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Autor: anton 15 July 2011
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The concept of sampling
Let us take a very simple example to explain the concept of sampling. Suppose you want to estimate the average age of the students in your class. There are two ways of doing this. The first method is to contact all students in the class, find out their ages, add them up and then divide this by the number of students (the definition of an average). The second method is to select a few students from the class, ask them their ages, add them up and then divide by the number of students you have asked. From this you can make an estimate of the average age of the class. Take another example: suppose you want to find out the average income of families living in a city. You could follow the procedures described above, but imagine the amount of effort and resources required to go to each family in the city to find out their income! You could follow the second method by selecting a few families to become the basis of your inquiry and then, from what you have found out from the few families, make an estimate of the average income of families in the city. A further example would be the outcome of an election: the result is decided after voting on Election Day, but predictions about the outcome are usually made on the basis of opinion polls. These polls are based upon a very small group of people who are questioned about their voting preferences. On the basis of these results, a prediction is made about the outcome.
Sampling, therefore, is the process of selecting a few (a sample) from a bigger group (the sampling population) to become the basis for estimating or predicting the prevalence of an unknown piece of information, situation or outcome regarding the bigger group. A sample is a subgroup of the population you are interested in. See Figure 12.1.
Figure 12.1 the concept of sampling
This process of selecting a sample from the total population has advantages and disadvantages. The advantages are that it saves time as well as financial and human resources. However, the disadvantage is that you do not find out the information about the populationÐ²Ð‚â„¢s characteristics of interest to you but only estimate or predict them. Hence, the possibility of an error in your estimation exists.
Sampling is thus a trade-off between certain gains and losses. While on the one hand you save time and resources, on the other hand you may compromise the level of accuracy in your findings. Through sampling you only make an estimate about the actual situation prevalent in the total population from which the sample is drawn. If you ascertain a piece of information from the total sampling population, and if your method of inquiry is correct, your findings should be reasonably accurate. However, if you select a sample and use this as the basis from which to estimate the situation in the total population, an error is possible. Tolerance of this possibility of error is an important consideration in selecting a sample.
The concept of sampling in qualitative research:-
In qualitative research the issue of sampling has little significance as the main aim of most qualitative inquiries is either to explore or describe the diversity in a situation, phenomenon or issue. Qualitative research does not make an attempt to either quantify or determine the extent of this diversity. You can select even one individual as your sample and describe whatever the aim of your inquiry is. A study based upon the information obtained from one individual, or undertaken to describe one event or situation is perfectly valid. In qualitative research, to explore the diversity, you need to reach what is known as saturation point in terms of your findings; for example, you go on interviewing or collecting information as long as you keep discovering new information. When you find that you are not obtaining any new data or the new information is negligible, you are assumed to have reached saturation point. Some researchers prefer to select a sample using non-probability designs and to collect data till they have reached saturation point. Keep in mind that saturation point is a subjective judgment which you, as a researcher, decide.
Let us, again, consider the examples used above. Our main aims are to find out the average age of the class, the average income of the families living in the city, and the likely election outcome for a particular state or country. Let us assume that we adopt the second methodÐ²Ð‚â€that is, we select a few students, families or electorates to achieve these aims. In this process there are a number of aspects:
Ð²Ð‚Ñž The class, families living the city or electorates from which you select a few students, families, electors to question in order to find answers to your research questions are called the population or study population, and are usually denoted by the letter (N).
Ð²Ð‚Ñž The small group of students, families or electors from whom you collect the required information to estimate the average age of the class, average income or the election outcome is called the sample.
Ð²Ð‚Ñž The number of students, families or electors from whom you obtain the required information is called the sample size and is usually denoted by the letter (n).
Ð²Ð‚Ñž The way you select students, families or electors is called the sampling design or strategy.
Ð²Ð‚Ñž Each student, family or elector that becomes the basis for selecting your sample is called the sampling unit or sampling element.
Ð²Ð‚Ñž A list identifying each student, family or elector in the study population is called the sampling frame. If all elements in a sampling population cannot be individually identified, you cannot have a sampling frame for that study population.
Ð²Ð‚Ñž Your findings based on the information obtained from your respondents (sample) are called sample statistics. Examples of this information are the average age of students (calculated from the information obtained from those students who responded to your question on age); the average income of a family (calculated from the relevant information obtained from those families who participated in your study); and the expected outcome (predicted on the basis of the information obtained from those who expressed their intention in voting). Your sample statistics become the basis of estimating the prevalence of the above characteristics in the study population.
