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The present paper is based on an earlier study of the author [11], and its primary objective is to present the methodology of a Bayesian design of stratified sampling. In order to avoid abstract mathematics, and make the exposition more concrete, we focus attention on a problem of estimating a linear function of the proportion defectives in k finite lots of a certain product. In particular, consider the following sampling inspection problem, k lots (k > 2) of a certain product are subject to sampling inspection. The sizes of the lots, N1, N2, * * * , Nk say, are known. Let Mi (i = 1, -.. , k) be the number of defective items in the i-th lot. The objective is to estimate the total number of defective items o =l.Mk, in the k lots. A total sample of a fixed size, n, is specified. The problem is how to allocate the sample over the k given lots. We study this allocation problem in a Bayesian framework with a squared-error loss. Thus, whatever sampling plan is adopted, the estimator of 0 to be used after observing the sample is the Bayes estimator for the squared-error loss. The question is, however, what sampling plan to adopt. The principle one following is that of determining a sampling plan which minimizes the associated prior risk. In a recent paper [12] the author discussed the problem of the optimal Bayes selection of a sample from a finite population. It has been shown there that the optimal Bayes sampling selection is nonrandomized, without replacement, and should often be performed sequentially. However, to obtain such an optimal Bayes sampling plan, it is required to specify a prior joint distribution to all the N variates (Xi, ... , XN) of the population. This requirement is very often impractical. It is rarely the case that the sampling designer can specify such a joint prior distribution. Bayesians tend often to specify a simple joint prior distribution, which can be conveniently manipulated. Under such simplified prior assumptions one is liable to attain trivial designs. A further elaboration of this point can be found in Solomon and Zacks [10]. We feel that most practitioners, especially, in the field of quality control, where they have to inspect

This paper deals with an empirical investigation of various stratified two-stage sampling systems for estimating totals of certain agricultural items of North Carolina. The 1940 Agricultural Census data were used for stratification, selection and estimation purposes. The observed data were the results of the 1945 Agricultural Census. Theory for the selection of n primary sampling units from a stratum with probability proportional to some measure of size but without replacement has already been developed by the senior author [11]. The principal contribution in this paper is the application of this theory to the selection of two primary sampling units without replacement from a stratum, where one of the units is selected with probability proportional to size and the other with equal probability. These results are compared with sampling systems (i) where both units are selected with probability proportional to size but with replacement and (ii) where an equal number of primary sampling units are sel...

Given a population of N units, it is required to draw a sample of n distinct units in such a way that the probability for the i th unit to be in the sample is proportional to its ‘size’ x. From the alternative methods of achieving this we consider here only the so-called systematic method which, to the best of our knowledge, was first developed by W. G. Madow (1949): The units in the population are listed in a ‘particular’ order, their xi accumulated and a systematic selection of n elements from a ‘random start’ is then made on the accumulation. In a more recent paper (H. O. Hartley and J. N. K. Rao (1962)) an asymptotic estimation theory (for large N) associated with this procedure was developed for the case when the order of the listed units is random. In this paper we draw attention to certain properties of Madow's estimator: We utilize the fact that with systematic sampling the total number of different samples is N (rather than ( N n ) as with completely random sampling). This simplification...

1.1. We are concerned in this paper with the problem of estimating the mean value ,u of a variable x in a population, by taking a sample which is in some way representative of the population. It has been realized since Bowley's paper (1926), and more particularly since Neyman's more comprehensive survey (1934), that a certain degree of precision in the estimate can often be obtained more economically by stratified random sampling (usually referred to merely as stratified sampling) than by unrestricted random sampling (usually called merely random sampling). In the stratified method, the population is divided into several strata, the sample size divided in some prearranged way among the strata, and sampling performed at random from each stratum. In unrestricted random sampling, a random selection is made from the whole population, and the method may be regarded as a particular case of stratification, where the number of strata is one. Some text-books deal briefly with stratified sampling. Wilks (1943) considers only infinite populations, and denotes by representative sampling what we should call a particular type of stratified sampling (see ? 1.2). The subject is treated by Kendall (1946, pp. 249-52), but he makes no comparison with unrestricted random sampling. We shall begin by introducing several well-known results which will be needed later.

