Annals of Library Science and Documentation 42,3; 1995; 96-100.

SAMPLE SIZE DETERMINATION: A COMPARISON OF ATTRIBUTE ANDCONTINUOUS VARIABLE METHODS

UDOFIA ITON UDOFIAEDET E. NKEREUWEMLibrary DepartmentUniversity of CalabarCalabarNigeria

Each of the two methods of sample size determi-nation - the Attribute and the Continuous VariableMethod has its use in the investigation of socialscience problems. The former allows the compu-tation of sample size with reterence to any param-eters of the variable and, therefore, can substitutefor the Continuous Variable Method, but, with andprobable increase in sample size. The later is veryuseful when data are collected in ratio form. How-ever, it demands estimates of dispersion from themean which may be primary purpose of the re-search in the first place. The Attribute Method ishighly recommended for library and informationscience since it can be substituted for ContinuousVariable Method.

INTRODUCTION

Researchers intending to use sampling proceduresfor studying library and information science sub-ject face many and diverse methodological prob-lems. One of these problems is the decision aboutthe sample size such as, the minimum amount ofdata to be collected or whether enough cases avail-able to be statistically valid.

To determine appropriate sample size, one musthave a thorough knowledge of the levels of mea-surement to be used, the hypotheses to be testedand the type of statistical tests most appropriateto the problem. Some knowledge of populationparameters is necessary, for example, percentageof occurrence or the standard deviation as well asdesired level of confidence and desired accuracyfor the sample, expressed as tolerance. Hence, athorough idea of statistical problems and trainingare required to decide which of the methodsnamely, the Attribute and Continuous VariableMethod is more suitable for library and informationscience surveys.

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One of the simple and informal method for deter-mination of sample size is to use the sample sizeused by others studying a similar problem. Forexample, Roscoe [4] states that in behavioral re-search there are few occasions when samplessmaller than 30 and larger than 500 in size can bejustified. While the range from 30 to 500 may ap-pear to be a large one, it does narrow the numberto some extent. Unfortunately, no rationale is givenfor this recommendation, and this view is sharedby Uko [5] and others.

The most frequently used statistical approachesare the Attribute and the Continuous Variable Meth-ods. In this paper a basis for comparison with thetwo approaches is given for calculation of samplesize for library and information science surveys.Convenience alone dictated the comparison of thetwo approaches.

STATISTICAL DETERMINATION OF SAMPLESIZE

To generalize from a sample to a universe or popu-lation, that is, to hypothesize that the mean of thepopulation variable falls within a certain range ofvalues at a certain level of confidence, statisticaltechniques must be used.

In discussions of the Attribute Method, and theContinuous Variable Method, found in the docu-ments on research methodology and statistics, twokey factors are always mentioned [3]. First, theneed to establish the level of confidence and thesecond, the need to establish the degree of accu-racy or tolerance that is required.

The Attribute Method deals with the significanceof proportions and requires an estimate of the per-centage of occurrence of the key variable in thestudy. The Continuous Variable Method requires

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an estimate of the dispersion within the key vari-able, usually, the standard deviation. In both cases,estimates based on prior knowledge are required.The attribute method does permit the selection ofthe largest possible sample size by estimating theoccurrence of the key variable at 50%.

The confidence levels establish the degree of sam-pling error that will be permitted in the study. Forexample, when a confidence level of 95% is used,it is said that the probability is 95%. That meansthat, most of the time, the confidence interval willcontain the true mean and in 5% of the time it willnot. Carpenter and Vasu [1] stated that the mostcommonly used confidence levels were the 95%level and might be used as a standard. The 99%level increases the sample size and the 90% levelis least used. To minimize total error that resultsfrom non-response, inaccuracy of recording re-sponses or copying figures from one file to anotherand so on, the confidence level should be set at aconservatively high level. The degree of accuracyrequired of a sample is translated into statementssuch as, ''The population mean falls with plus orminus 5 units of the sample mean". The researcherestablishes this level of tolerance through inspec-tion of the variable in question and a need for ac-curacy. But, as will be shown, sample size in-creases dramatically for higher level of tolerance.

THE ATTRIBUTE METHOD

The Attribute Method of sample size determina-tion requires an estimate of the proportion of oc-currence of a property or activity in the universe.Dougherty and Heinritz [2] used the example of"books that have been in circulation for fourteendays or more" to illustrate the concept of attribute.In this case, there are only two possible conditions- the case has the attribute or it does not. In effect,there are only two values. Other examples wouldbe books borrowed for five days or more than fivedays; students who use libraries heavily or notheavily; documents that contain ten or more cita-tions and those containing three or less.

