Among the most important decisions that a survey researcher must make, affecting time, budget, resource allocation, and the attainment of analytic objectives, is determination of the sample size. Despite the importance of this decision, a perusal of the internet and various software sources often show only the single sample size formula of \[\small n = z^2 \left(p(1-p) \right)/e^2 \] where \(z\) is a quantile from the standard normal distribution based on a confidence level and \(\small e\) is based on the required margin of error. Sometimes the \(\small t\)-statistic is referenced instead of the \(z\)-score. While this can be the best sample size formula to use, it should not be defaulted to. Instead, the sample size formula selected should reflect the sample design and the particular estimation goals. This vignette is designed to help the PracTools (Valliant, Dever, and Kreuter (2023)) user select the best sample size formula and the corresponding PracTools sample size function based on a series of survey design questions.

Considerations for Sample Size Determination

At a high level, all that is needed to determine a simple random sample size is the confidence level \(\small \alpha\), the required precision, the population variance or coefficient of variation, the population size \(\small N\), sample unit cost if this is a consideration, and \(\small \beta\) if power is needed. Furthermore, the population size can be treated as infinite for large \(\small N\). However, there are many other factors that should be considered prior to sample size determination.

PracTools Sample Size Functions

PracTools has many sample size functions, and the answers to a few basic questions should identify which are the best choice for the survey researcher:

For the first question, SRS vs. PPS, the issue as it relates to sample size is how the variance is calculated for use in any sample size calculation. For the remaining issues, the following table may be useful for determining which PracTools sample size functions are most appropriate given the survey design. The functions are listed in alphabetical order.

Impact of Design Effect (deff):

The deff is the ratio of the variance of the survey statistic under the complex design over the variance of the survey statistic under a simple random sample design. The effective sample size is the sample size in a complex sample divided by the deff for a particular statistic. The effective sample size is the size of a simple random sample needed to achieve the same variance as that obtained from the complex sample. For sample size calculation for a complex sample, one approach is to compute the SRSWR sample size then multiply it by a deff.

For better or worse, there are several ways of calculating the deff, and depending on the assumptions used, can produce very different deff results. A frequently used deff formula was proposed by Kish is 1965, where: \[\small deff_K = 1+relvar(w) = 1+ n^{-1} \sum_{i=1}^n (w_i- \bar{w})^2/\bar{w}^2\] and \(\small \{w_i\}_{i=1}^{n}\) is the set of sample weights. The Kish deff measures the increase in variances due to using variable weights when equal weights would be optimal.

The Kish formula assumes that a stratified SRS with proportional allocation is optimal. This will be true if all strata population variances and costs are equal. However, \(\small deff_K\) is not always relevant in surveys where variances differ across strata, where subgroups are intentionally sampled at different rates, and/or where different subgroups have substantially different response rates. Other design effect formulas take stratification, clustering, and unequal sampling probabilities more explicitly into account. In PracTools, these design effect functions can be calculated using deffK, deffH, deffS, or deffCR. Please see the PracTools documentation for more detail on how to use PracTool’s design effect functions. Cochran (1977), Lohr (1999), and Valliant, Dever, and Kreuter (2018) cover the mathematical detail behind the formulas evaluated by the functions.


Cochran, W. G. 1977. Sampling Techniques. New York: John Wiley & Sons, Inc.
Lohr, S. L. 1999. Sampling: Design and Analysis. Pacific Grove CA: Duxbury Press.
Valliant, R., J. A. Dever, and F. Kreuter. 2018. Practical Tools for Designing and Weighting Survey Samples. 2nd ed. New York: Springer-Verlag.
———. 2023. PracTools: Tools for Designing and Weighting Survey Samples, Version 1.4.