Sampling - Concepts and Definitions
The frequency distribution of the samples according to the values of the estimator t based on the sample estimates is termed as the sampling distribution of the estimator t. It is important to mention that though the population distribution may not be normal, the sampling distribution of the estimator t is usually close to normal, provided the sample size is sufficiently large. If the estimator t is unbiased and is normally distributed, the interval is expected to include the parameter in P% of the cases where P is the proportion of the area between –K and +K of the distribution of standard normal variate. The interval considered is said to be a confidence interval for the parameter with a confidence coefficient of P% with the confidence limit t – K SE(t) and t + K SE(t). For example, if a random sample of the records of batteries in routine use in a large factory shows an average life t = 394 days, with a standard error SE(t) = 4.6 days, the chances are 99 in 100 that the average life in the population of batteries lies between
The collection of all units of a specified type in a given region at a particular point or period of time is termed as a population or universe. Thus, we may consider a population of persons, families, farms, cattle in a region or a population of trees or birds in a forest or a population of fish in a tank etc. depending on the nature of data required.
b) Sampling Unit
Elementary units or group of such units which besides being clearly defined, identifiable and observable, are convenient for purpose of sampling are called sampling units. For instance, in a family budget enquiry, usually a family is considered as the sampling unit since it is found to be convenient for sampling and for ascertaining the required information. In a crop survey, a farm or a group of farms owned or operated by a household may be considered as the sampling unit.
c) Sampling Frame
A list of all the sampling units belonging to the population to be studied with their identification particulars or a map showing the boundaries of the sampling units is known as sampling frame. Examples of a frame are a list of farms and a list of suitable area segments like villages in India or counties in the United States. The frame should be up to date and free from errors of omission and duplication of sampling units.
d) Random Sample
One or more sampling units selected from a population according to some specified procedures are said to constitute a sample. The sample will be considered as random or probability sample, if its selection is governed by ascertainable laws of chance. In other words, a random or probability sample is a sample drawn in such a manner that each unit in the population has a predetermined probability of selection. For example, if a population consists of the N sampling units U1, U2,…,Ui,…,UN then we may select a sample of n units by selecting them unit by unit with equal probability for every unit at each draw with or without replacing the sampling units selected in the previous draws.
e) Non-random sample
A sample selected by a non-random process is termed as non-random sample. A Non-random sample, which is drawn using certain amount of judgment with a view to getting a representative sample is termed as judgment or purposive sample. In purposive sampling units are selected by considering the available auxiliary information more or less subjectively with a view to ensuring a reflection of the population in the sample. This type of sampling is seldom used in large-scale surveys mainly because it is not generally possible to get strictly valid estimates of the population parameters under consideration and of their sampling errors due to the risk of bias in subjective selection and the lack of information on the probabilities of selection of the units.
f) Population parameters
Suppose a finite population consists of the N units U1, U2,…,UN and let Yi be the value of the variable y, the characteristic under study, for the ith unit Ui, (i=1,2,…,N). For instance, the unit may be a farm and the characteristic under study may be the area under a particular crop. Any function of the values of all the population units (or of all the observations constituting a population) is known as a population parameter or simply a parameter. Some of the important parameters usually required to be estimated in surveys are population total and population mean
g) Statistic, Estimator and Estimate
Suppose a sample of n units is selected from a population of N units according to some probability scheme and let the sample observations be denoted by y1,y2,…,yn. Any function of these values which is free from unknown population parameters is called a statistic.
An estimator is a statistic obtained by a specified procedure for estimating a population parameter. The estimator is a random variable and its value differs from sample to sample and the samples are selected with specified probabilities. The particular value, which the estimator takes for a given sample, is known as an estimate.
h) Sample design
A clear specification of all possible samples of a given type with their corresponding probabilities is said to constitute a sample design. For example, suppose we select a sample of n units with equal probability with replacement, the sample design consists of Nn possible samples (taking into account the orders of selection and repetitions of units in the sample) with 1/Nn as the probability of selection for each of them, since in each of the n draws any one of the N units may get selected. Similarly, in sampling n units with equal probability without replacement, the number of possible samples (ignoring orders of selection of units) is and the probability of selecting each of the samples is .
i) Unbiased Estimator
Let the probability of getting the i-th sample be Pi and let ti be the estimate, that is, the value of an estimator t of the population parameter based on this sample (i=1,2,…,Mo), Mo being the total number of possible samples for the specified probability scheme. The expected value or the average of the estimator t is given by
An estimator t is said to be an unbiased estimator of the population parameter if its expected value is equal to irrespective of the y-values. In case expected value of the estimator is not equal to population parameter, the estimator t is said to be a biased estimator of . The estimator t is said to be positively or negatively biased for population parameter according as the value of the bias is positive or negative.
j) Measures of error
Since a sample design usually gives rise to different samples, the estimates based on the sample observations will, in general, differ from sample to sample and also from the value of the parameter under consideration. The difference between the estimate ti based on the i-th sample and the parameter, namely
(ti - ), may be called the error of the estimate and this error varies from sample to sample. An average measure of the divergence of the different estimates from the true value is given by the expected value of the squared error, which is
and this is known as mean square error (MSE) of the estimator. The MSE may be considered to be a measure of the accuracy with which the estimator t estimates the parameter.
The expected value of the squared deviation of the estimator from its expected value is termed sampling variance. It is a measure of the divergence of the estimator from its expected value and is given by
This measure of variability may be termed as the precision of the estimator t.
The MSE of t can be expressed as the sum of the sampling variance and the square of the bias. In case of unbiased estimator, the MSE and the sampling variance are same. The square root of the sampling variance σ(t) is termed as the standard error (SE) of the estimator t. In practice, the actual value of σ(t) is not generally known and hence it is usually estimated from the sample itself.
k) Confidence interval
tL = 394 - (2.58)(4.6) = 382 days
tU = 394 + (2.58)(4.6) = 406 days
The limits, 382 days and 406 days are called lower and upper confidence limits of 99% confidence interval for t. With a single estimate from a single survey, the statement “ lies between 382 and 406 days” is not certain to be correct. The “99% confidence” figure implies that if the same sampling plan were used may times in a population, a confidence statement being made from each sample, about 99% of these statements would be correct and 1% wrong.
l) Sampling and Non-sampling error
The error arising due to drawing inferences about the population on the basis of observations on a part (sample) of it is termed sampling error. The sampling error is non-existent in a complete enumeration survey since the whole population is surveyed.
The errors other than sampling errors such as those arising through non-response, in- completeness and inaccuracy of response are termed non-sampling errors and are likely to be more wide-spread and important in a complete enumeration survey than in a sample survey. Non-sampling errors arise due to various causes right from the beginning stage when the survey is planned and designed to the final stage when the data are processed and analyzed.
The sampling error usually decreases with increase in sample size (number of units selected in the sample) while the non-sampling error is likely to increase with increase in sample size.
As regards the non-sampling error, it is likely to be more in the case of a complete enumeration survey than in the case of a sample survey since it is possible to reduce the non-sampling error to a great extent by using better organization and suitably trained personnel at the field and tabulation stages in the latter than in the former.