The finite population correction
by Pamela Narins, Manager of Market Research
In the last column, we discussed a formula that could be used to determine the number of people that should be surveyed using a simple random sampling methodology. This formula works in many situations, but, of course, is not appropriate all of the time. Several readers asked for more information about a specific case, and that's what I'd like to address this time.
What happens if the population from which you are sampling is small? You might be doing research on city-wide elections, where the voting population is several thousand. Alternatively your study might address employee satisfaction for a company of 500 people. If your population seems small, you can employ the finite population correction to the equation that we discussed last time (see Keywords 54).
The sample size derived using the finite population correction will be smaller than that derived from the uncorrected equation. This makes sense, as one would naturally assume that you'd need to sample a smaller number of people from a smaller population.
Remember that this equation requires an estimate of the percent of responses to a dichotomous variable indicated by Py and Pn [usually set at (.50)(.50) for the most conservative estimate], and the standard error. The addition of the finite population correction is simply the addition of two other algebraic terms.
[This formula is from Kish, Survey Sampling, (Wiley, 1965). There are other, slightly different versions of this formula, although all principally do the same thing.]
Basically, this new formula adds in numbers that we already know. The only term in the equation that you have not yet seen is N1, the size of the true population. Let's remember that the formula for determining the standard error is:
The Z or t distribution coefficient is determined by the chosen level of confidence, that is, 1.96 for 95%, 2.58 for 99%, and so on. If we want to be 95% certain that our results are plus or minus 5 percentage points from the actual score, the Standard Error2 is (.05/1.96)2 or .0006507.
The new term in the denominator of Equation 2 is simply the proportion of those answering "Yes" times those answering "No" [the most conservative estimate again being (.5)(.5), or .25] divided by N1, the actual number in the population.
we arrive at the sample size
Using equation 1, the standard formula, the sample size estimate is 384.
A rule of thumb
If your sample size is more than 5% of the population size, consider using the finite population correction.
The municipal elections example provides us with a suggested sample of 384 which is about 4% of the total, known population of 10,000. In this example, the finite population correction, which reduces the sample size by 13, would save little time or cost.
The bad news is no, there is no easy way to do it. The good news is that because we can take the most conservative approach (by setting Py and Pn to .50), all things being equal, we are in relatively little danger of taking too few respondents if we use the dichotomous method.