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L4 theory of sampling

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L4 theory of sampling

  1. 1. Sampling fundamentals
  2. 2. INTRODUCTION The need for adequate and reliable data is ever increasing for taking wise decisions in different fields of human activity and business. There are two ways in which the required information may be obtained: 1. Complete enumeration survey or census method. 2. Sampling method. In the first case, data are collected for each and every unit. i.e Universe/ population (complete set of items).
  3. 3. What is Population? In any field of inquiry, all the items under consideration constitute ‘population’ or ‘universe’. A complete enumeration of all the items of ‘population’ is known as a census inquiry. In such an inquiry it is assumed that highest accuracy is obtained. But this type of inquiry involves a great deal of time, money and energy. Not only this, census inquiry is not possible in
  4. 4. Sample Hence, quite often we select only a few items from the universe for our study purposes. The items so selected is technically called sample.
  5. 5. What is Sampling Process? Sampling may be defined as the selection of some part of an aggregate or totality on the basis of which a judgment or inference about the population (aggregate or totality) is made. In other words, it is the process of obtaining information about an entire population by examining only a part of it.
  6. 6. Need for sampling 1. Sampling can save time and money. 2. Sampling may produce more accurate information if it is conducted by trained and experienced investigator. 3. Sampling becomes the only option when the population size is infinite.
  7. 7. Sample Design  Researcher must prepare a sample design for his study i.e., he must plan how a sample should be selected and of what size a sample would be.  Large and Small Sample: Let the population size be N and a part of size n ( which is less than N) of this population is selected according to some rule for some characteristics of the population. The group consisting of these n units is known as ‘sample’. Therefore n denotes sample size. If n>30 then it is considered as large sample, otherwise it is known as small sample.  The selection process i.e. the way the researcher decide to select a sample from the population is known as the ‘sample design’. In other words, it is a define plan ( determined by the researcher) before any data is collected for obtaining a sample from a given population. Eg : Research on pharmaceutical industry.
  8. 8. Sampling Method/Sampling Technique 1. Probability Sampling 2. Non-probability sampling
  9. 9. Probability Sampling  Probability sampling is also known as ‘random sampling’ or ‘chance sampling’. Samples selected according to some chance are known as random or probability samples i.e. every item in the population has known chance of being included in the sample.
  10. 10. Non-probability sampling On the other hand, non-random or non-probability samples are those where the selection of sample unit is based on the judgment of the researcher than randomness.
  11. 11. Important Sampling Designs Probability Sampling: i. Simple Random Sampling ii. Systematic Sampling iii. Stratified Sampling iv. Cluster and area Sampling
  12. 12. Major non-probability sampling are: i. Deliberate Sampling/ Purposive sampling/ Judgment sampling ii. Quota Sampling
  13. 13. Simple Random Sampling Method Under this sampling design , every item of the universe has an equal chance of inclusion in the sample. For example, if we have to select a sample of 300 items from a universe of 15,000 items, then we can put the names or numbers of all the 15,000 items on slips of paper and conduct a lottery.
  14. 14. Under this method, sampling is done without replacement, so that no unit can appear more than once in the sample. Thus, if from a population consisting of 4 members A, B, C and D, a simple random sample of n=2 is to be drawn, there would be 6 possible samples without replacement. They are AB, AC , AD, BC, BD, CD. Keeping in this view, we can say that a simple random sample of size n from population N results in N C n
  15. 15. Exercise Take a certain finite population of six elements ( say a, b, c, d, e, f). Suppose that we want to take a sample of size n=3 from it. Find out how many possible outcomes are there? Write the elements . Choose one sample out of it. What is their probabilities of getting into the sample?
  16. 16. Systematic Sampling In some instances, the most practical way of sampling is to select every ith item from the universe where ‘i’ refers to the sampling interval. The sampling interval can be determined by dividing the size of the population by the size of the sample to be chosen. For example, if we wish to draw 32 names out of the list of 320 names, the sampling interval will be 10. It means every 100th name will be selected. In this process a random start is always
  17. 17. Example In a class of 120 students , it was decided to constitute an Academic and cultural committee with 10 representatives. Use systematic sampling method to form the committee. Soln:
  18. 18. Merits and demerits Merits : a. It is a simple method b. It can be taken as an improvement over a simple random sample as it spread more evenly over the population. Demerits: a. It is not truly random in the strict sense. This is because all items selected for the sample ( except the first term) are pre-determined by the constant interval. b. There are certain dangers too in using this type of sampling. If there is a hidden periodicity in the population, systematic sampling will prove to be inefficient method of sampling. Example , quality
  19. 19. Stratified Sampling If a population from which a sample is to be drawn does not constitute a homogeneous group,( highly heterogeneous) stratified sampling technique is generally applied in order to obtain a representative sample. Under stratified sampling the population is divided into several sub-populations that are individually more homogeneous than the total population. The different sub-populations are called ‘strata’ . Then we select
  20. 20. The following three questions are highly relevant in the context of stratified sampling a) How to form strata? b) How should items be selected from each stratum? c) How many items be selected from each stratum or how to allocate the sample size of each stratum?