A formula for the computation of Attribute samplesize is given by Dougherty and Heinritz [2] andUko [5].

C2F == p(1-p).

t2

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SAMPLE SIZE DETERMINATION

where F ==C2 ==

the sample sizethe z-score squared represent-ing the desired confidence level.the desired tolerance expressedas a fraction or decimalthe estimated percentage occur-rence of the attribute being mea-sured.

p ==

The z-score squared and tolerance squared havebeen used for ease of computation. For those whoare more familiar with the z-score notation, the fol-lowing gives the value of the z-score, and C2, forsix typical confidence levels:

Confidence level Z-score C2

90% 2.58 6.656498% 2.33 5.428997% 2.27 5.152996% 2.05 4.202595% 1.96 3.841690% 1.65 2.7225

The percentage occurrence portion of the formula,p (t-p), has the property of maximizing sample sizeat the 50% level of occurrence for a given toler-ance. Hence, sample size is the same for both p ==40% and p == 60% or p == 30% and p == 70% [2,5].

This, however, eases the problem of deciding onthe criterion variable for sample size calculation ina multivariable study. For example, a hypotheticalstudy might have three variables - age, distanceand rate of visiting. Age could be defined as hav-ing the attribute of "greater than 18 years", distancehaving the attribute of "more than three kilometres"and rate of visiting as, "three or more visits permonth". A problem in some people's mind wouldbe to decide which of the three should be used incalculating sample size. In the Attribute Method,estimating the percentage of, occurrence at 50%would maximize sample size for any variable. If itis necessary to decrease sample size, the choiceshould go to that variable with the percentageclosest to 50%.

If in the above example, p == 65% for age, p == 30%for distance and p == 20% for rate of visiting, samplesize is highest for age because it is closest to 50%.

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UDOFIA ITON UDOFIA and EDET E. NKEREUWEM

Table 1 illustrates the results of the computationat the 95%, 98% and 99% confidence levels withtolerance set at :1:5 for all variables.

The rationale for choosing age as the criterion vari-able rather than distance or rate of visiting is simple.By choosing the greatest sample size, all the other

Table 1

Computed sample sizes for three variables with tolerance set at :1:5%.

95% confidence 98% confidence 99% confidence

Age 65% 350 494 606

Distance 30% 323 456 559

Rate of visiting 20% 246 347 426

Maximum size 50% 384 543 666

variables will be generalizable to the population. Ifthe maximum size is selected, it is not even nec-essary to estimate the percentage occurrence ofany of the other variables. In some cases, the dif-ference between the maximum size and the sizecomputed for a variable may be great enough tochoose the lower of the two. In Table 1, under theheading 95% confidence, there is a difference of34 cases between the maximum size of 384 andthe next highest age, 350. A difference of 34 casesmight be a rationale for choosing the lower figure.This rationale could be .used when the cost of col-lecting data for each case is high. If the cost ofcollecting data is low, choosing the maximumsample size would negate any errors in estimationof the percentage occurrence. This assumes, ofcourse, the same level of tolerance for all variables,but, such an assumption may not be warranted. Ifthe tolerance in Table 1 is changed to 3% for dis-tance, and remains at 5% for both age and rate ofvisiting, the sample size would be more than doubleto 896 and would result In the highest sample size.It is important to bear in mind that this interrela-tionship between tolerance and percentage occur-rence exists.

When the cost of data collection is low, the AttributeMethod is very good to be used, but when the costof data collection is high, estimates of the percent-

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age occurrence on each variable should be made.Standard tables can be constructed for various lev-els of confidence and tolerance with p = 50%.

THE CONTINUOUS VARIABLE METHOD

The Continuous Variable Method is similar to theAttribute Method with the substitution of a mea-sure of dispersion for the estimate of the percent-age occurrence.

The formula used by Uko [5] is typical:

n = where,

n =zq =s =t =

sample sizez-score for the confidence levelstandard deviationtolerance or degree of accuracy.

The Continuous Variable Method is used when thevariables are in the form of a ratio scale. In thiscase, an estimate is needed of the standard de-viation. The standard deviation can be determinedby using an electronic calculator and a small ran-dom sample of 10 or 20 cases from the intendedsampling frame. A sampling frame consists of alist, directory, index, maps and other records, list-ing the population elements from which the sample

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may be drawn. Some commonly used samplingframes are the telephone directory for telephoneinterviews, the staff nominal roll for sampling em-ployees of institutions, and membership lists forassociations and clubs.

