance to the variance of random sampling, and it is shown that the former variance is smaller than the latter if

>]Pkm < -(n - 1)/2(kn -- 1) -2k if n is large relative to k.

The paper then proceeds to consider the effect of periodicity in the data. If k is the period of the data, then the correlation between strata means of a systematic sample is ? 1 and the random sample is better- However, if the period is 2k then the systematic sample will probably have the smaller variance since the correlation between strata will be - 1 for adjacent strata, or those which differ in position by an odd number, and + 1 for those whose position differs by an even number. An interesting result is given that when the form of the population is linear, then the systematic sample is much more effective in removing the effect of an unknown trend than is the random sample, but less effective than the stratified random sample.

A simplified version of this paper was made by one of the authors (Madow, L. H., 1946). The approach is more definitely through cluster sampling (see Hansen and Hurwitz, 1943), the simplest case of which is where only one cluster is sampled with no sub-sampling within the clusters: the modifications necessary to embrace the systematic design are clearly set forth. The second example given in this paper shows that for systematic samples the variance for different sample sizes is not consistent in the sense that it does not decrease monotonically with increasing sample size. It follows, therefore, that the efficiency of the systematic sample design will be erratic.

Before discussing the next development in the theory, it is worth while looking at a group of three papers, which deal with important practical problems.

The first paper (Wadley, 1945) deals with the estimation of insect populations-questions of enumeration or organisms per unit area. If spatial differences are known to exist in the distribution, then a systematic sample is more representative than a random sample. The practical need for the advancement of systematic sampling theory and the interplay between practical needs and theoretical development is illustrated by the following quotation:

"We cannot correctly estimate the sampling variation of the systematic sample by standard methods. Any light which can be shed on the variance of such samples will be helpful. Madow and Madow

have discussed this problem, but their results are not yet in a form suited for the use of field workers". The next (Deming and Simmons, 1946) is an interesting case of sampling from a stratified

population with systematic selection within strata. It is evident that non-statistical considerations




were responsible for the fact that full use was not made of such developments in the theory as had been made at that time (1946). The third paper in this group (Bayley and Hammersley, 1946) arose from the authors' contribution to the discussion at the Symposium on Autocorrelation in Time. Series, held by the Research Section, Royal Statistical Society, in January, 1946. In order to provide a link between some of these papers now under review, it can be noted that expression No. 11 of Bayley and Hammersley's paper is formally the same as expression No. 4 in Madow and Madow (1944), where it is also reduced to an expression of very long standing-see formula 17.11 in Yule and Kendall, Introduction to the Theory of Statistics, 14th edition, page 404, and also to Yule's (1945) paper on Time Series in J.R. Statist. Soc., 108, 208. The problem attacked by Bayley and Hammersley was a case of two-stage systematic sampling in connection with instru- ments providing a continuous record of tests of anti-aircraft equipment.

The Correlogram in Systematic Sampling

The paper by Cochran (1946) was a most interesting development in connection with the form of the correlogram and its effect upon systematic sampling-it will be recalled that the variance of a systematic sample depends partly upon the sum of the serial correlations. It is also important, since it develops the idea o-f working with expected variances resulting from regarding the population elements as, random variables.

The author considers a population model where the variance within a group of contiguous elements increases with the size of the group-that is they are serially correlated. If this serial correlation is positive, then the general form of the correlogram is a monotonic decreasing function (concave upwards). On the average, the stratified random sample is always at least as efficient as the unrestricted random sample and its relative efficiency is a monotonic increasing function of the sample size, no such general result being available for the systematic sample. This con- firmation of parts of Madow (1946) is extended in that if the correlogram is concave upwards then the systematic sample is, on the average, more accurate than the stratified samplefor any size of sample. It may be noted in passing that such a restriction on the form of the correlogram is not as severe as it may appear. Wold (1938) used the linear form in the study of economic data, Osborne (1942) used a function of the same type in forestry and land-use surveys, and Fisher (1922) used a somewhat similar form in his study of the correlation between weekly rainfall at two weather stations.