The present paper is an attempt to define sufficiency in simple terms in the theory of sampling. This definition is a suitable version of the existing notion of sufficiency as defined by Fisher, Halmos and Savage, Bahadur and others. The paper gives justification for the use of sufficient statistics in sampling theory. Applications to interpenetrating subsampling and two-stage sampling are given. In interpenetrating subsampling, it is proved that for estimating the population mean when the subsamples are drawn by simple random sampling without replacement, an estimator better than the usual overall average of subsample means is given by the average of distinct sample units. An improved estimator of the population variance is derived. In two-stage sampling where the first-stage units are drawn with unequal probabilities and second-stage units by simple random sampling (without replacement), two estimators of the population mean which are better than the estimator in current use are given.

Seven major types of sampling for observational studies of social behavior have been found in the literature. These methods differ considerably in their suitability for providing unbiased data of various kinds. Below is a summary of the major recommended uses of each technique: In this paper, I have tried to point out the major strengths and weaknesses of each sampling method. Some methods are intrinsically biased with respect to many variables, others to fewer. In choosing a sampling method the main question is whether the procedure results in a biased sample of the variables under study. A method can produce a biased sample directly, as a result of intrinsic bias with respect to a study variable, or secondarily due to some degree of dependence (correlation) between the study variable and a directly-biased variable. In order to choose a sampling technique, the observer needs to consider carefully the characteristics of behavior and social interactions that are relevant to the study population and the research questions at hand. In most studies one will not have adequate empirical knowledge of the dependencies between relevant variables. Under the circumstances, the observer should avoid intrinsic biases to whatever extent possible, in particular those that direcly affect the variables under study. Finally, it will often be possible to use more than one sampling method in a study. Such samples can be taken successively or, under favorable conditions, even concurrently. For example, we have found it possible to take Instantaneous Samples of the identities and distances of nearest neighbors of a focal individual at five or ten minute intervals during Focal-Animal (behavior) Samples on that individual. Often during Focal-Animal Sampling one can also record All Occurrences of Some Behaviors, for the whole social group, for categories of conspicuous behavior, such as predation, intergroup contact, drinking, and so on. The extent to which concurrent multiple sampling is feasible will depend very much on the behavior categories and rate of occurrence, the observational conditions, etc. Where feasible, such multiple sampling can greatly aid in the efficient use of research time.

SUMMARY A comparison is made between play-the-winner sampling and vector-at-a-time sampling for selecting the better of two binomial populations when the selection requirement is defined in terms of the ratio of the single trial probabilities of success rather than the difference. Thus the probability of correct selection is to be at least P* when the true ratio of probabilities of success 0 is at least 0*, where P* and 0* are prescribed. Termination rules based on the difference in the number of successes and inverse sampling are considered. It is shown that play-the-winner sampling is uniformly preferable to vector-at-a-time sampling for both termination rules, and that for 0* close to one the success lead termination rule leads to a smaller expected number of trials on the poorer treatment unless 0 is even closer to one. The problem of selecting the better of two binomial populations, i.e. the one with the higher probability of success p on a single trial, has usually been stated using the framework of ranking and selection procedures as follows. If P* and A* are preassigned constants, with < P* < 1 and 0 < A* < 1, the probability of a correct selection is to be at least P* when the true difference in the p-values is at least A*. Sobel & Weiss (1970) consider two sequential procedures for this selection problem in which sampling is terminated when the absolute difference in the number of successes for the two populations first reaches a pre- determined integer. We will refer to this as the success-lead rule for termination. In the first procedure, play-the-winner sampling is used in which sampling continues with the same population after each success and switches to the other population after each failure. In the second procedure, vector-at-a-time sampling is used in which two observations are taken at each stage, one from each population. In many contexts, it is desirable to have a small expected number of trials on the poorer population or treatment but using the success-lead rule it is not true uniformly in the parameter space for the p's that play-the- winner sampling is superior to vector-at-a-time sampling in this respect. In a second paper Sobel & Weiss (1971) consider the use of an inverse sampling rule when sampling is termi- nated when any one population has r successes. They show that in this case play-the-winner sampling is uniformly preferable to vector-at-a-time sampling in the sense that for the same probability requirement, the expected number of trials on the poorer treatment is always smaller. Truncated versions of the inverse sampling rule and the success-lead rule for play- the-winner sampling were proposed by Hoel (1972) and Fushimi (1973). Their stopping rules include the number of failures as well as the number of successes and mean that excessive sample sizes are avoided when the population probabilities of success are small.

