Applied Biostatistics Essay
Article
An Update on Using the Range to Estimate p When Determining Sample Sizes
George Steven Rhiel and Edward Markowski Department of Information Technology and Decision
Sciences, Strome College of Business, Old Dominion
University, Norfolk, VA, USA
Abstract
In this research, we develop a strategy for using a range estimator of s when determining a sample size for estimating a mean. Previous research by Rhiel is
extended to provide dn values for use in calculating a range estimate of s when working with sampling frames up to size 1,000,000. This allows the use of the range
estimator of s with ‘‘big data.’’ A strategy is presented for using the range estimator of s for determining sample sizes based on the dn values developed in this study.
Keywords
Range estimator of s, sampling, sampling frame, sample size, standard deviation, standardized mean range
Introduction
In psychology, it is common to conduct research using survey sampling to collect data. In the process of conducting the survey, it is important to determine an appropriate sample size. If the objective is to estimate a mean, the researcher needs to decide on the maximum allowable error (e) in the estimate and the level of confidence that e is the maximum error. Also, the researcher must obtain either the population standard deviation (s) or an estimate of s. Given this information and a Z score determined from the confidence level, the sample size can be calculated using the following formula.
n ¼ Z 2�2
e2
Psychological Reports
2017, Vol. 120(2) 319–331
! The Author(s) 2017
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DOI: 10.1177/0033294116687311
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Corresponding Author:
George Steven Rhiel, Department of Information Technology and Decision Sciences, Strome College of
Business, Old Dominion University, Norfolk, VA 23529, USA.
Email: grhiel@odu.edu
A major challenge in determining an appropriate sample size is in obtaining the population standard deviation (s). An estimate of s can be obtained from a previous study involving the same or similar population or from a pilot study. These approaches for estimating s may have limitations. For example, when data from previous studies are not available or conducting a pilot study is not desired or is restrictive because it increases the time or cost of the research, an alternative approach may be needed.
Another technique for estimating the standard deviation when determining a sample size is to divide the range of the population or sampling frame by some factor. This can be accomplished without data collection if the range of the population or sampling frame is known or can be approximated. For example, if you are estimating the mean cost someone would be willing pay for specialized counseling, in many cases you may know the range of the cost or you may specify a range for the subjects to choose from in your survey instrument.
Several authors have suggested an appropriate factor to divide into the range to estimate s when determining a sample size. Black (2010) and Anderson, Sweeney, and Williams (2016) suggest dividing the range by 4 to estimate s. In the PASS Sample Size Software Documentation (NCSS, 2014) a range divisor of 4 is used to estimate s, as well, when determining a sample size. Others suggest dividing the range by 6 (see Daniel and Terrell (1995); Berenson, Levine, and Krebiel, 2013). Cochran (1977) suggests dividing the range by 2 if the population is V-shaped, by 3.47 if it is rectangular, by 4.22 if it is shaped like a right triangle, and by 4.88 if it is shaped like an isosceles triangle. Rhiel (1989) developed standardized mean ranges which could be divided into the range of a sampling frame to estimate s for determining sample sizes. Browne (2001) provides range divisors that give you a 95% chance of getting an estimate greater than or equal to s.
There are limitations to the above suggestions. Dividing the range of the population by 4 or 6 to estimate s relies on the concept that four standard deviations encompass 95.4% of the normal distribution and six standard devi- ations encompass 99.7% of the normal distribution. This requires that the popu- lation or sampling frame be normal and of sufficient size to have four or six standard deviations reasonably encompass the range. Cochran’s research is limited because of the types of distributions used to determine the divisors. The distributions are severely truncated and nonstandard. In Rhiel’s research, the use of the standardized mean range to estimate s from the sampling frame range is limited because the size of the sampling frames used in his study were 2000 or less. Browne’s research provides estimates of s and sample sizes that are large enough, but does not address the issue of situations where s is too large, resulting in corresponding sample sizes that are too large.
