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The Jonathan Dolhenty Archive |
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PAGE SIX by Jonathan Dolhenty, Ph.D
A population (or universe) is any defined aggregate of objects, persons, or events, the variables used as the basis for classification or measurement being specified. For instance, we may want to survey the attitude of adults over the age of 21 in the state of California towards the legalization of marijuana. Our population (or universe) for our survey would be all adults in the state of California over the age of 21. Of course, if we tried to survey each and every adult over the age of 21 in California, our task would be virtually impossible. How would we allow for adults over 21 who left the state just before or after we took our survey? What would we do about the adults who are moving to the state on a daily basis? How would we ever complete our survey? This is precisely where statistical sampling comes into the picture. A sample is any subaggregate drawn from the population. It is a subset of the population. It is through using these samples, drawn from a defined population, that we can begin and complete a survey even while the population remains unstable.
You probably have seen statistics used during a news broadcast on television. The reporter will state, for instance, that, "according to a survey just completed, 78% of Americans indicated they oppose the legalization of marijuana." Now obviously, whoever did the survey did not talk to each and every American citizen in the country. This is where statistical sampling enters the picture. A subgroup of the population, called the sample, is assumed to approximate the larger group (the population) on whatever is to be studied, such as attitudes toward the legalization of marijuana. Once the sample has been drawn and surveyed, it ceases to be of any interest, since whatever attitude is found is assumed to reflect that of the population, the larger group. There is, of course, some potential error here, and that is allowed for by indicating a measure of error of so-many points. For instance, our reporter above may say, "these results are accurate within an error of plus or minus 3 points." The important particular here is that whatever is found out about the sample is generalized to the defined population. Important Considerations There are some important considerations to be aware of in statistical sampling. The first task of the people involved in any study is to define the population of the study both in terms of numbers involved and the distribution of the characteristic that will be involved in the study. The second task is to obtain a sample that approximates the targeted population and here two conditions must be met:
Equal chance means that every member in the defined population must be given an equal chance of becoming a part of the sample. The sample must be representative of the defined population. If certain portions of the population are excluded from being chosen, the sample becomes biased. Independence means that the selection of one individual for a sample must not be dependent on the selection of another individual. This latter condition is usually not a problem if equal chance is strictly adhered to in drawing the sample. Sample Size First, it needs to be noted that any sample that numbers less than the total population will produce some degree of error. That being said, it follows that precision and accuracy increase as the size of the sample approaches the size of the total population. The purpose of sampling is to get a mini-population that is representative of the larger population. The sample must contain a distribution of the characteristic under study approximately the same as it actually exists in the total population. Therefore, the size of the sample must be large enough to insure the probability of including the extremes in the population. There is a mathematical formula which statisticians can use to help them generate a table for determining sample size from a given population. Methods of Sampling Several methods are available to insure the probability of obtaining a representative sample in a statistical study. We will briefly describe three of them here. Random sampling is the most common and efficient method of obtaining a representative sample if the characteristics under study are assumed to be normally distributed throughout the population. The important consideration in random sampling is that each member of the defined population must be given an equal chance of being selected for the sample. There are several ways to accomplish this. The least time consuming is to use a table of random numbers. This is a randomly generated set of numbers which has no order or structure. First, each member of the population is assigned a number. Then, after the appropriate size of the sample has been determined, you select those members of the population whose numbers occur first as you read down a column or across a row of the table of random numbers. If the population size was 1,000, the numbering would range from 001 to 1,000. The actual sample selection would involve beginning with any three digits and proceeding through the table in an orderly fashion. The justification for using a table of random numbers is that the numbers do not follow any set pattern. Each number has an equal chance of being selected. There are also other ways of obtaining a random sample, including drawing cards from a container with numbers written on them. But using a table of random numbers has proven to be efficient and easy to implement. Stratified random sampling is another method used to draw a sample. This is especially appropriate where variables under study may not be normally distributed in the population. The procedure here is to divide the population into natural groupings prior to sampling. It is hoped that in this way a more representative sample is obtained. Let's go back to our legalization of marijuana example. In an attempt to determine the attitude of adults over 21 in California towards the legalization of marijuana, the age of a person might have something to do with the way he or she feels about legalization. If this is the case, the population could be subdivided according to age groups prior to the sampling. Many surveys attempting to gauge attitudes about the performance of the sitting U.S. president break down the population into, for instance, Democrats, Republicans, and Independents. Then a random sample (hopefully!) is drawn from each subgroup. Cluster sampling involves subdividing the population into clusters or large blocks of individuals prior to sampling. A cluster might be defined as neighborhoods within a city, or rural regions within a geographical area, or fire districts within a county.
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