The theme of this paper is the extent to which the assumptions of the general linear hypothesis are justified by purposeful random sampling from a finite population. For three elementary models it is found that the square of the sample size divided by the population size must approach zero to insure substantial agreement between the two approaches. Usually, when the asymptotic condition is not fulfilled only symmetry among the random variables remains. But even symmetry is not invariably present; it is lacking for the stratified one‐way classification

In some situations, it is desired to estimate multinomial proportions from data which have been misclassified. One such area is the sampling inspection area of quality control. In this paper, it is assumed that two measuring devices are available to classify units into one of r mutually exclusive categories. The first device is an expensive procedure which classifies units correctly; the second device is a cheaper procedure which tends to misclassify units. In order to estimate the proportions pi (i = 1,2, …, r) a double sampling scheme is presented. At the first stage, a sample of N units is taken and the fallible classifications are obtained; at the second stage a subsample of n units is drawn from the main sample and the true classifications are obtained. The maximum likelihood estimates of the pi are derived along with their asymptotic variances. Optimum values of n and N which minimize the measurement costs for a fixed precision of estimation and which minimize the precision for fixed cost are derive...

Summary For estimating the mean of a finite population on the second of two successive occasions from a simple random sample, the authors [1] have proposedelsewhere an estimate which utilizes the data obtained from the sample on the first occasion as ancillary information. In this paper, it is shown that this estimate is more efficient than the one similar to that of Pathak and Rao [2] in all situations where the well known ratio estimate in simple random sampling is no less efficient than the usual regression estimate or the Rao-Hartley-Cochran estimate in sampling with varying probabilities and without replacement.

JN SAMPLING, the sampling-units are usually chosen so as to be similar in size and structure. With some types of population, however, it is convenient or necessary to use sampling-units that differ in size. Thus the farm is often the sampling-unit for collecting agricultural data, though farms in the same county may vary in land acreage from a few acres to over 1,000 acres. Similarly, when obtaining information about sales or prices, the sampling-unit may be a dealer or store, these ranging from small to large concerns. In such cases the question arises: Should differences between the sizes of the sampling-units be ignored or taken into account in selecting the sample and in making estimates from the results of the sample? This paper contains a preliminary discussion of the problem, though further research is needed, many of the results given below being only large-sample approximations. It is convenient to consider first the problem of estimation, since it appears that the best method of distributing the sample depends on the process of estimation that is to be used.

This paper reports the results of a survey of vaccination status and smallpox history undertaken in the city of Lahore West Pakistan in 1968-69 6 months before introduction of an eradication program. An aim of the survey was to develop reliable estimates of the immunization status of a typical urban population that could be used as a baseline for assessment of the subsequent eradication program. The sampling procedure used in this study was a 3-stage cluster sample of the population. This provided an acceptable degree of precision with a minimum utilization of field resources. 6952 (86%) individuals from the final enumerated sample of 8064 were examined and interviewed. Overall 90.3% of the population was found to have vaccination scars; there were no significant sex differences. On the basis of interviews 93.4% had a positive vaccination history although 24.4% had received only primary vaccination. An estimated 68.3% of the population had a history of vaccination or revaccination within the last 3 years and 80.2% within the last 10 years. Vaccination status was directly related to location of the city area; lower rates of reported vaccination were obtained in peripheral areas and agricultural neighborhoods. An estimated 4.7% of the population had a history of previous smallpox. When the 90.3% with visible evidence of successful immunization were combined with the 2.8% with a history of past smallpox and no vaccination scars an overall level of past experience with pox virus antigens of 93.1% was obtained. These results suggest a relatively high level of vaccination in the Lahore area. The persistence of endemic smallpox despite this achievement is attributed to the considerable numbers of younger children who are not vaccinated and the waning of immunity in older age groups.

1). For determination of grain size distribution, it is necessary to sift some volume of deposits. The sample size is discussed in this paper. The most useful is the treanlent as follow. At first, deposits aer stratified Into several strata and then cluster sampling method is used in every stratum. 2). And if we wish to know the ratio of rocks on minerals, sample size is also the important and. fundamental problem. Where p is the population ratio and n sample size, then the variance S2 is S2=pq/n in this case, q=1-p. Let the coefficient of variance of p be smaller than a, so we have the sample size n defind by n_??_pq l/a2•q/p. Fig. 1 shows n and p, in case a=0.10 and a=0.05.

