Sample Size Formula

Sample is the part of the population that helps us to draw inferences about the population. Collecting research of the complete information about the population is not possible and it is time consuming and expensive. Thus, we need an appropriate sample size so that we can make inferences about the population based on that sample.

One of the most frequent problems in statistical analysis is the determination of the appropriate sample size. One may ask why sample size is so important. The answer to this is that an appropriate sample size is required for validity. If the sample size it too small, it will not yield valid results. An appropriate sample size can produce accuracy of results. Moreover, the results from the small sample size will be questionable. A sample size that is too large will result in wasting money and time. It is also unethical to choose too large a sample size. There is no certain rule of thumb to determine the sample size. Some researchers do, however, support a rule of thumb when using the sample size. For example, in regression analysis, many researchers say that there should be at least 10 observations per variable. If we are using three independent variables, then a clear rule would be to have a minimum sample size of 30. Some researchers follow a statistical formula to calculate the sample size.

Calculation of the sample size:

Sample size based on confidence intervals: In calculating the sample size, we are interested in calculating the population parameter. Thus, we should determine the confidence intervals, so that all the values of the sample lie within that interval range. The population mean upper bound confidence interval can be calculated by using the following formula:

Where

= sample mean

= value from the normal distribution

s = sample standard deviation
n = sample size
When

is subtracted from both sides of the equation, the result can be solved for n as follows:

When the sample proportion is known to us, then we can calculate the sample size by using this formula:

Where n = sample size and q= 1-p

When the sample size is calculated from two sets of populations, then the above mentioned formula will be the same, except for the variance factor. Sample size can be calculated from two populations using the following formula:

Sample size calculation based on the effect size:

An alternative approach of calculating the sample size is effect size. Effect size is known as the difference between the sample statistics divided by the standard error. More efficiently it is as follows:

Once an effect size has been estimated, the following table can be used to estimate a sample:

As mentioned above, the alpha is equal to the acceptable probability of the type I error and beta is the acceptable probability of type two errors and 1-beta equal to the power. As the power will increase with different levels of alpha, sample size will also increase.