Cochran suggested that the deciding factor in determining the relative accuracies of the systematic and random sample designs were the second differences of the serial coefficients rather than the first differences. This represented a significant advance in the interpretation of some inequalities given in Madow and Madow (1944), but it still could not be inferred that the relative efficiency of the systematic sample to the stratified random sample was a monotonic function of the sample size. Following his principle of working with expected variances, Cochran, gave expressions for the variances of the mean of random, stratified random and sysiematic samples, and showed how the relative efficiencies are bound up with the linear functions of the serial correlations which appear in these formulae. The paper closes with some results for the different types of sample selection in relation to populations with (i) exponential and (ii) linear correlograms.

It will again be convenient to note a group of papers which report attempts to use systematic sampling methods before proceeding with the next main phase in the development of the theory.

The first (Nordskog and Crump, 1948) is a paper from the field of animal husbandry. In connection with the economics of poultry farming, the authors compared the results obtained with egg-sampling schemes based upon

(1) random days, (2) consecutive days, (3) interval days,

and conclude that the differences in accuracy between the three methods were small. A systematic sample (interval days) was slightly more accurate when the egg production of groups of poultry had to be checked on different days-a situation which frequently occurs on large poultry farms. The second paper in this group (Finney, 1948) is of importance, since it serves to underline many points which are frequently overlpoked in the general press and rush of events. In the assessment



212 BUCKLAND-Review of the Literature of Systematic Sampling [No. 2,

of standing timber, the time and labour for a complete enumeration makes the consideration of sampling techniques of great practical value. This paper is rather like Hasel (1938) above in that it is also a general review of sampling techniques as applied to forestry problems and serves as an introduction to a later paper (Finney, 1949). The third of these papers (Shaul and Myburgh, 1948) deals with proposals for holding a sample census of African populations. A formula is quoted for the calculation of sample size to allow for a given sampling error, and the comment is made that it assumes unrestricted random sampling: a rider is added that a stratified sample would give greater precision. In practice, however, it was proposed to increase the precision by taking a systematic sample from lists and to treat the results by the theory of random sampling. This appears to be a recent case of the approach used by Bowley and others, and is clearly not systematic sampling of the more rigorous kihd we have been considering in this review. In this connection it should be noted that the paper by Gray and Corlett (1950) outlines a method of systematic sampling from ordered lists which offers some improvement on these methods.

Contributions by the English School It is now possible to turn to a review of the main contributions by statisticians in this country

-most of the earlier references having come from North America. It may be that war pre- occupations prevented an earlier participation, but one result of this is that the group of papers now to be considered contain some of the most practically useful developments, especially in the field of two-dimensional systematic sampling. In considering papers written at a point in time as near as September, 1948, it becomes difficult to allocate an order of priority, since much de- pended upon the accident of publication as between the various journals. A way round the difficulty might be to consider groups of related papers, and it is proposed first to deal with those relating to the work of Yates (1946, 1948 and 1949).

The principal paper is that entitled "Systematic Sampling", appearing in Phil. Trans. Roy. Soc. (A) (1948). Some of its results were briefly reviewed in paragraphs of the earlier, and more general, paper (Yates, 1946), and it is summarized and placed into perspective in the book Sampling Methods for Census and Surveys (1949), which has quickly become a standard text in this field. Yates' paper in 1948 was an important landmark in the development of systematic sampling for many reasons, not least of which is the immediate sense it gives the reader of the practical aspects and utility of systematic samples for dealing with sequences of quantitative values. The treatment in this paper, however, was not restricted to quantitative data, since one of its sections deals with the systematic sampling of attributes-presented as a sample of a two-valued function. The author cites many fields of practical application, including that of meteorology, where, in passing, we may note that recent papers give the impression that the existence of problems of systematic sampling is recognized, but beyond that very little is done. A case in point is the paper under Carruthers (1949), where the author is clearly advocating a systematic sample of temperature readings, but takes no action to avoid the difficulties which arise from taking a single observation at a fixed hour each day.