SUMMARY IN previous discussions of the problem of balancing sampling costs against the cost of choosing the wrong population it has been assumed that the cost of sampling is equal to the sample size multiplied by a constant known unit cost. In the present paper the cost of sampling is assumed to be the cost of an incorrect choice for half of the sample programme (which is divided evenly between the two populations). The resulting loss function is considered when the choice is between two normal populations of known equal variance, the standard of performance being determined by the mean. Wald's (1950) minimax principle is used to determine fixed size and sequential sampling procedures, and the performance of these procedures is examined.

Random time sampling of activities (work sampling) is successfully applied to everyone in a division of 350 employees for the purpose of budget planning and cost control. To insure this success several technical and administrative problems had to be solved, especially those relating to the design and management of the sample. An appropriate model is described together with methods of sampling, estimating, and calculating sampling errors. Bias is reduced by using an employee instruction booklet, anonymous reporting, and by classifying and coding activities. Cost of the plan is reduced by using a stratified cluster sample, a check-type data sheet, a simplified method of estimation, and by calling random times by telephone. * This paper was prepared and the project described herein was carried out under the general supervision of E. J. Engquist, Jr., Director of the Statistics Division.

This paper presents the results of a study with two main objectives: (1) to investigate an extension of some principles recently developed in the theory of cluster sampling, and (2) to determine the efficiency of sample segments containing specified numbers of farms, as indicated by highway maps, for estimating numbers of commercial orchards in a commercial peach area.

This is the fourth paper in a series on the epidemiology of psychoses in Croatia, Yugoslavia. Data collected from 1960-1975 on a representative sample of the population of the study area, including the Istrian Peninsula and the northern Adriatic littoral, and the control area, the rest of Croatia, indicate that functional psychotic illnesses are more frequent in the study area, with the highest rates in older population groups. The finding is not associated with coastal or inland residence, with educational level, or with occupational group. Other diseases in excess in the study area are diabetes mellitus, psoriasis, and alcoholism. Nutritional disorders are about equally distributed between study and control areas. Data on extent of cigarette smoking was more prevalent in the study area, cases of functional psychoses smoked no more than the general population.

Usually auxiliary information based on just one variate is used to improve the precision of estimators of population totals, means, etc. In this paper a method is proposed of using information on several variates to achieve higher precision. The technique of difference estimation is employed throughout. It is shown that the variances of difference estimators are comparable to those of ratio estimators. The results are extended to double sampling procedures and sampling over two occasions.

SUMMARY Sampford (1962) has considered the following sampling scheme to select a sample with n distinct population units. Sampling with unequal probabilities with replacement is carried out until (n + 1) different population units are selected, the last unit is not recorded in the sample to insure some simplicity in the estimation procedure. The method is called inverse sampling with unequal probabilities. In this paper it is shown that this method of sampling is equivalent to sampling with unequal probabilities without replacement in some sense. The estimator of the population total given by Sampford is compared with other existing estimators. It is shown that there exist estimators which are uniformly better than the estimator given by Sampford. Finally, it is demonstrated that Des Raj's estimator (1956) has certain definite advantages over Sampford's estimator suggesting that sampling with unequal probabilities (without replacement) with Des Raj's estimator probably gives a better estimation procedure than inverse sampling with unequal probabilities. 1. INTRODUCTION AND PRELIMINARIES Although sampling with unequal probabilities with replacement has the merit of simplicity, it has a major drawback that the number of distinct population units selected in a sample is a random variable and varies from sample to sample. Sampling with unequal probabilities without replacement does not have this drawback, but most estimators of the population total in this sampling scheme are either unwieldy or biased. Recently some workers have developed sampling schemes wherein the number of distinct population units selected in a sample is constant from sample to sample, and also the estimator of the population total is easy to compute and has some desirable properties like possessing a nonnegative estimator of its variance, etc. Of special interest in this connexion are the works of Des Raj (1956), Stevens (1958), Rao, Hartley & Cochran (1962) and others. In cluster sampling, Sampford (1962) has considered a method of sampling with probabilities proportional to cluster size (with replacement) by which a fixed number of distinct clusters are included in the sample and has given an unbiased estimator of the population total and an estimator of its variance. A slightly different version of this method of sampling is considered here in detail and is referred to as inverse sampling with unequal probabilities. The following notation and definitions are used in the subsequent sections. Number of population units: N.