Traditionally, many survey studies have involved sampling from phone books or other extensive lists. Today, large lists of populations may be found online. In these cases, the sampling frame may be much larger than that found in Rhiel’s
320 Psychological Reports 120(2)
research (N¼ 2000). With today’s computerized survey research, panels consist- ing of individuals who are willing to participate in surveys are often much larger than 2000 people. One example of this is provided by the company Data Intelligence. Greenbook Market Research Directory (2008) reported that one of the Data Intelligence panels consists of 400,000 households within 25 miles of several malls for purposes of conducting surveys. Another example is from a company called Precision Sample (2011). They maintain panels that exceed 2,800,000 participants.
The purpose of this study is to determine standardized mean ranges for sampling frame sizes up to 1,000,000 for various shaped distributions. This will facilitate the use of range estimates of s for determining sample sizes when sampling from the large sampling frames that are available in today’s computerized world. Using these results and the historical range divisors, com- parisons are made among the different techniques for determining sample sizes. From this, a clear strategy for using range divisors to estimate s to determine sample sizes will emerge.
Methodology
Computer simulation is used to generate standardized mean ranges (dn) for sampling frame sizes from 500 to 1,000,000 for a variety of distribution shapes. Once the dn values are simulated, a comparison is made among the dn values in this study and the historical range divisors. Further, the effect of using the various range divisors on sample sizes is investigated. Finally, a strategy is formulated for using range divisors to estimate s when determining sample sizes.
Distributions
Samples are simulated from eight symmetric distributions with kurtoses (b2) from 2.00 to 6.48. Singh (1976) found that kurtosis is the distribution parameter that affects the sampling distribution of the mean range. Thus, in this article, we study symmetric distributions of varying kurtosis but not skewness to determine the effect of distribution shape on the standardized mean range.
We wanted to include a variety of symmetric distributions with kurtosis rep- resenting short-tailed distributions (kurtosis< 3) and kurtosis representing heavy-tailed distributions (kurtosis> 3). For example, in some simulation stu- dies utilizing symmetric distributions, the extreme distributions on each end are the uniform (kurtosis¼ 1.8) to the double exponential (kurtosis¼ 6). We used this to create a range of kurtosis close to these limits. We choose the exponential power distribution family for our short-tailed distributions as a convenient way to adjust kurtosis by manipulating one parameter while keeping everything else constant. In the same way, the contaminated normal family allowed for a con- venient way to adjust kurtosis while manipulating only the parameters of
Rhiel and Markowski 321
that family of distributions. In the simulation study, we incorporated eight dis- tributions which have infinite ranges. These were the normal distribution, four contaminated normal distributions with high kurtosis, and three exponential power distributions with low kurtosis. In addition, samples are simulated from five truncated-normal distributions.
Although populations with infinite ranges are not usual, we wanted to simu- late dn values associated with sampling frames from distributions with infinite ranges to provide lower limits on sample sizes. We will use these to guide the researcher in making choices based on the size of the sample and its effect on the cost of using panels for survey research and when conducting double sampling. For the distributions with infinite ranges, we compare the dn values as kurtosis varies to determine its effect on the sample size.
The five truncated-normal distributions provide dn values that are compared with the dn values for the normal distribution with an infinite range to determine the effect that restricted ranges have on sample sizes. For the most part, real-life distributions do not fit the infinite range model. As examples, the distribution of age or family income would not be consistent with such a model. Thus, it is important to consider the effect that restricted ranges have on using range estimators of s for determining sample samples. The truncated-normal distri- butions are truncated from �2 to �4 standard deviations from the mean to allow a comparison of the dn values from several truncated-normal distributions to the dn values from the normal distribution with infinite range.
Results
Values of dn for infinite range distributions
Table 1 contains the standardized mean ranges simulated from the distributions with infinite ranges for sampling frame sizes from 500 to 1,000,000. By dividing dn into the range of a sampling frame, an unbiased estimate of s is obtained (i.e., E(R/dn)¼s) and would be useful in determining s when a sample size is desired (Rhiel, 1989).