Ð²Ð‚Ñž Your main aim is to find answers to your research questions in the study population, not in the sample you collected information from. In the examples we have been talking about, our aims are to find Out the average age of students in a class, the average income of families living in a city and the expected outcome of the election respectively. From sample statistics we make an estimate of the answers to our research questions in the study population. The estimates arrived at from sample statistics are called population parameters or the population mean.
Ð²Ð‚Ñž As mentioned earlier, in qualitative research, when you reach a stage where no new information is coming from your respondents, this is called saturation point.
Principles of sampling:-
The theory of sampling is guided by three principles. To effectively explain these, we will take an extremely simple example. Suppose there are four individuals: A, B, C and D. A is 18 years of age, B is 20, C is 23 and D is 25. As you know their ages, you can find out (calculate) their average age by simply adding 18 + 20 + 23 + 25 = 86 and dividing by 4. This gives the average age of A, B, C and D as 21.5 years.
Now let us suppose that you want to select a sample of two individuals to make an estimate of the average age of the four individuals. If you adopt the theory of probability, we can have six possible combinations of two: A and B; A and C; A and D; B and C; B and D; and C and D. Let us take each of these pairs to calculate the average age of the sample:
1. A + B = 18 + 20 = 38 / 2 = 19.0 years;
2. A + C = 18 + 23 = 41 / 2 = 20.5years;
3. A + D = 18 + 25 = 43 / 2 = 21.5years;
4. B + C = 20+ 23 = 43 / 2 = 21.5years;
5. B + D = 20 + 25 = 45 / 2 = 22.5years;
6. C + D = 23 + 25 = 48 / 2 = 24.0 years.
Notice that in most cases the average age calculated on the basis of these samples of two (sample statistics) is different. Now compare these sample statistics with the average of all four individualsÐ²Ð‚â€the population mean (population parameter) of 21.5 years. Out of a total of six possible sample combinations, only in the case of two is there no difference between the sample statistics and the population mean. Where there is a difference, this is attributed to the sample and is known as sampling error. Again, the size of the sampling error varies markedly. Let us consider the difference in the sample statistics and the population mean for each of the six samples (Table 12.1).
This analysis suggests a very important principle of sampling:
Principle oneÐ²Ð‚â€in a majority of cases of sampling there will be a difference between the sample statistics and the true population mean, which is attributable to the selection of the units in the sample
To understand the second principle, let us continue with the above example, but instead of a sample of two individuals take a sample of three. There are four possible combinations of three that can be drawn.
1. A + B + C = 18 + 20 + 23 = 61/3 = 20.33years;
2. A + B + D = 18 + 20 + 25 = 63/3 = 21.00 years;
3. A + C + D = 18 + 23 + 25 = 66/3 = 22.00 years;
4. B + C + D = 20 + 23 + 25 = 68/3 = 22.67 years.
Now, let us compare the difference between the sample statistics and the populations mean (Table 12.2).
Compare the difference between the differences calculated in Table 12.1 and Table 12.2. In Table 12.1 the difference between the sample statistics and the population mean lies between Ð²Ð‚â€2.5 and +2.5 years, whereas in the second, it is between Ð²Ð‚â€1.17 and + 1.17 years. The gap between the sample statistics and the population mean is reduced in Table 12.2. This reduction is attributed to the increase in the sample size. This, therefore, leads to the second principle:
Principle two Ð²Ð‚â€œ The greater the sample size, the more accurate will be the estimate of the true population mean.
The third principle of sampling is particularly important as a number of sampling strategies, such as stratified and cluster sampling, are based on it. To understand this principle, let us continue with the same example but use slightly different data. Suppose the ages of four individuals are markedly different: A = 18, B 26, C = 32 and D 40. In other words, we are visualizing a population where the individuals with respect to ageÐ²Ð‚â€the variable we are interested inÐ²Ð‚â€are markedly different.
Let us follow the same procedure, selecting samples of two individuals at a time and then three. If we work through the same procedures (described above) we will find that the difference in the average age in the case of samples of two ranges between Ð²Ð‚â€7.00 and + 7.00 years and in the case of the sample of three ranges between Ð²Ð‚â€3.67 and +3.67. In both cases the range of the difference is greater than previously calculated. This is attributable to the greater difference in the ages of the four individualsÐ²Ð‚â€the sampling population. In other words, the sampling population is more heterogeneous in regard to age.