Yates deals with one-dimensional systematic sampling, and evolves new methods for estimating the sampling error. Simpie end-corrections are proposed for eliminating the trend errors which are inherent in randomly located systematic samples. As foreshadowed in his earlier paper (Yates, 1946), it is shown that it is impossible to make a fully reliable estimate of the sampling error from the systematic sampling results alone, and methods are described for using additional samples at half and quarter spacing to provide supplementary evidence. The performance of systematic sampling is investigated for the following population models:

(1) A two-valued function, i.e. attributes. (2) Material normally distributed. (3) A one-term autoregressive function.

Following Cochran (1946), Yates also uses the concept of expected variances. He also adds his quota of emphasis on the danger of using systematic samples for material about which nothing is known or which may be periodic.

In sampling a two-valued function along a line divided into sections, if the proportion of the sections possessing the attribute, is small, or nearly unity, and if the majority of the sections are




short relative to the sampling interval, there is little to choose between systematic sampling and forms of random sampling. But if the majority of sections are greater than the sampling interval, then systematic sampling may be expected to be substantially more accurate than any form of random sampling. For the intermediate case more work has still to be done, but it was expected that the same general results would hold. The case of sampling material normally distributed is shown to have a two-dimensional analogue in sampling from vertical air photographs-a field which has yet to be fully developed. The third case investigated is that of sampling the single- term autoregressive function (a simple Markoff chain):

Yr+1 = byr + ar+.l

It is pointed out that the properties of this kind of function are essentially different from the continuous variate type: its correlogram is exponential, which at once gives a link with references quoted in connection with Cochran's paper (1946). If the material is known to be truly autoregressive, then no difficulty arises on estimating the variance of systematic samples. This is frequently not the case, however, and if the departure from true autoregression is on account of a long-term variation, then the effect will not be serious. But if the departure is on account of superimposed random variation, then a very serious underestimation will occur. To overcome this difficulty it is suggested that recourse be made to the device of obtaining supplementary information through partial systematic samples. Some results are given of sampling studies which make interesting reading, and afford practical illustration of this branch of sampling theory.

The second main contribution of the English school was the paper by the late A. E. Jones (1948), which was continued by M. G. Kendall (1948) owing to Dr. Jones' most unfortunate fatal accident in May, 1948. The problem considered in this paper is that of estimating by a systematic sample the average value of a random variable from a one-dimensional homogeneous population depending on a continuous parameter. The core of this problem is the best method of distributing the sample members according to simple criteria relating to the degree of correlation between successive values-an autoregressive scheme similar to that investigated by Yates (1948). If this correlation is very small then Jones suggested distributing the n sample points at distances

T/(n + 1), 2T/(n + 1) *

where T is the length of the "strip" being sampled. On the other hand, if the correlation is greater than 0-25 the best distribution will be at distances

TI2n, 3T/2n, -5T/2n.

In either case the optimum distribution is obtained when the points are equidistant, i.e. systematically arranged.

In his continuation Kendall (1948) points out an essential distinction between more intensive sampling and an extension of the sampled "strip", and comments that in the Jatter case it is an unnecessary refinement to worry unduly about the lengths of the end intervals. This remark is echoed by Yates (1949) on page 175 of his book. While many of the earlier references under review had commented upon the effect of intensive systematic sampling, which is formally equivalent to increasing the sampling fraction, none had brought out its counterpart in the extension of the sampled "strip". In the more general problem where the form of the correlogram is, not exponential, Kendall states that any conclusion based upon the intuitive approach of equal weights and distances for, and between, the observations would be wrong. However, the general equations necessary to allow for this do not appear to offer prospect of an immediate solution. In his turn, Kendall concludes by raising his standard against the use of systematic sampling where there is any marked periodicity in the material.