As the distributions increase in kurtosis, the dn values become larger. This is because the higher the kurtosis of a distribution, the heavier the tails, allowing more values in the sampling frames to come from the tails of the population, increasing the mean range. What is stark about this increase is the considerable difference between the dn values of the lowest to the highest kurtosis distribu- tions. Even the difference between the dn values for the normal distribution and the lowest kurtosis distribution and the normal and the highest kurtosis distri- bution is substantial. What we see from this study is that the differences tend to be more emphatic as the sampling frame size increases. The consequence of having these substantial differences is that when using range divisors to estimate s for determining sample sizes, the estimate of s and the sample size could be
322 Psychological Reports 120(2)
quite different depending on your distribution choice and sampling frame size, especially with large populations or sample frames.
Values of dn for finite range distributions
Table 2 contains the dn values for the truncated-normal distributions. It can be seen from this table that the dn values converge to the truncated value as the sampling frame size increases. For the normal distribution, which is truncated at �2 standard deviation units, creating a range of 4s, the dn values have already converged to 4 at a sampling frame size of 500. For the normal distribution which is truncated at a range of 5s, the dn values converge quickly to 5. The same is true for the normal distribution truncated at 6s where the dn values converge to 6 or near 6, quickly. As the truncated ranges increase to seven and eight standard deviation units, much larger sampling frame sizes are needed before the standardized mean range converges to the truncated value.
By comparing Table 1 with Table 2, we can assess the effect of truncation on the dn values. For example, when comparing the dn values from the infinite- normal distribution to the dn values for the truncated-normal distributions, a couple of things should be noted. As one might expect, the most severely trun- cated distributions generate dn values which are much smaller than the dn values
Table 1. Standardized mean range (dn) for normal, low kurtosis, and
high kurtosis infinite distributions.
N
b2
2.00 2.40 2.80 3.00 3.52 4.57 5.62 6.48
500 4.3 5.1 5.8 6.1 6.8 7.7 8.2 8.6
1000 4.4 5.4 6.1 6.5 7.4 8.4 9.0 9.4
1500 4.5 5.5 6.3 6.7 7.7 8.7 9.4 9.8
2000 4.5 5.6 6.5 6.9 7.9 9.0 9.7 10.1
3000 4.6 5.8 6.7 7.1 8.2 9.3 10.0 10.5
4000 4.6 5.9 6.8 7.2 8.5 9.6 10.3 10.7
5000 4.7 5.9 6.9 7.3 8.6 9.8 10.5 10.9
10,000 4.7 6.1 7.2 7.7 9.1 10.3 11.1 11.6
50,000 4.9 6.5 7.9 8.5 10.1 11.5 12.4 12.9
100,000 5.0 6.7 8.1 8.8 10.5 11.9 12.8 13.3
500,000 5.1 7.00 8.5 9.4 11.1 12.6 13.5 14.1
1,000,000 5.2 7.00 8.6 9.7 11.3 12.8 13.7 14.3
dn values for eight distributions of varying kurtosis values (b2) from sampling frame sizes (N).
Rhiel and Markowski 323
for the infinite-normal distribution. As the truncation becomes less severe, the dn values converge toward the dn values for the infinite-normal distribution. For the distribution that is truncated at 8s, the dn values are quite similar to those values for the infinite normal with this similarity being less prominent for larger sam- pling frame sizes. Of course, the real question is what effect does this have on determining an appropriate sample size?
Effect on sample size of using a range estimator of 4 or 6
The most often suggested range divisors for estimating s when determining a sample size has been either 4 or 6 (see Black, 2010 and Berenson et al., 2013). Figure 1 provides information concerning the effect on the sample size of using range divisors of 4 or 6 when compared with using the standardized mean ranges (dn) from the infinite-normal distribution as the divisors. This is of interest since in this article we provide dn values up to a sampling frame size of 1,000,000 whereas previously these values were only provided for sampling frame sizes up to 2000.
In determining the sample sizes that are in Figure 1, we are assuming the correct range divisors should come from the infinite-normal distribution. Thus,
Table 2. Standardized mean range (dn) for truncated-normal
distributions.
