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  1. 1. Chapter 11 Sampling
  2. 2. Sampling? The process of selecting the right individuals, objects or events for study is known as sampling.
  3. 3. • • • • Population – The entire group of people , events , or things of interest that the researcher wishes to investigate. Element – The element is a single member of the population Population frame – Lisiting of all the elements in the population from which the sample is drawn Sample – Subset of population. It comprises some members selected from it – Element of population would form sample – Subgroup or subset of population • Subject – A single member of the sample
  4. 4. Basic Concepts in Samples and Sampling What is a sample? A sample is a finite part of a statistical population whose properties are studied to gain information about the whole(Webster, 1985). When dealing with people, it can be defined as a set of respondents(people) selected from a larger population for the purpose of a survey.
  5. 5. Basic Concepts in Samples and Sampling What is sampling? Sampling is the act, process, or technique of selecting a suitable sample, or a representative part of a population for the purpose of determining parameters or characteristics of the whole population.
  6. 6. Basic Concepts in Samples and Sampling What is the purpose of sampling?
  7. 7. Basic Concepts in Samples and Sampling There would be no need for statistical theory if a census rather than a sample was always used to obtain information about populations. But a census may not be practical and is almost never economical. There are six main reasons for sampling instead of doing a census. These are; 1-Economy – 2-Timeliness – 3-The large size of many populations – 4-Inaccessibility of some of the population – 5-Destructiveness of the observation – 6-Accuracy
  8. 8. Basic Concepts in Samples and Sampling -1- Economy •Taking a sample requires fewer resources than a census. For example (students) •Unrealistic •complete the interviews and analysis on your own
  9. 9. -2-The time factor. • A sample may provide you with needed information quickly. – For example, you are a Doctor and a disease has broken out in a village within your area of jurisdiction, the disease is contagious and it is killing within hours nobody knows what it is. You are required to conduct quick tests to help save the situation. If you try a census of those affected, they may be long dead when you arrive with your results. In such a case just a few of those already infected could be used to provide the required information.
  10. 10. -3-The very large populations . • Many populations about which inferences must be made are quite large. – For example, Consider the population of high school seniors in United States of America, a group numbering 4,000,000. The responsible agency in the government has to plan for how they will be absorbed into the different departments and even the private sector. The employers would like to have specific knowledge about the student's plans in order to make compatible plans to absorb them during the coming year. But the big size of the population makes it physically impossible to conduct a census. In such a case, selecting a representative sample may be the only way to get the information required from high school seniors.
  11. 11. -4-The partly accessible populations • There are Some populations that are so difficult to get access to that only a sample can be used. – Like people in prison, like crashed aeroplanes in the deep seas, presidents e.t.c. The inaccessibility may be economic or time related. Like a particular study population may be so costly to reach like the population of planets that only a sample can be used. In other cases, a population of some events may be taking too long to occur that only sample information can be relied on. For example natural disasters like a flood that occurs every 100 years.
  12. 12. -5- destructive nature of the observation • The destructive nature of the observation Sometimes the very act of observing the desired characteristic of a unit of the population destroys it for the intended use. – Good examples of this occur in quality control. For example to test the quality of a fuse, to determine whether it is defective, it must be destroyed. To obtain a census of the quality of a lorry load of fuses, you have to destroy all of them. This is contrary to the purpose served by quality-control testing. In this case, only a sample should be used to assess the quality of the fuses.
  13. 13. -6-Accuracy • Accuracy and sampling A sample may be more accurate than a census. A sloppily conducted census can provide less reliable information than a carefully obtained sample.
  14. 14. BIAS AND ERROR IN SAMPLING • A sample is expected to mirror the population from which it comes, however, there is no guarantee that any sample will be precisely representative of the population from which it comes. – Chance may dictate that a disproportionate number of untypical observations will be made like for the case of testing fuses, the sample of fuses may consist of more or less faulty fuses than the real population proportion of faulty cases.