Sampling errors are generally a function of two factors the time spent making an observation, and the total number of observations. In addition, if sampling error costs are not symmetrically distributed about the true value, there is a problem as to whether the sampling mean should be used as a basis for making a decision, or if a biased estimate should be used. A general model is developed in this paper for the total expected cost of a sampling procedure as a function of the sample size, observation time per sample unit, and the use of a biased estimate. The general model is then applied to a specific situation to illustrate the application of the model. A method is also given for evaluating the observational variance as a function of observation time.

In this paper it is assumed that, using a sample survey, two factors are to be studied, comparisons between the “levels” of the factors are of greatest interest, and there is “interaction” between the factors. Attention is concentrated on situations in which only two levels of each factor are to be compared, but extensions to more complex surveys are discussed. Assuming independent sampling, optimal sample size allocations are obtained. Where these allocations require recourse to programming algorithms, approximate solutions are given. If independent sampling is not feasible, a double sampling procedure is suggested. To indicate how sub-sampling from the first phase sample is to be carried out, a sampling rule (possessing optimal conditional precision properties) is derived. Then, a procedure to determine the optimal first phase sample size is given. Finally, it is demonstrated that this double sampling procedure can be applied to estimation of the (finite) population mean when double sampling wi...

In this paper, several observations are presented to point out the behavior of systematic sampling in forest inventory. A brief literature review and the results of empirical studies indicate that the relative precision of two-dimensional systematic sampling for a given forest stand varies with the variable of interest, spatial distribution of trees, plot size, and sampling intensity. More specifically, results of this study indicate that systematic sampling performs better than or as well as simple random procedures for clustered or near-randomly distributed forest populations for tree frequency estimation. For uniformly spaced populations such as plantations, however, simple random sampling should be most precise for tree frequency estimation. With basal area, results were less clear but considering this study plus others reported in the literature, systematic sampling should usually perform as well as or better than simple random procedures for most tree populations.

In this paper, we will present sampling theory pertinent to the following type of sample design. Each one of the B boxes is shaken and classified into one of L strata on the basis of the auditory observation made concerning the quality of the content. A sample of boxes is selected from each stratum, and an accurate observation is made by inspecting the content of each box selected for the sample. The efficiency of thii design will depend upon the accuracy of the auditory observation, and also on the costs involved.

In sample surveys the final estimate is prepared from information collected for sample units of definite size (area) located at random. Large-scale work involves journeys from one sample unit to another so that both cost and precision of the result depend on size (area) as well as the number (density per sq. mile) of sample units. The object of planning is to settle these two quantities in such a way that ( a ) the precision is a maximum for any assigned cost, or ( b ) the cost is a minimum for any assigned precision. The present paper discusses the solution for (1) uni-stage sampling (with randomization in one single stage) both in the abstract and in the concrete; and for (2) multi-stage sample (with randomization in more than one stage) mostly in the abstract. The whole area is considered here as a statistical field consisting of a large number of basic cells each having a definite value of the variate under study. These values (with suitable grouping) form an abstract frequency distribution corresponding to which there exists a set of associated space distributions (of which the observed field is but one) generated by allocating the variate values to different cells in different ways. This raises novel problems which are space generalizations of the classical theory of sampling distribution and estimation. On the applied side it also enables classification of the technique into two types: ( a ) ‘individual’ or ( b ) ‘grid’ sampling depending on whether each sample unit consists of only one or more than one basic cell. For most space distribution precision of the result is nearly equal for both types of sampling; these are called fields of random type. For certain fields (including those usually observed in nature) precision depends on sampling type; these are fields of non-random type. Application to estimating acreage under jute covering 60,000 sq. miles in Bengal in 1941-2 is described with numerical data. The margin of error of the sample estimate was about 2 %, while cost was only a fifteenth of that of a complete census made in the same year by an official agency.

SUMMARY In this paper a test method for time study developed from a theoretical basis of sequential analysis of mathematical statistics is presented. This method admits a statistical decision on the regions which contain a certain time ratio. In the course of the test process, the instantaneous observations of the activities made at the work places are assigned by small integers which yield a simple sequential statistic. The observations need not be recorded. The sample size is, on the average, considerably smaller than that of comparable studies by normal work sampling. The design and application of sequential work sampling tests are described with regard to the estimation method of work sampling and their most important features are discussed.