Principle three Ð²Ð‚â€œ The greater the difference in the variable under study in a population for a given sample size, the greater will be the difference between the sample statistics and the true population mean.
These principles are crucial to keep in mind when you are determining the sample size needed for a particular level of accuracy, and in selecting the sampling strategy best suited to your study.
Factors affecting the inferences drawn from a sample:-
The above principles suggest that two factors may influence the degree of certainty about the inferences drawn from a sample:
1. The size of the sampleÐ²Ð‚â€findings based upon larger samples have more certainty than those based on smaller ones. As a rule, the larger the sample size, the more accurate will be the findings.
2. The extent of variation in the sampling populationÐ²Ð‚â€the greater the variation in the study population with respect to the characteristics under study for a given sample size, the greater will be the uncertainty. (In technical terms, the greater the standard deviation, the higher will be the standard error for a given sample size in your estimates.) If a population is homogeneous with respect to the characteristics under study, a small sample can provide a reasonably good estimate, but if it is heterogeneous, you need to select a larger sample to obtain the same level of accuracy. Of course, if all the elements in a population are identical, then the selection of even one will provide an absolutely accurate estimate. As a rule, the higher the variation with respect to the characteristics under study in the study population, the greater will be the uncertainty for a given sample size.
Aims in selecting a sample:-
The aims in selecting a sample are to:
Ð²Ð‚Ñž Achieve maximum precision in your estimates within a given sample size;
Ð²Ð‚Ñž Avoid bias in the selection of your sample.
Bias in the selection of a sample can occur if:
Ð²Ð‚Ñž sampling is done by a non-random methodÐ²Ð‚â€that is, if the selection is consciously or unconsciously influenced by human choice;
Ð²Ð‚Ñž the sampling frameÐ²Ð‚â€list, index or other population recordsÐ²Ð‚â€which serves as the basis of selection, does not cover the sampling population accurately and completely;
Ð²Ð‚Ñž a section of a sampling population is impossible to find or refuses to cooperate.
Types of sampling:-
The various sampling strategies can be categorized as follows (Figure 12.2):
Ð²Ð‚Ñž Random/probability sampling designs;
Ð²Ð‚Ñž Non n random/non n probability sampling designs;
Ð²Ð‚Ñž MixedÐ²Ð‚â„¢ sampling designs
To understand these designs, we will discuss each individually.
Random/probability sampling designs:-
For a sampling design to be called a random or probability sample, it is imperative that each element in the population has an equal and independent chance of selection in the sample. Equal implies that the probability of selection of each element in the population is the same; that is, the choice of an element in the sample is not influenced by other considerations such as personal preference. The concept of independence means that the choice of one element is not dependent upon the choice of another element in the sampling; that is, the selection or rejection of one element does not affect the inclusion or exclusion of another. To explain these concepts let us return to our example of the class.
Suppose there are 80 students in the class. Assume 20 of these refuse to participate in your study. You want the entire population of 80 students in your study but as 20 refuse to participate, you can only use a sample of 60 students. The 20 students who refuse to participate could have strong feelings about the issues you wish to explore, but your findings will not reflect their opinions. Their exclusion from your study means that each of the 80 students does not have an equal chance of selection. Therefore, your sample does not represent the total class.
The same could apply to a community. In a community, in addition to the refusal to participate, let us assume that you are unable to identify all the residents living in the community. If a significant proportion of people cannot be included in the sampling population because they either cannot be identified or refuse to participate, then any sample drawn will not give each element in the sampling population an equal chance of being selected in the sample. Hence, the sample will not be representative of the total community.
To understand the concept of an independent chance of selection, let us assume that there are five students in the class who are extremely close friends. If one of them is selected but refuses to participate because the other four are not chosen, and you are therefore forced to select either the five or none, then your sample will not be considered an independent sample since the selection of one is dependent upon the selection of others. The same could happen in the community where a small group says that either all of them or none of them will participate in the study. In these situations where you are forced either to include or to exclude a part of the sampling population, the sample is not considered to be independent, and hence is not representative of the sampling population.
A sample can only be considered a random/probability sample and therefore representative of the population under study if both these conditions are met. If not, bias can be introduced into the study.
There are two main advantages of random/probability samples:
Ð²Ð‚Ñž As they represent the total sampling population, the inferences drawn from such samples can be generalized to the total sampling population.
