Introduction Concepts
Population: In a statistical investigation the interest usually lies in studying the various characteristics relating to items or individuals belonging to a particular group. This group of individuals under study is known as the population or Universe. The units are denoted as capital letters Ex: X1, X2, ……..,XNand population size represented as N
Sample: A finite subset of the Universe is called a Sample. Generally we will collect information from the sample and based on that information, we try to take decisions about the Universe. Sample units are denoted as small letters Ex: x1, x2, . . . . ,xn.
Sample Size: The number of units in the population is called sample Size. It is denoted as n.
Sampling Frame: In order to cover the population decided upon, there should be some list, map or other acceptable material, called the frame.
Random sample: Suppose X is a random variable with distribution function F(x), and X1, X2, ….., Xn is a set of n i.i.d s distributed random variables having the same distribution as that of X. Then X1, X2, ….., Xn is called Random Sample of size n.
Parameter: Population constants are called parameters. Eg: Mean (µ), Standard Deviation (σ), Variance (σ2), Proportion (P), etc….
Statistics: Suppose X is a random variable with distribution function F(x), and X1, X2, ….., Xn is a random sample of size n. Let t=t(X1, X2, ….., Xn) be a function of sample observations that does not depend on any population parameter, then ‘t’ is called Statistics. In other words, sample constants are called statistics.
Sampling Distribution: Since ‘t’ is a function of certain random variables, it itself is a random variable. The probability distribution of ‘t’ is called the sampling distribution of ‘t’. In other words, suppose k samples are drawn from the population and k samples are having mean, now we can arrange these sample mean in a frequency distribution called sampling distribution of that statistic (i.e. mean).
Sampling Error: The error involved in approximation of population characteristics by using sample information is known as Sampling error.
Standard Error: The standard deviation of the sampling distribution of a statistic is known as Standard Error.
Ex:
Sampling Vs Census
Sampling is a tool which enables us to draw conclusions about the characteristics of population after studying only those objects/items that are included in the sample.
Census is a tool which enables you to draw conclusions by studying each and every characteristic of population units.
The main merits of sampling technique over the complete enumeration survey may be outlined as follows:
1. Less Time: Since only a part of the population has to be examined, there is considerable saving in time and labour. The sampling results can be obtained more rapidly and the data can be analysed much faster since relatively fewer data have to be collected and processed.
2. Reduced Cost of the Survey: Sampling usually results in reduction in cost in terms of money and in terms of man hours. Although the amount of labour and expenses involved in collecting information are generally greater per unit of sample than in complete Census. Since in most of the cases our resources are limited in terms of money and the time within which the results of the survey should be obtained, it is usually imperative to resort to sampling rather than complete enumeration.
3. Greater Accuracy of Results: The results of sample survey are usually much more reliable than those obtained from complete Census due to following reasons:
It is always possible to determine the extent of the sampling errors.
The non- sampling errors due to factors such as training of field workers, measuring and records observation, location of units, incompleteness of returns, bias due to interviewers etc., are likely to be of a serious nature in complete Census rather than in sample survey. In a sample survey non-sampling errors can be controlled more effectively by employing qualified and trained personnel, better supervision and better equipment for processing and analysis of data. Moreover it is easier to follow up non-response and incomplete responses.
4. Greater Scope: Sample Survey has generally greater scope as compared with complete Census. The complete enumeration is impracticable, rather inconvenient. Through sample survey, it is possible to conduct intensive enquiry, so detailed information can be obtained from a small group of respondents.
5. If the population is too large, as for example, of trees in a jungle.
6. If testing is destructive i.e., if the quality of an article can be determined only by destroying the articles in the process of testing.
testing the quality of milk or chemical salt by analysis
testing the breaking strength of chalks.
testing of crackers and explosives
testing the life of electric bulbs etc…………..
7. If the population is hypothetical, as for example in coin testing problems where the process was continued indefinitely, sampling method is the only scientific method to estimate parameters of the Universe.
Limitations of Sample Survey:
The advantages of sample survey over complete enumeration can be derived only if
· sampling units are drawn in a scientific manner
· appropriate sampling technique is used
· the sample size is adequate
The sample survey has its own limitations and problems which may be briefly outlined as:
1. Proper care should be taken in planning and execution of the sample survey, otherwise the results obtained might be inaccurate and misleading.
2. Sampling theory requires the services of trained and qualified personnel and sophisticated equipment for its planning, execution and analysis. In the absence of these, the results of the sample survey are not trustworthy.
3. However, if the information required about each and every unit of the UNIVERSE, there is no way but to resort to complete enumeration. More over time and money are not constraints, then complete enumeration may be better than any sampling method.
Sources: Fundamentals of Applied Statistics, S.C.Gupta & V.K.Kapoor, Fundamentals of Mathematical Statistics, S.C.Gupta & V.K.Kapoor, Sampling Techniques, W.G.Cochran
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