When is n large enough? Looking for the right sample size to estimate proportions

ABSTRACT

The central limit theorem indicates that when the sample size goes to infinite, the sampling distribution of means tends to follow a normal distribution; it is the basis for the most usual confidence interval and sample size formulas. This study analyzes what sample size is large enough to assume that the distribution of the estimator of a proportion follows a Normal distribution. Also, we propose the use of a correction factor in sample size formulas to ensure a confidence level even when the central limit theorem does not apply for these distributions.

KEYWORDS:

• Sample size
• proportion
• central limit theorem
• confidence interval
• Bernoulli distribution

Acknowledgements

We thank Lafayette Eaton, Lucia Cifuentes and Mauricio Canals for their valuable comments. We also thank our students, who asked us many times when is n large enough, motivating this research.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 There have been some efforts to provide a sample size estimation avoiding the formula based on asymptotic approximation, like the proposed by Vallejo et al. [15]. However, probably their approximation is not simple enough to be broadly used and might fail to ensure the desired confidence level when the variance is very high. See Appendix for further analysis.

2 When the variance of the variable is unknown, instead of using a normal distribution we usually use a Student’s t distribution. However, when the sample size is greater than equal to 30 this formula is a good approximation [2].

3 We fitted a power function using this method with a maximum of 400 iterations, 1e⁻⁰⁸ and 0.1 as minimum and maximum change, respectively, in variables for finite differencing, and a tolerance of 1e⁻⁰⁶.

4 We also adjusted a model with the form $n^{\ast \ast }=A+f_c\cdot Z_{\alpha /2}^2\left(p\right(1-p\left)/e^2\right)$, being A a constant, but its R² was lower than imposing A = 0.

5 We did not analyze the sample size for error = 1% because the cited web page only estimate sample size for the cases of sample size lower than 2500, and most of sample size required with 1% of error, are greater than 2500. So we do not have enough data to make a comparison.