Ð²Ð‚Ñž Some statistical tests based upon the theory of probability can be applied only to data collected from random samples. Some of these tests are important for establishing conclusive correlations
Methods of drawing a random sample:-
Of the methods that you can adopt to select a random sample the three most common are:
Ð²Ð‚Ñž The fishbowl drawÐ²Ð‚â€if your total population is small, an easy procedure is to number each element using separate slips of paper for each element, put all the slips into a box, and then pick them out one by one without looking, until the number of slips selected equals the sample size you decided upon. This method is used in some lotteries.
Ð²Ð‚Ñž Computer programÐ²Ð‚â€there are a number of programs that can help you to select a random sample.
Ð²Ð‚Ñž a table of random numbersÐ²Ð‚â€most books on research methodology and statistics include a table of randomly generated numbers in their appendices (see, for example, Table 12.3). You can select your sample using these tables according to the procedure described in Figure 12.3. The procedure for selecting a sample using a table of random numbers is as follows:
Figure 12.3 the procedure for using a table of random numbers
Step 1 Identify the total number of elements in the study population, for example 50, 100, 430, 795 or 1265. The total number of elements in a study population may run up to four or more digits ( if your total sampling population is 9 or less, it is one digit; if it is 99 or less, it is two digits.
Step 2 Number each element starting from 1.
Step 3 If the table for random numbers is on more than one page; choose the starting page by a random procedure. Again select a column or row that will be your starting point with a random procedure and proceed from there in a predetermined direction.
Step 4 Corresponding to a number of digits to which the total population runs; select the same number, randomly, of columns or rows of digits from the table.
Step 5 Decide on your sample size
Step 6 Select the required number of elements for your sample from the table. If you happen to select the same number twice, discard it and go to the next. This can happen as the table for random numbers is generated by sampling with replacement.
Let us take an example to illustrate the use of Table 12.3 for random numbers. Let us assume that your sampling population consists of 256 individuals. Number each individual, from 1 to 256. Randomly select the starting page, set of column (1 to 10) or row from the table and then identify three columns or rows of numbers.
Suppose you identify the ninth column of numbers and the last three digits of this column (underlined). Assume that you are selecting 10 per cent of the total population as your sample (25 elements). Lt us go through the numbers underlined in the ninth set of columns. The first number is 049 which is below 256 (total population); hence, the 49th element becomes a part of your sample. The second number, 319, is more than the total elements in your population (256); hence, you cannot accept the 319th element in the sample. The same applies to the next element, 758, and indeed the next five elements, 589, 507, 483, 487 and 540. After 540 is 232, and as this number is within the sampling frame, it can be accepted as a part of the sample. Similarly, if you follow down the same three digits in the same column, you select 052, 029, 065, 246 and 161, before you come to the element 029 again. As the 27th element has already been selected, go to the next number, and so on until 25 elements have been chosen. Once you have reached the end of a column, you can either move to the next set of columns or randomly select another one in order to continue the process of selection. For example the 25 elements shown in Table 12.4 are selected from the ninth, tenth and second columns of Table 12.3:
Different systems of drawing a random sample:-
There are two ways of selecting a random sample:
Ð²Ð‚Ñž Sampling without replacement;
Ð²Ð‚Ñž Sampling with replacement.
Suppose you want to select a sample of 20 students out of a total of 80. The first student is selected out of the total class, and so the probability of selection for the first student is 1/80. When you select the second student there are only 79 left in the class and the probability of selection for the second student is not 1/80 but 1/79. The probability of selecting the next student is 1/78. By the time you select the 20th student, the probability of his or her selection is 1/61. This type of sampling is called sampling without replacement. But this is contrary to our basic definition of randomization; that is, each element has an equal and independent chance of selection. In the second system, called sampling with replacement, the selected element is replaced in the sampling population and if it is selected again, it is discarded and the next one is selected. If the sampling population is fairly large, the probability of selecting the same element twice is fairly remote.
Specific random/probability sampling designs:-
There are three commonly used types of random sampling design.
Ð²Ð‚Ñž Simple random sampling (SRS)Ð²Ð‚â€the most commonly used method of selecting a probability sample. In line with the definition of randomization, whereby each element in the population is given an equal and independent chance of selection, a simple random sample is selected by the procedure presented in Figure 12.4.
Step 1 Identify by a number all elements or sampling, units in the population
Step 2 Decide on the sample size (n)
Step 3 Select (n) using either the fishbowl draw, the table of random numbers or a computer program.