Recent years have seen an increased demand for sampling methods which permit simple computations of means and variances. For efficient designs the calculation of variances though theoretically possible may be difficult enough that busy statisticians omit to do it, and so a basic advantage of probability sampling is lost. A practical method for the simple treatment of cluster designs with slight loss of efficiency has been devised by Deming; it involves an ingenious arrangement of the population in such fashion that both the sampling units and the strata in which they are grouped are effectively equal in size, so that equal weights are appropriate for estimating both totals and variances. The present paper describes a simplification in the different direction of restricting selection to two units from each stratum. To the extent that this involves some sacrifice of efficiency we recall that the decision on which sampling method to use ought to take account of simplicity of calculation of mean and ...

Summary A new two-phase sampling procedure is suggested in this paper. The information on the ancillary variate in the preliminary sample is used for selecting the units for sub-sample with unequal probabilities. Unbiased estimators for the population mean and their variance expressions are obtained for sub-sampling procedures, both with and without replacement.

The author [1], [2] has given an algorithm for finding that stratified allocation which minimizes the posterior variance of the overall population mean subject to a budget constraint under a model in which a normal prior distribution and independent normal sampling distributions were assumed. The budget constraint assumed a variable per unit cost of observation. In the present paper these results are extended to cover the case where there are fixed costs, as well as variable costs, associated with sampling in the ith stratum. The resulting algorithm is noted to be applicable in finding the optimal allocation of sampling effort (with fixed and variable sampling costs) under a variety of distributional assumptions. An interpretation is also given to two and higher stage design questions when there is differential prior information regarding the first stage units.

This paper deals with an empirical investigation of 5 single-stage sampling systems for estimating the average kill of ducks per hunter at the provincial and national levels for Canada. The Canadian migratory hunting permit holders during 1966-67 were used as the sampling universe for selection and estimation purposes. The observed data were the results of the 1967-68 Canadian harvest survey. It is shown that none of the sampling systems could be chosen as uniformly the best for all the provinces of Canada. A design based on a cluster sample of post offices selected with probability proportional to their sales proved most efficient in a large majority of the cases. In the remaining cases a design based on a stratified sample of hunters with post offices as strata showed the maximum efficiency. Economic considerations may, however, weigh in favour of retaining the current method which consists in selecting a simple random sample of permit holders. Sample sizes are obtained for the various sampling systems to provide estimates of the mean kill with coefficient of variation of 5 percent for the different provinces of Canada.

Deals with the descriptive and sampling statistics of the distribution of absorbed dose in a volume of tissue. It treats the dose distribution over one or more strata as a cluster sample of the dose distribution in a volume of tissue and employs the relative frequency of 'percentage position against dose' representation rather than the conventional isodose of 'percentage dose against position' representation. The relative advantages of the former are three: (1) it permits description of the dose distribution over one or more strata of the anatomical target volume in terms of a few familiar attributes or summary statistics; (2) it permits estimation of the values of the corresponding attributes of the distribution of dose over the target volume of tissue together with a numerical estimate of the confidence which may be placed in these estimates and (3) it permits estimation of the likelihood that two different distributions of radiation dose over selected anatomical strata are samples from the same volume distribution. This last is described in detail in this paper.

mechanism of stratification, it may still be necessary to further control the selection of the sample within each stratum. The techniques of cluster sampling and subsampling can be used within a stratum to minimize the selection of non-preferred samples. However, they may result in loss of precision of the estimated characteristic under study. Attempts were therefore made to propose new methods of controlled selection. Goodman and Kish were the first to suggest a technique of controlled selection. The idea of controlled selection is doubtless attractive but needs to be further developed before it can be used to obtain estimates of known precision. The authors [2] have developed a technique of controlled sampling with equal probabilities and without replacement which reduces the risk of obtaining a non-preferred sample to the minimum possible extent and yet provides an estimate which is at least as efficient as in the case of simple random sampling. To further enhance the applicability of this technique, the authors [3] have proposed some simplified procedures of controlled selection. At times, even these may appear to be time consuming. Avadhani [4] has suggested a new approach to obviate this difficulty altogether. The results based on this approach which ultimately lead to the method of controlled simple random sampling, are consolidated in this paper in an integrated fashion and utilised to present a suitable sampling mechanism for controlled selection. 2. Notation and Definitions

