Determining Sample Size Page 2 - Tarleton

1 Determining Sample SizePage 2 Figure of Means for Of VariabilityThe third criterion, thedegree of variabilityin theattributes being measured refers to the distribution ofattributes in the population. The more heterogeneousa population, the larger the Sample size required toobtain a given level of precision. The less variable(more homogeneous) a population, the smaller thesample size . Note that a proportion of 50% indicatesa greater level of variability than either 20% or 80%.This is because 20% and 80% indicate that a largemajority do not or do, respectively, have the attributeof interest. Because a proportion of .5 indicates themaximum variability in a population, it is often usedin Determining a more conservative Sample size , thatis, the Sample size may be larger than if the truevariability of the population attribute were FOR DETERMININGSAMPLE SIZET here are several approaches to Determining thesample size .

2 These include using a census for smallpopulations, imitating a Sample size of similar studies,using published tables, and applying formulas tocalculate a Sample size . Each strategy is A Census For Small PopulationsOne approach is to use the entire population asthe Sample . Although cost considerations make thisimpossible for large populations, a census is attractivefor small populations ( , 200 or less). A censuseliminates sampling error and provides data on all theindividuals in the population. In addition, some costssuch as questionnaire design and developing thesampling frame are "fixed," that is, they will be thesame for samples of 50 or 200. Finally, virtually theentire population would have to be sampled in smallpopulations to achieve a desirable level of A Sample size Of A Similar StudyAnother approach is to use the same Sample sizeas those of studies similar to the one you reviewing the procedures employed in thesestudies you may run the risk of repeating errors thatwere made in Determining the Sample size for anotherstudy.

3 However, a review of the literature in yourdiscipline can provide guidance about "typical" samplesizes which are Published TablesA third way to determine Sample size is to rely onpublished tables which provide the Sample size for agiven set of criteria. Table 1 and Table 2 presentsample sizes that would be necessary for givencombinations of precision, confidence levels, note two , thesesample sizes reflect the number ofobtainedresponses,and not necessarily the number of surveys mailed orinterviews planned (this number is often increased tocompensate for nonresponse). Second, the samplesizes in Table 2 presume that the attributes beingmeasured are distributed normally or nearly so. Ifthis assumption cannot be met, then the entirepopulation may need to be Formulas To Calculate A Sample SizeAlthough tables can provide a useful guide fordetermining the Sample size , you may need tocalculate the necessary Sample size for a differentcombination of levels of precision, confidence, fourth approach to determiningsample size is the application of one of severalformulas (Equation 5 was used to calculate thesample sizes in Table 1 and Table 2).

4 Determining Sample SizePage 3 Table size for 3%, 5%, 7% and 10%Precision Levels Where Confidence Level is 95% andP=. ofSample size (n) for Precision (e) of:Population 3% 5% 7% 10%500a22214583600a24015286700a255158888 00a26716389900a277166901,000a286169912,0 00714333185953,000811353191974,000870364 194985,000909370196986,000938375197987,0 00959378198998,000976381199999,000989383 2009910,0001,0003852009915,0001,03439020 19920,0001,05339220410025,0001,064394204 10050,0001,087397204100100,0001,09939820 4100>100,0001,111400204100a = Assumption of normal population is poor (Yamane,1967). The entire population should be For Calculating A Sample ForProportionsFor populations that are large, Cochran (1963:75)developed the Equation 1 to yield a representativesample for is valid where n0is the Sample size , Z2is theabscissa of the normal curve that cuts off an area atthe tails (1 - equals the desired confidence level, , 95%)1, e is the desired level of precision, p is theestimated proportion of an attribute that is present inthe population, and q is value for Z isfound in statistical tables which contain the areaunder the normal illustrate, suppose we wish to evaluate a state-Table size for 5%, 7% and 10% PrecisionLevels Where Confidence Level is 95% and P=.