To illustrate, let us again take our example of the class. There are 80 students in the class, and so the first step is to identify each student by a number from 1 to 80. Suppose you decide to select a sample of 20 using the simple random sampling technique. Use the fishbowl draw, the table for random numbers or a computer program to select the 20 students. These 20 students become the basis of your inquiry.
Ð²Ð‚Ñž Stratified random samplingÐ²Ð‚â€as discussed, the accuracy of your estimate largely depends on the extent of variability or heterogeneity of the study population with respect to the characteristics that have a strong correlation with what you are trying to ascertain (Principle three), It follows, therefore, that if the heterogeneity in the population can be reduced by some means for a given sample size you can achieve greater accuracy in your estimate. Stratified random sampling is based upon this logic.
In stratified random sampling the researcher attempts to stratify the population in such a way that the population within a stratum is homogeneous with respect to the characteristic on the basis of which it is being stratified. It is important that the characteristics chosen as the basis of stratification are clearly identifiable in the study population. For example, it is much easier to stratify a population on the basis of gender than on the basis of age, income or attitude. It is also important for the characteristic that becomes the basis of stratification to be related to the main variable that you are exploring. Once the sampling population has been separated into non-overlapping groups you select the required number of elements from each stratum, using the simple random sampling technique. There are two types of stratified sampling: proportionate and disproportionate stratified sampling. With proportionate stratified sampling, the number of elements from each stratum in relation to its proportion in the total population is selected, whereas in disproportionate stratified sampling, consideration is not given to the size of the stratum. The procedure for selecting a stratified sample is schematically presented in Figure 12.5.
Ð²Ð‚Ñž Cluster samplingÐ²Ð‚â€simple random and stratified sampling techniques are based on a researcherÐ²Ð‚â„¢s ability to identify each element in a population. It is easy to do this if the total sampling population is small, but if the population is large, as in the case of a city, state or country, it becomes difficult and expensive to identify each sampling unit. In such cases the use of cluster sampling is more appropriate.
Cluster sampling is based on the ability of the researcher to divide the sampling population into groups, called clusters, and then to select
Step 1 Identify all elements or sampling units in the sampling population
Step 2 Decide upon the different strata (k) into which you want to stratify the population
Step 3 Place each element into the appropriate stratum
Step 4 Number every element in each stratum separately.
Step 5 Decide the total sample size (n)
Step 6 Decide whether you want to select proportionate or disproportionate stratified sampling and follow the steps below.
Elements within each cluster, using the SRS technique clusters can be formed on the basis of geographical proximity or a common characteristic that has a correlation with the main variable of the study (as in stratified sampling). Depending on the level of clustering, sometimes sampling may be done at different levels. These levels constitute the different stages (single, double or multi) of clustering, which will be explained later.
Imagine you want to investigate the attitude of post-secondary students in Australia towards problems in higher education in the country. Higher education institutions are in every state and territory of Australia. In addition, there are different types of institutions, for example universities, universities of technology, colleges of advanced education and technical and further education (TAFE) colleges (Figure 12.6). Within each institution various courses are offered both at undergraduate and postgraduate levels. Each academic course could take three to four years. You can imagine the magnitude of the task. In such situations cluster sampling is extremely useful in selecting a random sample.
The first level of cluster sampling could be at the state or territory level. Clusters could be grouped according to similar characteristics that ensure their comparability in terms of student population. If this is not easy, you may decide to select all the states and territories and then select a sample at the institutional level. For example, with a simple random technique, one institution from each category within each state could be selected (one university, one university of technology, one college of advanced education and one TAFE college). This is based upon the assumption that institutions within a category are fairly similar with regards to student profile. Then, within an institution on a random basis, one or more academic programs could be selected, depending on resources. Within each study program selected, students studying in a particular year could be selected. Further selection of a proportion of students could then occur. The process of selecting a sample in this manner is called multi-stage cluster sampling.
