Sample Size Formula

The sample is part of the population in some data collection. It helps us to draw inferences about the whole population. Collecting the research of the complete information about the population is not possible and also it is time-consuming as well as expensive. Thus, we need an appropriate sample size so that we can make proper inferences about the population. Thus, we can understand that sample size computation is very necessary for statistics based on some samples. Let us learn more about the sample size formula.

What is Sample Size?

The sample size formula helps us find the accurate sample size through the difference between the values of the population and the sample. As we know that the number of observations in a given sample population is known as sample size. Since it is not possible to survey the whole population, we have to take a sample from the population and then conduct a survey or research accordingly. The sample size is denoted by n or  N. It is also written as SS.

We should know that the sample size, we are taking from the population, will not hold good for the whole sample. We have a level of confidence and margin of error to determine that the sample size is accurate or not. The confidence level will help to describe how sure we are about the results of the survey hold true or accurate.

One of the very frequent problems in statistical analysis is the determination of the desired sample size. We may ask why the sample size is so important. Its answer is that the appropriate sample size is required for validity. If the sample size is very small, it will not give valid results. An appropriate sample size will help to produce accurate results. Moreover, the results from the small sample size will not be reliable.

Source:en.wikipedia.org

A sample size which is very large will result in wasting time and effort. It is also unethical to choose a very large sample size. There is no specific rule of thumb to determine the sample size. But, some researchers support a rule of thumb when using the sample size. For example, for regression analysis, it is believed that there should be at least 10 observations per variable. Therefore, for three independent variables, we need a minimum sample size of 30. Some researchers follow the formula for finding out the sample size.

Sample Size Formula

The sample size for an infinite population as well as for a finite population is:

For Infinite Sample Size: \(SS1 = \frac{Z^2 p(1-p)}{C^2}\)

For Finite sample size: \(SS = \frac {SS1}{1+ \frac {SS1}{pop} }\)

SS	The sample size for infinite sample
SS1	The sample size for finite sample
Z	Given Z value
p	Percentage of population
C	Confidence level
pop	Population

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.

Solved Examples for Sample Size Formula

Q.1: Find the sample size for some finite and infinite population when the percentage of 4300 population is given as 0.05. Here take confidence level as 99 and confidence interval as 0.01?

Solution: First we need to find Z value for the z-table which is 2.58.

p = 0.05

C= 0.01

pop = 4300

Now we will apply the given data in the formula:

\(SS1 = \frac{Z^2 p(1-p)}{C^2}\)

\(SS1 = \frac {{2.58}^2{(.05)}{(1-.05)}}{{.01}^2}\)

Thus SS1 = 3161.8

\(SS = \frac {SS1}{1+ \frac {SS1}{pop} }\)

\(So, SS = \frac {SS1}{1+ \frac {SS1}{pop} }\)

\(SS = \frac {3161.8}{1 + \frac {3161.8}{4300}}\)

SS = 852