THE theoretical distribution of many statistics calculated from small samples is known when the population is normal, but when it is not normal we know very little about the distribution of such statistics. Such work as has been done has generally assumed population forms of standard types, but we may occasionally come up against samples from populations which do not appear to fit into any known type. This has led to many attempts being made to build up, by experimental sampling from non-normal data, partial populations of samples from which can be inferred in an empirical way the laws of distribution followed by derived statistics. A list of papers dealing with this subject which have come to the author's notice is given on pp. 79, 80 below. In many cases it has been found that in sampling from curves with one mode not at the end of the range, the distributions of statistics such as "Student's " "t", the correlation coefficient and, in certain cases, Fisher's "z ", differ very slightly from one population to another. On the whole these investigations have suggested that in such cases we can neglect the departure of the population from normality without introducing serious error into our tests of significance. The possibility of further theoretical work must not be overlooked, but unless our results are independent of population form (as, for instance, in recent work by Pitman and Welch) it is unlikely that we shall be able to make much practical use of the results. It is customary to designate a non-normal population by the values of 1 and f2; but in the case of samples of 100 or less from a normal population the range of values of ,81 and /2 excluding 5 % of the total at each end is comparable with the range of 1 and /2 in the non-normal populations which have been used for sampling experiments. Further, this range of populations is considered by E. S. Pearson to cover most cases which will be found to occur in practice. On these grounds I think that conclusions of practical value are most likely to be reached by further sampling. No attempt appears to have been made to carry out an experimental sampling from a bivariate population in which the distribution surface is not normal and in which the correlation coefficient is high, or to take sets of samples from a univariate non-normal population and to assign the samples to blocks and treatments in a randomized block experiment, taking a completely fresh

This paper introduces a new method of obtaining multifactor uniform regions. Those fusion techniques such as 'centroid' and 'group average', which are based on the imposition of minimum-variance constraints, may well generate artificial classifications, while the step-wise clustering procedure and its corresponding dendrogram facility is inefficient when dealing with large data sets. The new method, termed the dense-space method, searches initially for dense spheres which signal the presence of important uniform regions or dense space, and then derives distinct regions by linking any dense spheres which intersect. Three classification levels are suggested: nuclear, basic and complete. The nucleus of each distinct region is described by a set of intersecting dense spheres of radius Jr, each of which has the property that it does not intersect any dense sphere from any other distinct region. The subset of sample points inside the dense spheres are termed nuclei points and constitute a cluster. Those points which lie outside the dense spheres but at a distance not greater than r from the centre of a dense sphere are included with the classification for the sphere at the basic level. Points which are unclassifiable at the basic level are relatively remote and their classification at the complete level, into the regions which contain the points' nearest dense spheres, should only be used when a best fit is demanded for every sample. The dense-space method achieves more 'natural' classifications and demands less computation and memory storage. Its advantages over other methods are shown by a reworking of U.S.A. census data, first used by B. J. L. Berry, and by reference to an urban survey of Middlesbrough. THE PROBLEM of regional classification in geography is essentially the classification problem common to all the behavioural sciences. A set of samples (observation units) is divided into a small number of subsets or clusters so that each subset represents a grouping of samples which have a basic common similarity with respect to the survey variables. Classifications involving total uniformity in the cluster samples, that is, every variable having uniform values for each subset, are derived by imposing constraints on the cluster's overall variance. Two such techniques, centroid and group-average, are compared here before introducing a third method, that of dense space. The new method, it is claimed, achieves more 'natural' classifications and is suitable for rapid analysis of large surveys. Computation details are given for data used by B. J. L. Berry in a recent paper1 from nine census divisions of the U.S.A., and the speed facility of the dense-space method when used on large surveys is demonstrated with reference to a 23 I-sample urban survey. A formal presentation of the methods is given in a mathematical appendix.

Ranked set sampling employs judgment ordering to obtain an estimate of a population mean. The method is most useful when the measurement or quantification of an element is difficult but the elements of a set of given size are easily drawn and ranked with reasonable success by judgment. In each set all elements are ranked but only one is quantified. Sufficient sets are processed to yield a specified number of quantified elements and a mean for each rank. The average of these means is an unbiased estimate of the population mean regardless of errors in ranking. Precision relative to random sampling, with the same number of units quantified, depends upon properties of the population and success in ranking. In this paper the ranked set concept is reviewed with particular consideration of errors in judgment ordering.