Non-random/non-probability sampling designs:-
Non-probability sampling designs do not follow the theory of probability in the choice of elements from the sampling population. Non-probability sampling designs are used when the number of elements in a population is either unknown or cannot be individually identified. In such situations the selection of elements is dependent upon other considerations. There are four non-random designs, each based on a different consideration, which are commonly used in qualitative and quantitative research:
Ð²Ð‚Ñž Quota sampling;
Ð²Ð‚Ñž Accidental sampling;
Ð²Ð‚Ñž Judgmental or purpose sampling;
Ð²Ð‚Ñž Snowball sampling
The main consideration directing quota sampling is the researcherÐ²Ð‚â„¢s ease of access to the sample population. In addition to convenience, s/he is guided by some visible characteristic, such as gender or race, of the study population that is of interest to him/her. The sample is selected from a location convenient to the researcher, and whenever a person with this visible relevant characteristic is seen that person is asked to participate in the study. The process continues until the researcher has been able to contact the required number of respondents (quota).
Let-us suppose that you want to select a sample of 20 male students in order to find out the average age of the male students in your class. You decide to stand at the entrance to the classroom, as this is convenient, and whenever a male student enters the classroom, you ask his age. This process continues until you have asked 20 students their age. Alternatively, you might want to find out about the attitudes of Aboriginal and Torres Strait Islander students towards the facilities provided to them in your university. Stand at a convenient location and whenever you see such a student, collect the required information through whatever method you have adopted for the study.
There are advantages and disadvantages with this design. The advantages are: it is the least expensive way of selecting a sample; you do not need any information, such as a sampling frame, the total number of elements, their location, or other information about the sampling population; and it guarantees the inclusion of the type of people you need. The disadvantages are: as the resulting sample is not a probability one, the findings cannot be generalized to the total sampling population; and the most accessible individuals might have characteristics that are unique to them and hence might not be truly representative of the total sampling population.
Accidental sampling is also based upon convenience in accessing the sampling population. Whereas quota sampling attempts to include people possessing an obvious/visible characteristic, accidental sampling makes no such attempt.
This method of sampling is common among market research and newspaper reporters. It has more or less the same advantages and disadvantages as quota sampling. As you are not guided by any obvious characteristics, some people contacted may not have the required information.
Judgmental or purposive sampling:-
The primary consideration in purposive sampling is the judgment of the researcher as to who can provide the best information to achieve the objectives of the study. The researcher only goes to those people who in his/her opinion are likely to have the required information and be willing to share it.
This type of sampling is extremely useful when you want to construct a historical reality, describe a phenomenon or develop something about which only a little is known.
Snowball sampling is the process of selecting a sample using networks. To start with, a few individuals in a group or organization are selected and the required information is collected from them. They are then asked to identify other people in the group or organization, and the people selected by them become a part of the sample. Information is collected from them, and then these people are asked to identify other members of the group and, in turn, those identified become the basis of further data collection (Figure 12.7). This process is continued until the required number or a saturation point has been reached, in terms of the information being sought.
This sampling technique is useful if you know little about the group or organization you wish to study, as you need only to make contact with a few individuals, who can then direct you to the other members of the group. This method of selecting a sample is useful for studying communication patterns, decision making or diffusion of knowledge within a group. There are disadvantages to this technique, however. The choice of the entire sample rests upon the choice of individuals at the first stage. If they belong to a particular faction or have strong biases, the study may be biased. Also, it is difficult to use this technique when the sample becomes fairly large.
Ð²Ð‚?MixedÐ²Ð‚â„¢ sampling designs
Systematic sampling design:-
Systematic sampling has been classified under the Ð²Ð‚?mixedÐ²Ð‚â„¢ sampling category because it has the characteristics of both random and non-random sampling designs.
In systematic sampling the sampling frame is first divided into a number of segments called intervals. Then, from the first interval, using the SRS technique, one element is selected. The selection of subsequent elements from other intervals is dependent upon the order of the element selected in the first interval, if in the first interval it is the fifth element, the fifth element of each subsequent interval will be chosen. Notice that from the first interval the choice of an element is on a random basis, but the choice of the elements from subsequent intervals is dependent upon the choice from the first, and hence cannot be classified as a random sample. For this reason it has been classified here as Ð²Ð‚?mixedÐ²Ð‚â„¢ sampling. The procedure used in systematic sampling is presented in Figure 12.8.
Figure 12.8 the procedure for selecting a systematic sample
Step 1 Prepare a list of all the elements in the study population (N).
Step 2 Decide on the sample size (n)
Step 3 Determine the width of the interval (k) =__total population__
Step 4 Using the SRS; select an element from the first interval (nth order)
Step 5 Select the same order element from each subsequent interval.
Although the general procedure for selecting a sample by the systematic sampling technique is described above, one can deviate from it by selecting a different element from each interval with the SRS technique. By adopting this, systematic sampling can be classified under probability sampling designs.