To our example, we might estimate the standarddeviations of distance as 0.5 kilometres, age as 5years and rate of visiting as 0.65 visits per month.

SAMPLE SIZE DETERMINATION

Deciding on the tolerance level is also somewhatdifferent than the procedure used in AttributeMethod. A percentage of accuracy was decidedupon in the previous method. In the ContinuousVariable Method, an absolute value for the degreeof accuracy is required such as, ±0.5 kilometresor ± 5 years ot age. The desired degree of accu-racy affects sample size considerably.

Table 2

Variation in sample size given different tolerance and standard deviations for three variables.

Tolerance Confidence Standard Sample(±) level deviation size

Age 4 Years 95% 15.5 583 Years 95% 15.5 1032 Years 95% 15.5 2311 Year 95% 15.5 923

Distance 0.50 Kilometres 95% 1.5 350.25 Kilometres 95% 1.5 1380.20 Kilometres 95% 1.5 2160.10 Kilometres 95% 1.5 864

Visits 0.50 visits 95% 1.05 170.25 visits 95% 1.05 680.20 visits 95% 1.05 1060.10 visits 95% 1.05 435

Table 2 illustrates the differences in sample size,given various hypothetical degrees of accuracy.With the Attribute Method, the sample size of thestudy should be based on the variable that yieldsthe maximum sample size. In the example shownin Table 2, the tolerance of ±0.10 visits should beused as our criterion variable, as this would allowus to generalize to the population for distance aswell as age.

GENERALIZATION

The formulae for calculating the sample size canbe used to solve other aspects such as, tolerance,confidence level or proportion accounted for when

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"n", the sample size, is known. For example, theattribute formula for sample size is given as :

F = C2 p(1-p).t2 [2]

solving for t

...1 C2t = V F P (1 - p)

The tolerance can be determined for a givensample size (F) ,confidence level (C 2), and per-centage of occurrence. Taking the sample size of

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UDOFIA IT ON UDOFIA and EDET E. NKEREUWEM

500, a 95% confidence level, and the 50% occur-rence level, tolerance is

t = V 3.~~6 (.5) (.5)

= ± 4.4%

Tolerances can be computed for various level ofconfidence, sample sizes and confidence rates,using this formula.

Continuous Variable Method can be used to cal-culate either confidence level or tolerance, givenan estimate of standard deviation and a knownsample size. The Continuous Variables samplesize formula is given as :

solving for t

where zq is the z-score, s is the standard devia-tion, and n is the sample size.

From the example in Table 2, where the standarddeviation for age was 15.5 and with a 95% confi-dence level, and a sample size of 500, tolerancecan be calculated thus:

1.96 (15.5)= r500

= ± 1.36 years

It is possible to compute the confidence level givena tolerance, sample size and standard deviation.

In using the Continuous Variable Method with 95%confidence, ±2 units of tolerance, and a standarddeviation of 15.5 yields a sample size of only 231.A calculation of sample size by the Attribute Method

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with 95% confidence, ±5% tolerance, and 50%occurrence would yield a sample size of 384 whichis 153 cases larger.

CONCLUSION

The main intention of the study is to compareAttribute Method with Continuous Variable Method.The Attribute Method is highly recommended forlibrary and information science surveys as it al-lows the computation of sample size with refer-ence to any parameters of the variable and can besubstituted for Continuous Variable Method.

The choice of the 95% level of confidence is arbi-trary in many respects and the estimates of thedesired tolerance or the standard deviation are of-ten equally arbitrary depending on the knowledgethe researcher has of the population parameters.

The ultimate criterion for choosing sample size iscost. If the costs of gathering each unit of informa-tion are high, the method that keeps sample sizeat a minimum can be used and when the costs perunit are low, methods that increase understand-ing, planning, and flexibility are to be considered.It is the author's contention that statistically trainedresearchers will probably continue to use AttributeMethod as previously recommended.

REFERENCES

1. CARPENTER (R L) and VASU (E S). Statisti-cal methods for librarians. 1978. American Li-brary Association; Chicago. p. 39.

2. DOUGHERTY (R M) and HEINRITZ (F J). Sci-entific management of library operations. 1982.The Scarecrow Press; Metuchen. pp.212-216.

3. KERLINGER (F N). Foundations ofbehavioural research. 1986.Holt, Rinehart andWinston; New York. pp. 34-36,148-149.

4. ROSCOE (J T). Fundamental research statis-tics for behavioural science. 1975. Holt,Rinehart and Winston; New York. p. 184.

5. UKO (J U). Basic research method; GDMHandout. 1994. University of Calabar; Cala-bar. pp. 20-24.

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