What is the formula for calculating a sample size?

Answer 1

Your question is a bit ambiguous, but I will provide the equation commonly used to calculate the necessary sample size in estimating the proportion of a distribution:

n = (z @ half-alpha/(2*Error))^2

z is commonly called the "Z score" and the error is given in terms of +/-. I refer this as z @ half alpha, so if my confidence level is 0.99, alpha is 0.01 and Z is evaluated at p= 1-0.1/2 of 0.995. The z value is found from tables of the normal distribution.

Example: I want to take a survey if people like Republicans or Democrats, I want to a 99% confidence limit that my survey proportion is correct within +/- 0.06.

z for .995 is 2.576, so n = (2.578/.12)^2 = 461 samples.

Note that this is the minimum size given the parameters established for my survey. If the survey question is biased, or if the selection of respondants is non-random, the above relationship beween error and sample size does not hold.

Sampling distributions of statistics are at the heart of both confidence intervals and sample size. Generally, the underlying concept is that the errors that make up the uncertainty in estimates are small and independent (random), so as the sample size increases, the accuracy of results will increase. This is not always true in nature, as in the case where more data leads to a higher percentage of flawed data.

I hope this answers your question. If not, please rephrase. Also, you may find a lot more information under "sample size" and "statistics" by searching wikipedia and other internet sites.