N
Truncation
� 2s(4s) � 2.5s(5s) � 3s(6s) � 3.5s(7s) � 4s(8s)
b2¼ 2.42 b2¼ 2.53 b2¼ 2.78 b2¼ 2.93 b2¼ 2.98
500 4.0 4.9 5.5 5.9 6.0
1000 4.0 4.9 5.7 6.2 6.4
1500 4.0 4.9 5.8 6.4 6.6
2000 4.0 5.0 5.9 6.5 6.8
3000 4.0 5.0 5.9 6.6 6.9
4000 4.0 5.0 5.9 6.7 7.1
5000 4.0 5.0 5.9 6.7 7.2
10,000 4.0 5.0 5.9 6.8 7.4
50,000 4.0 5.0 6.0 7.0 7.8
100,000 4.0 5.0 6.0 7.0 7.9
500,000 4.0 5.0 6.0 7.0 8.0
1,000,000 4.0 5.0 6.0 7.0 8.0
dn values for five truncated-normal distributions of varying kurtosis values (b2) from sampling frame sizes (N). The truncations were done to a (0, 1) normal distribution.
324 Psychological Reports 120(2)
we see the effect of using range divisors of 4 or 6 when dn values should be from the normal distribution. When using the dn values from the normal distribution (see b2¼ 3 in Figure 1), the correct sample size is around 100. If we use 4 or 6 as range divisors, the calculated sample sizes are much larger than 100 and this effect becomes more severe as the size of the sampling frame or population increases. For example, using a range divisor of 4 results in a sample size which is 475% larger than using the standardized mean range (dn) when the sampling-frame/population size is 1,000,000.
So why have range divisors of 4 and 6 been popular in textbooks? Following the procedure that is used for determining sample sizes when estimating a pro- portion, if p is not known, p¼ .5 is used because it maximizes the sample size. The same is most likely true when dividing the range by 4; it would maximize or nearly maximize the sample size since a small divisor produces a larger estimate of s and a larger sample size. This makes sense with our simulated values where 4 is the minimum dn value for sampling frame sizes of 500 or more (see Tables 1 and 2). We see from this study that using a range divisor of 4 or 6 to maximize sample sizes can produce sample sizes substantially larger than needed.
Is it a good procedure to always maximize the sample size? If the object is to make sure that your sample size is large enough, probably. However, if you are concerned about the cost of conducting surveys, a sample size that is 475% larger than needed could be undesirable. It seems that wise choices are important in those situations.
The obvious alternative to using range divisors of 4 or 6 would be to use dn values from the infinite-normal distribution (see Table 1) based on the sampling frame size to estimate s for determining a sample size. Two questions arise from this approach. First, what if you use the dn values from the normal and the distribution is not normal and, second, what if you use the dn values from the
Figure 1. Sample sizes when using range divisors of 1 and 6 when dn values should be
from the normal. For determining the sample sizes, Z¼ 1.96 and e¼ .2. Value of s is deter- mined from range divisors.
Rhiel and Markowski 325
normal distribution with an infinite range, but the range of the variable in ques- tion is not infinite?
Let’s look at the first question. Figure 2 contains the sample sizes calculated using the dn values from Table 1 for the normal distribution when the true distributions are of higher kurtosis. When we assume that the high kurtosis distribution with b2¼ 4.57 is the appropriate distribution, we see the following. Dividing the range of the sampling frame by the dn values from the higher kurtosis distributions produces a desired sample size of around 100 for all sampling frame sizes. In comparison, using the dn values from the normal dis- tribution when the true distribution has b2¼ 4.57, the sample sizes are around 150 (see, Figure 2, b2¼ 4.57N) or about 50% higher than desired for all sam- pling frame sizes.
Now, looking at a distribution with even higher kurtosis (b2¼ 6.48), the effect of using the normal dn instead of the dn from the higher kurtosis distribution is much more dramatic. The dn values from the distribution with b2¼ 6.48 provide the desired sample size of slightly larger than 200. However, using the dn values from the normal results in sample sizes between 400 and 500. This is more than a 100% increase in the sample size when using the normal dn values when the true distribution is of higher kurtosis (b2¼ 6.48).