  15. 15. Sampling error • What can make a sample unrepresentative of its population? One of the most frequent causes is sampling error. • Sampling error comprises the differences between the sample and the population that are due solely to the particular units that happen to have been selected. – For example, suppose that a sample of 100 American women are measured and are all found to be taller than six feet. It is very clear even without any statistical prove that this would be a highly unrepresentative sample leading to invalid conclusions. This is a very unlikely occurrence because naturally such rare cases are widely distributed among the population. But it can occur. Luckily, this is a very obvious error and can be detected very easily.
  16. 16. • There are two basic causes for sampling error. • One is chance: That is the error that occurs just because of bad luck. This may result in untypical choices. Unusual units in a population do exist and there is always a possibility that an abnormally large number of them will be chosen. – For example, in a recent study in which I was looking at the number of trees, I selected a sample of households randomly but strange enough, the two households in the whole population, which had the highest number of trees (10,018 and 6345 ) were both selected making the sample average higher than it should be. The average with these two extremes removed was 828 trees. The main protection against this kind of error is to use a large enough sample. The second cause of sampling is sampling bias.
  17. 17. • Sampling bias is a tendency to favour the selection of units that have particular characteristics. • Sampling bias is usually the result of a poor sampling plan. The most notable is the bias of non response when for some reason some units have no chance of appearing in the sample. – For example, take a hypothetical case where a survey was conducted recently by Cornell Graduate school to find out the level of stress that graduate students were going through. A mail questionnaire was sent to 100 randomly selected graduate students. Only 52 responded and the results were that students were not under strees at that time when the actual case was that it was the highest time of stress for all students except those who were writing their thesis at their own pace.
  18. 18. • Apparently, this is the group that had the time to respond. The researcher who was conducting the study went back to the questionnaire to find out what the problem was and found that all those who had responded were third and fourth PhD students. Bias can be very costly and has to be guarded against as much as possible. – For this case, $2000.00 had been spent and there were no reliable results in addition, it cost the researcher his job since his employer thought if he was qualified, he should have known that before hand and planned on how to avoid it. A means of selecting the units of analysis must be designed to avoid the more obvious forms of bias. Another example would be where you would like to know the average income of some community and you decide to use the telephone numbers to select a sample of the total population in a locality where only the rich and middle class households have telephone lines. You will end up with high average income which will lead to the wrong policy decisions.
  19. 19. Non sampling error (measurement error) • A non sampling error is an error that results solely from the manner in which the observations are made. • Like sampling error, non sampling error may either be produced by participants in the statistical study or be an innocent by product of the sampling plans and procedures.
  20. 20. • The simplest example of non sampling error is inaccurate measurements due to malfunctioning instruments or poor procedures. – For example, Consider the observation of human weights. If persons are asked to state their own weights themselves, no two answers will be of equal reliability. The people will have weighed themselves on different scales in various states of poor calibration. An individual's weight fluctuates diurnally by several pounds, so that the time of weighing will affect the answer. The scale reading will also vary with the person's state of undress. Responses therefore will not be of comparable validity unless all persons are weighed under the same circumstances.
  21. 21. • Biased observations due to inaccurate measurement can be innocent but very devastating. – A story is told of a French astronomer who once proposed a new theory based on spectroscopic measurements of light emitted by a particular star. When his colleagues discovered that the measuring instrument had been contaminated by cigarette smoke, they rejected his findings. • In surveys of personal characteristics, unintended errors may result from: -The manner in which the response is elicited -The social desirability of the persons surveyed -The purpose of the study -The personal biases of the interviewer or survey writer
  22. 22. Unintended errors during survey • • • • The interviews effect The respondent effect Knowing the study purpose Induced bias
  23. 23. Basic Sampling Classifications • Probability samples: ones in which members of the population have a known chance (probability) of being selected • Non-probability samples: instances in which the chances (probability) of selecting members from the population are unknown
  24. 24. • A random sample allows a known probability that each elementary unit will be chosen. For this reason, it is sometimes referred to as a probability sample. This is the type of sampling that is used in lotteries and raffles. For example, if you want to select 10 players randomly from a population of 100, you can write their names, fold them up, mix them thoroughly then pick ten. In this case, every name had any equal chance of being picked. Random numbers can also be used
  25. 25. • • • • A probability sample is one in which each element of the population has a known non-zero probability of selection. Not a probability sample of some elements of population cannot be selected (have zero probability) • Not a probability sample if probabilities of selection are not known.