The paper by Finney (i949) was the sequel to the more general paper in the previous year. It is a searching investigation into the supposed greater precision of systematic sampling in forest surveys, based upon some extensive data drawn from various sources. It may be noted that this paper also affords evidence of the manner in which the interplay of statistical techniques is brought to bear upon a problem.



214 BUCKLAND-Review of the Literature of Systematic Sampling [No. 2,

The Latest Developments The second instalment of the work in this field by W. G. Madow appeared in late 1949. In

this paper he extends results of earlier papers to the systematic sampling of clusters of equal and unequal sizes: some comments are also included on systematic sampling in two dimensions, but the major work on this topic is the paper by Quenouille (1949). For the systematic sampling of clusters of equal size there are two proposed methods:

(i) enumeration of all elements in the sampled cluster; (ii) stratification and sub-sampling in the sampled cluster;

and it appears that the particular design would be more suited to sampling physical populations rather than human populations. The most important conclusion is that systematic selection of clusters, even when this is desirable for non-statistical reasons, may not compensate for the increase in variance caused by the use of clusters.

On the question of systematic sampling in two dimensions two problems are cited: (1) The "serpentine" numbering of blocks in a city can be inefficient if there is correla-

tion between blocks: a systematic sample could be columns of blocks in the city. The correlogram could not be concave upwards.

(2) A systematic sample of rows and columns on a grid might be inefficient if there is a fertility gradient along the rows or columns.

But, as noted above, it is the next, and concluding, paper in this review which sets the stage for dealing with the problem of systematic sampling in two dimensions.

In his paper entitled "Problems in Plane Sampling", Quenouille (1949) brings the subject of systematic sampling a great deal nearer to practice, which has hitherto been largely intuitive. After commenting that certain expressions in Cochran's paper (1946) require the addition of a superposed variation, the author gives the integral equivalent expressions for dealing with the sampling of a continuous process. He shows how the differencQ between the sampling variances of the systematic and the stratified random designs, when expressed as a ratio of the random sampling variance, can be used graphically with the correlogram to investigate the relative efficiencies of these sampling schemes.

With the aid -of some useful diagrams, the topic of two-dimensional sampling is explained for the random, stratified random and systematic schemes as well as for samples which are aligned or independent along one, or both, of the axes-a situation which offers a great many combinations. The effect of alignment is usually to increase the variance in the case of the two random sampling schemes. As might be anticipated, for systematic samples it is the form of the correlogram which determines whether the variance is increased or decreased. Limiting expressions are given for continuous processes, and a table is provided of the relative efficiencies of four specimen-sampling schemes according to different values of the internal correlations along each axis. A general conclusion is that, over a wide variety of cases, the independent systematic sample in two dimensions gives a more accurate result than does random sampling. The methods suggested for the estimation of sampling errors of linear schemes can be adapted for plane schemes:

(i) Sets of systematic samples, randomly located with respect to each other: the error variance is calculated from variances of the systematic samples in each block.

(ii) One set of systematic samples, randomly placed: the error variance is calculated from the variances of the portions of the systematic sample.

(iii) Using one systematic sample, which is then broken into several systematic samples by using a wider spacing: the error variance is calculated from variances of each sub- sample.

These three are increasingly accurate in their estimation of a mean value, but increasingly biased in their estimation of sampling variance and also increasingly difficult to put into practice. Thus the nature of the problem will to some extent govern the choice of method.