Figure 3 contains the sample sizes calculated using the dn values from the normal and low kurtosis distributions when the true distributions are of low kurtosis. The effect of using the dn values from the normal distribution when the distribution is of lower kurtosis produces sample sizes that are too small. For example when b2¼ 2.4, we see that the correct sample size should be around 43 (see Figure 3, b2¼ 2.4). However, for b2¼ 2.4N, where we used the dn values from
Figure 2. Sample sizes when using the normal dn values when high kurtosis dn values
should be used. For determining the sample sizes, Z¼ 1.96 and e¼ .4. Value of s is deter- mined from range divisors.
326 Psychological Reports 120(2)
the normal distribution but the range is from the low kurtosis distribution, we see that the sample sizes decrease from 44 to between 18 and 25, which is around a 50% decrease. Likewise when b2¼ 2.0, using the dn values from the normal rather than the true distribution results in around a 60% decrease in the size of the sample. In summary, using the dn values from the normal when the true distribu- tions are of lower kurtosis results in sample sizes that are too small.
Figure 4 contains the sample sizes generated when estimating s by using the dn values from normal distributions truncated at a range of five and eight stand- ard deviation units when the true distributions are truncated (see 5s and 8s lines in Figure 4). In addition, the figure contains the sample sizes generated when dividing the range of the sampling frame from the truncated distribution by the dn values from the infinite-normal distribution (see Use Normal with 5s and Use Normal with 8s lines). This illustrates the effect on the sample size of using the dn values from the infinite-normal distribution when the true distributions are truncated normal.
We see from Figure 4 that using the dn values from the infinite-normal dis- tribution when the true ranges are from the truncated-normal distributions results in sample sizes that are too small, especially for the larger sampling frame sizes. The biggest difference is for the normal distribution truncated at 5s where the sample size decreases from about 65 to 22. This is a decrease of 66% in the sample size. For the normal distribution truncated at 8s, we see that the effect of using the dn values from the infinite-normal distribution does not have a big effect on the sample size until the sampling frame sizes are larger than 4000. At that point, the sample sizes become increasingly smaller as the sample frame sizes increase.
Figure 3. Sample sizes when using the normal dn values when low kurtosis dn values
should be used. For determining the sample sizes, Z¼ 1.96 and e¼ .2. Value of s is deter- mined from range divisors.
Rhiel and Markowski 327
Discussion
Given the above information, the question is what strategy do we use in choos- ing a range divisor to provide an estimate of s when estimating a sample size? To develop a strategy, we must consider several issues, including the extent to which the population is truncated, the tail weight of the population, the size of our population or sampling frame, and how aggressive we would like to be in estimating the appropriate sample size.
First, the researcher should consider how much he or she knows about the population. If the researcher has knowledge about the population shape and knows that the range of the population is wide enough that it fits the infinite distribution model, the appropriate dn divisor could be selected from Table 1 in this study. This would require that the researcher knows the size of the sampling frame or panel and knows or can estimate the range of the variable of interest. In some cases, experienced researchers may know the extent to which a popu- lation has a truncated range and would be able to choose the appropriate dn values from Table 2.
Second, if the researcher is concerned about obtaining a sample size which is large enough, but is not concerned if the sample size is too large, a range divisor of 4 would be the choice if the size of the sample frame is at least 500. This technique would maximize the sample size. However, if the cost of
Figure 4. Using the normal dn values with the truncated-normal distributions. For determining
the sample sizes, Z¼ 1.96 and e¼ .2. Value of s is determined from range divisors.
328 Psychological Reports 120(2)
obtaining the sample values is of importance, maximizing the sample size may not be desirable.
Third, the researcher could choose to be very conservative by choosing the largest dn value based on the size of the sampling frame. For example if the sam- pling frame size is 50,000, from Table 1, the researcher would choose a dn¼ 12.9. This would give a fairly small sample size, butwould protect against oversampling. In this case, a double sampling technique (see Cox, 1952) may be needed to add a second sample to the first one to achieve the appropriate sample size.