  26. 26. Probability Sampling Methods Simple Random Sampling • Simple random sampling: the probability of being selected is “known and equal” for all members of the population • Blind Draw Method (e.g. names “placed in a hat” and then drawn randomly) • Random Numbers Method (all items in the sampling frame given numbers, numbers then drawn using table or computer program) • Advantages: • Known and equal chance of selection • Easy method when there is an electronic database
  27. 27. Probability Sampling Methods Simple Random Sampling • Disadvantages: (Overcome with electronic database) • Complete accounting of population needed • Cumbersome to provide unique designations to every population member • Very inefficient when applied to skewed population distribution (over- and under-sampling problems) – this is not “overcome with the use of an electronic database)
  28. 28. A simple random sample • A simple random sample is obtained by choosing elementary units in search a way that each unit in the population has an equal chance of being selected. A simple random sample is free from sampling bias. However, using a random number table to choose the elementary units can be cumbersome. If the sample is to be collected by a person untrained in statistics, then instructions may be misinterpreted and selections may be made improperly.
  29. 29. Probability Sampling Methods Systematic Sampling (A Cluster Method) • Systematic sampling: way to select a probabilitybased sample from a directory or list. This method is at times more efficient than simple random sampling. This is a type of cluster sampling method. • Sampling interval (SI) = population list size (N) divided by a pre-determined sample size (n) • How to draw: 1) calculate SI, 2) select a number between 1 and SI randomly, 3) go to this number as the starting point and the item on the list here is the first in the sample, 4) add SI to the position number of this item and the new position will be the second sampled item, 5) continue this process until desired sample size is reached.
  30. 30. Probability Sampling Methods Systematic Sampling • Advantages: • Known and equal chance of any of the SI “clusters” being selected • not need to designate (assign a number to) every population member, just those early on on the list (unless there is a very large sampling frame). • Less expensive…faster than SRS • Disadvantages: • Small loss in sampling precision • Potential “periodicity” problems
  31. 31. • A systematic random sample is obtained by selecting one unit on a random basis and choosing additional elementary units at evenly spaced intervals until the desired number of units is obtained. For example, there are 100 students in your class. You want a sample of 20 from these 100 and you have their names listed on a piece of paper may be in an alphabetical order. If you choose to use systematic random sampling, divide 100 by 20, you will get 5. Randomly select any number between 1 and five. Suppose the number you have picked is 4, that will be your starting number. So student number 4 has been selected. From there you will select every 5th name until you reach the last one, number one hundred. You will end up with 20 selected students.
  32. 32. Probability Sampling Methods Cluster Sampling • Cluster sampling: method by which the population is divided into groups (clusters), any of which can be considered a representative sample. These clusters are mini-populations and therefore are heterogeneous. Once clusters are established a random draw is done to select one (or more) clusters to represent the population. Area and systematic sampling (discussed earlier) are two common methods. • Area sampling
  33. 33. Probability Sampling Methods Cluster Sampling • Advantages • Economic efficiency … faster and less expensive than SRS • Does not require a list of all members of the universe • Disadvantage: • Cluster specification error…the more homogeneous the cluster chosen, the more imprecise the sample results
  34. 34. Probability Sampling Methods Cluster Sampling – Area Method • Drawing the area sample: • Divide the geo area into sectors (subareas) and give them names/numbers, determine how many sectors are to be sampled (typically a judgment call), randomly select these subareas. Do either a census or a systematic draw within each area. • To determine the total geo area estimate add the counts in the subareas together and multiply this number by the ratio of the total number of subareas divided by number of subareas.