  21. 21. Regarding the first question, we can say that the items which are homogeneous ( i.e. of common characteristics) should be put in the same group or strata. In other words strata be formed in such a way that elements are most homogeneous within the strata and most heterogeneous between the different strata.
  22. 22. In respect to 2nd question, we can say that to choose the items from each strata we normally adopt simple random sampling.
  23. 23. To answer the 3rd question we have to understand the following concepts: Stratified sampling can be of two types: proportionate and disproportionate. In proportionate stratified sampling the number of sample units in various strata are in the same proportion as found in the population. Thus, larger the particular stratum, the more weight it receives in the analysis.
  24. 24. Example 1 A sample of 30 students is to be drawn from a population consisting of 300 students belonging to two colleges A and B. How would you draw the desired sample by using proportionate stratified random sampling?College Number of students A 200 B 100
  25. 25. Disproportionate Stratified sampling Here the strata are represented in the total sample in a proportion other than the one with which they are found in the population. In Disproportionate Stratified sampling the sample proportion for each stratum will be determined by using the following rule: The proportion of the ith stratum will be ni = Ni . σi i=1, 2 …k N1σ1 + N2σ2 +….. Nkσk
  26. 26. Example 2 A population is dived into three strata with N1=5000 ,N2=2000 and N3=3000 . Respective standard deviations are σ1 =15, σ2 =18 and σ3 =5. How should a sample of size 84 be selected from the three strata . Use proportionate and disproportionate Stratified sampling technique.
  27. 27. Example To know the customer demand for expensive and luxurious item (say diamond Jewelry), among the followings which sampling technique will you chose? Justify your answer. a) Simple random sampling b) Systematic sampling c) Stratified Sampling
  28. 28. Cluster Sampling /Area Sampling In Cluster sampling first we divide the population into groups called ‘clusters’ and then select some units from the groups or the clusters for sample. Cluster sampling is totally opposite to stratified sampling in the sense that , a. The units within each cluster should be as heterogeneous as possible. b. There should be small difference between the clusters.
  29. 29. Ex. If a market research team is attempting to study the preference of TV- Brand in a large city.
  30. 30. Area Sampling Since geographical area of interest happens to be a big one. Under this sampling we divide the total area into smaller non-overlapping areas, generally called geographical clusters, then certain areas are randomly selected and all households in the selected area are would be interviewed to get the information.
  31. 31. Non-probability sampling i. Deliberate Sampling/ Purposive sampling/ Judgment sampling: Selection made by choice not by chance where investigator is highly experienced and skilled. This method is seldom used and cannot be recommended for general use since it suffers from the drawback of favoritism depending on the beliefs and prejudices of the investigator. i. Quota Sampling: Here the interviewer got some quota based on gender, age , income , occupation etc. to be filled where actual selection of units totally depends on the interviewers judgment.
  32. 32. iii) Convenience Sampling Determination of sample size: ‘ smaller but properly selected samples are superior to large but badly selected samples’ 1. Resources available 2. Nature of study 3. Method of sampling used 4. Nature of respondents( response rate) 5. Nature of population ( existence of heterogeneity)
  33. 33. Statistic and Parameter  A statistic is a characteristics of a sample , whereas a parameter is a characteristic of a population.  Thus, when we work out certain measures such as mean, median, mode, standard deviation from sample, they are called statistic as they will describe the characteristic of the population.  Eg Sample mean= x . Sample s.d. = s  Eg of parameter , population mean= μ ,population s.d. = σp
  34. 34. Formula  Sample Mean Formula: x =∑x n  Sample variance Formula s2= ∑(X-X)2 (n-1) n= sample size  Population Mean Formula: μ =∑x N  Population variance Formula σ2= ∑(X-X)2 N N= population size.
  35. 35. Sampling error
  36. 36. The law of large numbers Draw observations at random from any population with finite mean μ. As the number of observations drawn increases, the mean of the observed values gets closer and closer to the mean μ of the population. ` x
  37. 37. The central limit theorem Take a large (30 or more) random sample of size n from any population with mean μ and standard deviation σ. The sample mean, X is approaches the normal distribution with mean μ and standard deviation . n         n NX  ,~