5 OfSample size (n) for Precision (e) of:Population 5% 7% 10%1008167511259678561501108661175122946 4200134101672251441077025015411272275163 1177430017212176325180125773501871297837 5194132804002011358142520713882450212140 82wide Extension program in which farmers wereencouraged to adopt a new practice. Assume there isa large population but that we do not know thevariability in the proportion that will adopt thepractice;therefore,assumep=.5(maximum variability). Furthermore, suppose we desire a 95%confidence level and 5% precision. The resultingsample size is demonstrated in Equation Population Correction For ProportionsIf the population is small then the Sample size canbe reduced slightly. This is because a given samplesize provides proportionately more information for asmall population than for a large population.

6 Thesample size (n0) can be adjusted using Equation n is the Sample size and N is the Sample SizePage 4 Suppose our evaluation of farmers adoption ofthe new practice only affected 2,000 farmers. Thesample size that would now be necessary is shown inEquation you can see, this adjustment (called the finitepopulation correction) can substantially reduce thenecessary Sample size for small Simplified Formula For ProportionsYamane (1967:886) provides a simplified formulato calculate Sample sizes. This formula was used tocalculate the Sample sizes in Tables 2 and 3 and isshown below. A 95% confidence level andP=.5areassumed for Equation n is the Sample size , N is the population size ,and e is the level of precision. When this formula isapplied to the above Sample , we get Equation For Sample size For The MeanThe use of tables and formulas to determinesample size in the above discussion employedproportions that assume a dichotomous response forthe attributes being are twomethods to determine Sample size for variables thatare polytomous or continuous.

7 One method is tocombine responses into two categories and then usea Sample size based on proportion (Smith, 1983).The second method is to use the formula for thesample size for the mean. The formula of the samplesize for the mean is similar to that of the proportion,except for the measure of variability. The formula forthe mean employs 2instead of (p x q), as shown inEquation n0is the Sample size , z is the abscissa of thenormal curve that cuts off an area at the tails, e isthe desired level of precision (in the same unit ofmeasure as the variance), and 2is the variance of anattribute in the disadvantage of the Sample size based on themean is that a "good" estimate of the populationvariance is , an estimate is , the Sample size can varywidely from one attribute to another because each islikely to have a different variance.

8 Because of theseproblems, the Sample size for the proportion isfrequently CONSIDERATIONSIn completing this discussion of determiningsample size , there are three additional issues. First,the above approaches to Determining Sample size haveassumed that a simple random Sample is the samplingdesign. More complex designs, , stratified randomsamples, must take into account the variances ofsubpopulations, strata, or clusters before an estimateof the variability in the population as a whole can consideration with Sample size is thenumber needed for the data analysis. If descriptivestatistics are to be used, , mean, frequencies, thennearly any Sample size will the otherhand, a good size Sample , , 200-500, is needed formultiple regression, analysis of covariance, or log-linear analysis, which might be performed for morerigorous state impact evaluations.

9 The Sample sizeshould be appropriate for the analysis that is addition, an adjustment in the Sample size maybe needed to accommodate a comparative analysis ofsubgroups ( , such as an evaluation of programparticipants with nonparticipants). Sudman (1976)suggests that a minimum of 100 elements is neededfor each major group or subgroup in the Sample andfor each minor subgroup, a Sample of 20 to 50elements is necessary. Similarly, Kish (1965) says that30 to 200 elements are sufficient when the attribute ispresent 20 to 80 percent of the time ( , thedistribution approaches normality).On the otherhand, skewed distributions can result in seriousdepartures from normality even for moderate sizesamples (Kish, 1965:17). Then a larger Sample or acensus is , the Sample size formulas provide thenumber of responses that need to be obtained.

10 Manyresearchers commonly add 10% to the Sample size tocompensate for persons that the researcher is unableDetermining Sample SizePage 5to contact. The Sample size also is often increased by30% to compensate for , thenumber of mailed surveys or planned interviews canbe substantially larger than the number required fora desired level of confidence and The area corresponds to the shaded areas inthe sampling distribution shown in Figure The use of the level of maximum variability(P=.5) in the calculation of the Sample size forthe proportion generally will produce a moreconservative Sample size ( , a larger one) thanwill be calculated by the Sample size of the , W. G. Techniques, 2nd Ed.,New York: John Wiley and Sons, , Glenn D. The Evidence OfExtension Program Impact.