  35. 35. • • A cluster sample A cluster sample is obtained by selecting clusters from the population on the basis of simple random sampling. The sample comprises a census of each random cluster selected. For example, a cluster may be some thing like a village or a school, a state. So you decide all the elementary schools in Newyork State are clusters. You want 20 schools selected. You can use simple or systematic random sampling to select the schools, then every school selected becomes a cluster. If you interest is to interview teachers on their opinion of some new program which has been introduced, then all the teachers in a cluster must be interviewed. Though very economical cluster sampling is very susceptible to sampling bias. Like for the above case, you are likely to get similar responses from teachers in one school due to the fact that they interact with one another.
  36. 36. A two-step area cluster sample (sampling several clusters) is preferable to a one-step (selecting only one cluster) sample unless the clusters are homogeneous
  37. 37. Probability Sampling Methods Stratified Sampling Method • This method is used when the population distribution of items is skewed. It allows us to draw a more representative sample. Hence if there are more of certain type of item in the population the sample has more of this type and if there are fewer of another type, there are fewer in the sample.
  38. 38. • A stratified sample is obtained by independently selecting a separate simple random sample from each population stratum. A population can be divided into different groups may be based on some characteristic or variable like income of education. Like any body with ten years of education will be in group A, between 10 and 20 group B and between 20 and 30 group C. These groups are referred to as strata. You can then randomly select from each stratum a given number of units which may be based on proportion like if group A has 100 persons while group B has 50, and C has 30 you may decide you will take 10% of each. So you end up with 10 from group A, 5 from group B and 3 from group C.
  39. 39. Probability Sampling Methods Stratified Sampling • Stratified sampling: the population is separated into homogeneous groups/segments/strata and a sample is taken from each. The results are then combined to get the picture of the total population. • Sample stratum size determination • Proportional method (stratum share of total sample is stratum share of total population) • Disproportionate method (variances among strata affect sample size for each stratum)
  40. 40. Probability Sampling Methods Stratified Sampling • Advantage: • More accurate overall sample of skewed population…see next slide for WHY • Disadvantage: • More complex sampling plan requiring different sample sizes for each stratum
  41. 41. Why is Stratified Sampling more accurate when there are skewed populations? The less the variance in a group, the smaller the sample size it takes to produce a precise answer. Why? If 99% of the population (low variance) agreed on the choice of brand A, it would be easy to make a precise estimate that the population preferred brand A even with a small sample size. But, if 33% chose brand A, and 23% chose B, and so on (high variance) it would be difficult to make a precise estimate of the population’s preferred brand…it would take a larger sample size….
  42. 42. Why is Stratified Sampling more accurate when there are skewed populations? Continued.. Stratified sampling allows the researcher to allocate a larger sample size to strata with more variance and smaller sample size to strata with less variance. Thus, for the same sample size, more precision is achieved. This is normally accomplished by disproportionate sampling.
  43. 43. Nonprobability Sampling Methods Convenience Sampling Method • Convenience samples: samples drawn at the convenience of the interviewer. People tend to make the selection at familiar locations and to choose respondents who are like themselves. • Error occurs 1) in the form of members of the population who are infrequent or nonusers of that location and 2) who are not typical in the population
  44. 44. Nonprobability Sampling Methods Judgment Sampling Method • Judgment samples: samples that require a judgment or an “educated guess” on the part of the interviewer as to who should represent the population. Also, “judges” (informed individuals) may be asked to suggest who should be in the sample. • Subjectivity enters in here, and certain members of the population will have a smaller or no chance of selection compared to others
  45. 45. Nonprobabilty Sampling Methods Referral and Quota Sampling Methods • Referral samples (snowball samples): samples which require respondents to provide the names of additional respondents • Members of the population who are less known, disliked, or whose opinions conflict with the respondent have a low probability of being selected. • Quota samples: samples that set a specific number of certain types of individuals to be interviewed • Often used to ensure that convenience samples will have desired proportion of different respondent classes