Finally, what would be the approach if the only information the researcher has is the range and size of the sampling frame when cost is a major factor in collecting the data? This is an important question because the cost to obtain survey data can vary. For example, online surveys can have different costs based on the quality of their panels. Two companies which provide panels for surveys are Qualtrics and Amazon Mechanical Turk. Let’s say you want to conduct a survey and your pricing for the two services is $5.50 per survey for Qualtrics and $1.00 per survey for Amazon Mechanical Turk. Although you prefer Qualtrics, your budget is limited, so you must consider cost in making your choice between the two. Let’s say that you plan on employing a double sampling technique (see Cox, 1952), but would like to factor in cost when choosing which company to employ for the survey.
We suggest determining a range of sample sizes from which you can determine a range of the cost for the survey. For example, if you know that the population has a near infinite range and you know the panel size for each database is 50,000, you would do the following to get the range of the sample size. Choose the smallest and largest dn values from Table 1 for a sampling frame size of 50,000 and use the formula below to calculate your smallest and largest sample size.
Z2 R=dnlgð Þ2
e2 � n � Z
2 R=dnsmð Þ2
e2
Assuming R¼ 100, Z¼ 1.96, and the sampling frame size is 50,000 and using the dn values from Table 1 of 12.9 and 4.9, the range for the sample size is calculated below.
1:962ð100=12:9Þ2
22 � n � 1:96
2ð100=4:9Þ2
22
58 � n � 400
This provides a range that contains the true sample size for the study. From this, you can calculate the range of possible costs for both Qualtrics and Amazon Mechanical Turk by multiplying the sample sizes times the cost. For this example, the Qualtrics range is calculated as follows: 5.50� 54 to
Rhiel and Markowski 329
5.50� 400, which results in a range of the cost of $297–$2200. For Amazon Mechanical Turk, it will be 1� 58 to 1� 400, which results in a cost range of $58–$400. This provides you the maximum and minimum possible costs for the survey for each company. This could be a factor you would consider when making your decision about which service to use.
If the distribution that you are sampling from has significant truncation, dn values from Table 2 can be used to determine the sample sizes and costs. Or, to be very conservative, you could use a dn sm value of 4 from the truncated dis- tributions (Table 2) and the dn lg from the infinite-normal distribution (Table 1) when determining a sample size range.
Limitations
There are two main limitations to this study. These include the researchers’ ability to estimate the range of the sampling frame for the variable of interest and the researchers’ ability to choose the appropriate dn value. In some research projects, this may not be a problem. In research where it is a problem, the technique provides more of an approximation of s and the sample size rather than a close estimate of the sample size. Of note, however, is the use of the technique in calculating minimum and maximum sample sizes. This allows researchers to calculate upper and lower limits of the cost for the sample as illustrated in the Discussion section of this article. Also of note is using the technique to provide a maximum sample size or to provide a minimum sample size to be used as the first step in a double sampling procedure. I believe that as researchers use this technique for determining sample sizes, they will discover what works best for their research needs.
Conclusions
To summarize, we have proposed four options to consider if you are using the range to estimate s when determining a sample size. These four options provide a framework for using the range to estimate s when the objective is to determine an appropriate sample size for estimating a mean. Selection of the best option to use is based on the sampling needs of the researcher.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publica- tion of this article.
330 Psychological Reports 120(2)
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Author Biographies
George Steven Rhiel is a University Professor and former Chair of IT/Decisions Sciences in the Strome College of Business at Old Dominion University in Norfolk, Virginia. His research has been focused in the areas of sampling, estima- tion, robust statistics, and service to online programs. He has published articles in the Journal of Statistical Computation and Simulation, Communications in Statistics-Simulation and Computation, Industrial Relations, Journal of Statistics Education, Psychological Reports, and Corporate Reputation Review.
Edward Markowski is University Professor of Decision Sciences in the Department of Information Technology and Decision Sciences at Old Dominion University. He has published in a variety of journals in areas such as nonparametric statistics, robust statistical methods, statistical education, and mathematical programming formulations of statistical problems. Currently, his research focuses on application of advanced statistical methods such as struc- tural equation modeling and hierarchical linear modeling to social science research.
Rhiel and Markowski 331
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