# 8.2.2.3 - Computing Necessary Sample Size

Calculating the sample size necessary for estimating a population mean with a given margin of error and level of confidence is similar to that for estimating a population proportion. However, since the $$t$$ distribution is not as “neat” as the standard normal distribution, the process can be iterative. (Recall, the shape of the $$t$$ distribution is different for each degree of freedom). This means that we would solve, reset, solve, reset, etc. until we reached a conclusion. Yet, we can avoid this iterative process if we employ an approximate method based on $$t$$ distribution approaching the standard normal distribution as the sample size increases. This approximate method invokes the following formula:

Finding the Sample Size for Estimating a Population Mean
$$n=\frac{z^{2}\widetilde{\sigma}^{2}}{M^{2}}=\left ( \frac{z\widetilde{\sigma}}{M} \right )^2$$

$$z$$ = z multiplier for given confidence level
$$\widetilde{\sigma}$$ = estimated population standard deviation
$$M$$ = margin of error

The sample standard deviation may be estimated on the basis of prior research studies.