Conclusion

During the past decade much has been done to furnish a background of theory for the intuitive attempts to use systematic sampling methods in practice. On all sides there is strong evidence




that the practical problems are sufficiently important for there to be a continuing incentive to further developments of the theory. We have seen, however, that there is a latent difficulty over nomenclature where two kinds of sampling schemes are being called by the same name when, in fact, they bear only slight resemblance to each other. Some may be inclined to go further and say that all- sampling schemes which do not embrace a random selection at some stage should be called systematic. This would undoubtedly embrace the situation of cluster sampling from continuous parameter populations (see Kendall, J.R. Statist. Soc., 108, 230, paragraph 11), but this is clearly not the kind of sampling scheme we have been reviewing in these notes. The time may be approaching when some attempt should be made to arrive at a standard classification of sampling schemes.

Select Bibliography on Systematic Sampling BAYLEY, G. V., and HAMMERSLEY, J. M. (1946), "The effective number of 'independent' observations in an

autocorrelated time series", J.R. Statist. Soc. Suppl., 8, 184. CARRUTHERS, N. (1949), "Accuracy of mean of 'n' temperature observations", Met. Mag., 78, 65.* COCHRAN, W. G. (1946), "Relative accuracy of systemnatic and stratified random samples for a certain class

of population", Ann. Math. Statist., 17, 164. DEMING, W. E., and SIMMONS, W. R. (1946), "On the design of a sample for de- ler inventories", J. Amer.

Stat. Assoc., 41, 16*. FINNEY, D. J. (1948), "Volume estimation of standing timber by survey", Forestry, 21, 179.*

- (1949), "Random and systematic sampling in timber surveys", ibid., 22, 64.* FISHER, R. A., and MACKENZIE, W. A. (1922), "The correlation of weekly rainfall", Q. J. Met. Soc., 48, 234. GRAY, P. G., and CORLETT, F. (1950), "Sampling for The Social Survey", J.R. Statist. Soc. (A),-113, 150. HANSEN, M. H., and HuRwrrz, W. N. (1943), "On the theory of sampling from finite populations", Ann.

Math. Statist., 14, 333. HASEL, A. A. (1938), "Sampling error in timber surveys", J. Agric. Res., 57, 713.* JONES, A. E. (1948), "Systematic sampling of continuous parameter populations", Biometrika, 35, 283. KENDALL, M. G. (1948), "A continuation of Dr. Jones' paper", ibid., 35, 291. MADOW, L. H. (1946), "Systematic sampling and its relation to other sampling designs", J. Amer. Stat.

Assoc., 41, 204. MADOW, W. G. (1949), "Systematic sampling: II", Ann. Math. Statist., 20, 333. - and MADOW, L. H. (1944), "On the theory of systematic sampling: I", ibid., 15, 1. NoRDSKOG, A. W., and CRUMP, S. L. (1948), "Systematic and random sampling for estimating egg pro-

duction in poultry", Biometrics, 4, 223.* OSBORNE, J. G. (1942), "Sampling errors of systematic and random surveys of cover-type areas", J. Amer.

Stat. Assoc., 37, 256.* PROUDFOOT, M. J. (1942), "Sampling with transverse traverse lines", ibid., 37, 265.* QUENOUILLE, M. H. (1949), "Problems in plane sampling", Ann. Math. Statist., 20, 355. SHAUL, J. R. H., and MYBURGH, C. A. L. (1948), "Sample survey of the African population of Southern

Rhodesia", Population Studies, 2, 339.* STEPHAN, F. F. (1948), "History of the uses of modern sampling procedures", J. Amer. Stat. Assoc., 43, 12. WADLEY, F. M. (1945), "An. application of the Poisson series to some problems of enumeration", ibid.,

40, 93.* WOLD, H. (1938), A Study in Analysis of Stationary Time Series. Uppsala: Almqvist & Wicksells. YATES, F. (1946),."A review of recent statistical developments in sampling and sampling surveys", J.R.

Statist. Soc., 109, 12. (1948), "Systematic sampling", Philos. Trans. (A), 241, 347. (1949), Sampling Methods for Census and Surveys. London: Griffin & Co..

The asterisk (*) denotes papers devoted mainly to applications.

VOL. XIII. NO. 2. Q



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A Review of the Literature of Systematic Sampling