6 Sample Size

6.2.1 Cochran’s Sample Size Formula

Used to compute an ideal sample size for a desired level of precision, it is recommended to be used for studies with infinite populations (Cochran ⊕1977Cochran, W.G. 1977. Sampling Techniques. 3rd ed. New York: John Wiley & Sons.).

\(n_{0}=\dfrac{z^{2}\cdot p\cdot(1-p)}{e^{2}}\)

e: desired level of precision, the margin of error

p: the fraction of the population (as percentage) that displays the attribute

z: the z-value, extracted from a z-table6262 The entry for z in a z-table represents the area under the normal distribution curve to the left of z (Figure 6.1)..

Figure 6.1: Area represented by the z-value.

Let’s consider an example. Think of a study of students in a large university campus for which we don’t know the campus size6363 For example a large campus may have 10 - 15 K students. We are interested in finding the percentage of students who eat lunch at the campus dinner halls but we do not have insider information. The question is how many students would we need to ask that question to be able to determine, with reasonable confidence, what percentage of students conform to the sought behavior. Given the lack of information we start by considering that 50% of the students eat lunch at the school dining halls, which provides the largest variability. Then we consider a 95% confidence level (leading to an \(\alpha\)=0.05) and a ±5% precision. From the z-tables, the value for z is 1.96. Therefore, the theoretical sample would be:

\(n_{0}=\dfrac{1.96^{2}\cdot0.5\cdot(1-0.5)}{0.05^{2}}=384.16\approx385\)

How to find the value of z from a z-table. The procedure is:

1. Convert the confidence level from percent form to decimal form as value between 0 and 1. (95% \(\rightarrow\) 0.95)
2. Subtract the value from 1 and divide by 2 to find out how much is half (1 - 0.95 = 0.05; 0.05/2 = 0.025)
3. Add the value from 2) to the value from 1) (0.95 + 0.025 = 0.975)
4. Look for the value obtained in step 3) in table values. In Table 6.1 the value sits at the intersection of row labeled 1.9 and column labeled 0.06.
5. Determine the value of z by adding the value for the column with the value for the row obtained in step 4 (1.9 + 0.06 = 1.96).

6.2.2 Cochran’s Modified Formula for Finite Populations

A slightly modified formula can be used if the size of the population is known.

\(n=\dfrac{n_{0}}{1+\dfrac{n_{0}-1}{N}}\)

n₀: Cochran’s sample size computed using the formula for ideal sample size;

N: the size of the population6464 The sample size is dependent on the size of the population until the population reaches about 40-50 K, after which the increase is almost none. Therefore, if the estimated population is this large or larger, the theoretical sample size, as computed for an unknown population, is about equal to the one generated by the modified formula..

As an example, let’s look at the same problem as before but for a much smaller campus of N = 600 students. While we can still use the theoretical sample of 385 participants computed before, do we need to? The necessary sample size may be smaller.

\(n=\dfrac{385}{1+\dfrac{385-1}{600}}=234.76\approx235\)

The result of this computation indicates that for smaller populations the number of subjects (sample size) can be smaller (235 vs. 385) for the researchers to be reasonably confident of the findings.

6.2.3 Yamane’s Simplified Formula for Sample Size

To make it simpler to compute the sample size without over estimating it when the population is known Yamane (1967) proposed the following formula:

\(n=\dfrac{N}{1+N\cdot e^{2}}\)

N - population size

e - level of precision

Using the same example as before, Yamane’s formula would suggest a sample size of 240 subjects for a student population of 600.

\(n=\dfrac{600}{1+600\cdot0.05^{2}}=240\)

Table 6.1: z-table