To select a random sample one must have a sampling frame (Figure 12.9). Sometimes this is impossible or obtaining one may be too expensive. However, in real life there are situations when a kind of sampling frame exists, for example records of clients in an agency, enrolment lists of students in a school or university, electoral lists of people living in an area, or records of the staff employed in an organization. All these can be used as a sampling frame to select a sample with the systematic sampling technique. This convenience of having a Ð²Ð‚?ready madeÐ²Ð‚â„¢ sampling frame may be at a price: in some cases it may not truly be a random listing. Mostly these lists are in alphabetical order, based upon a number assigned to a case, or arranged in a way that is convenient to the users of the records. If the Ð²Ð‚?width of an intervalÐ²Ð‚â„¢ is large, say, 1 in 30 cases, and if the cases are arranged in alphabetical order, you could preclude some whose surnames start with the same letter or some adjoining letter may not be included at all.
Suppose there are 50 students in a class and you want to select 10 students using the systematic sampling technique. The first step is to determine the width of the interval (50/10 = 5). This means that from every five you need to select one element. Using the SRS technique, from the first interval (1Ð²Ð‚â€5 elements), select one of the elements suppose you selected the third element. From the rest of the intervals you would select every third element.
The calculation of sample size:-
Ð²Ð‚?How big a sample should I select?Ð²Ð‚â„¢ Ð²Ð‚?What should be my sample size?Ð²Ð‚â„¢ and Ð²Ð‚?How many cases do I need?Ð²Ð‚â„¢ These are the most common questions asked. Basically, it depends on what you want to do with the findings and what type of relationships you want to establish. Your purpose in undertaking research is the main determinant of the level of accuracy required in the results, and this level of accuracy is an important determinant of sample size. However, in qualitative research, as the main focus is to explore or describe a situation, issue, process or phenomenon, the question of sample size is less important. You usually collect data till you think you have reached saturation point in terms of discovering new information. Once you think you are not getting much new data from your respondents, you stop collecting further information. Of course, the diversity or heterogeneity in what you are trying to find out about plays an important role in how fast you will reach saturation point. And remember: the greater the heterogeneity or diversity in what you are t7ying to find out about, the greater the number of respondents you need to contact to reach saturation point. In determining the size of your sample for quantitative studies and in particular for cause-and effect studies, you need to consider the following:
Ð²Ð‚Ñž At what level of confidence do you want to test your results, findings or hypotheses?
Ð²Ð‚Ñž With what degree of accuracy do you wish to estimate the population parameters?
Ð²Ð‚Ñž What is the estimated level of variation (standard deviation), with respect to the main variable you are studying, in the study population?
Answering these questions is necessary regardless of whether you intend to determine the sample size yourself or have an expert do it for you. The size of the sample is important for testing a hypothesis or establishing an association, but for other studies the general rule is the larger the sample size, the more accurate will be your estimates. In practice, your budget determines the size of your sample. Your skills in selecting a sample, within the constraints of your budget, lie in the way you select your elements so that they effectively and adequately represent your sampling population.
To illustrate this procedure let us take the example of a class. Suppose you want to find out the average age of the students within an accuracy of 0.5 of a year; that is, you can tolerate an error of half a year on either side of the true average age. Let us also assume that you want to find the average age within half a year of accuracy at 95 per cent confidence level; that is, you want to be 95 per cent confident about your findings.
The formula (from statistics) for determining the confidence limits is:
There is only one unknown quantity in the above equation, that is a. Now the main problem is to find the value of a without having to collect data. This is the biggest problem in estimating the sample size. Because of this it is important to know as much as possible about the study population.
The value of a can be found by one of the following:
2. Consulting an expert;
3. Obtaining the value of a from previous comparable studies; or
4. Carrying out a pilot study to calculate the value
Hence, to determine the average age of the class at a level of 95 per cent accuracy (assuming a = 1 year) with 1/2 year of error, a sample of at least 16 students is necessary.
Now assume that instead of 95 per cent, you want to be 99 per cent confident about the estimated age, tolerating an error of 1/2 year.
Hence, if you want to be 99 per cent confident and are willing to tolerate an error of 1/2 year, you need to select a sample of 27 students. Similarly, you can calculate the sample size with varying values of a. Remember the golden rule: the greater the sample size, the more accurately your findings will reflect the Ð²Ð‚?trueÐ²Ð‚â„¢ picture.