Sample Size Formula

Sample size is nothing but the number of observations that constitute the experiment /  research. It is generally denoted by n. 

Sample size determination is closely related to estimation. Sometimes, we may need to know how large a sample is necessary in order to make an accurate estimate. The answer depends on 

  • The margin of error.
  • The degree of confidence.
In considering large sample confidence intervals for the mean, since the error of estimate is given by E = E = Zα/2 (σ / √n), we can solve to find n, the sample size. 
Solving, we have
n = [(Zα/2 x σ) / E]2


Example for Sample Size Formula


Q :   What sample size should be selected to estimate the mean age of workers in the large factory to with in ±1 year at a 95 percent confidence level if the standard deviation for the ages is 3.5 years?

Sol :  We are given that α=0.05, Zα/2 = 1.96 {the value is obtained from the normal distribution chart}, and E=1.

Substituting into the formula, we get the sample size as 

n = [(Zα/2 x σ) / E]2 = [(1.96 x 3.5)/1]2 = 47.0596 ≈ 48

That is, in order to be 95 percent certain that the estimate is with in 1 year of the true mean age, a sample of at least 48 is selected.

Note: For a large enough sample size (n ≥ 30), when the population standard deviation σ is unknown, we can replace σ with s in the above equations.


Minimum Sample Size:


The equation for the minimum sample size is:

n = 0.5 x [Zα/2/ E]2


Example for Sample Size Formula :


Q : The researcher wants to determine the sample size for his research study. If a margin of error of ±0.02 is acceptable at 95 percent confidence interval, what is the minimum sample size that should be taken?

Sol :  We are given α= 0.05 and E= 0.02. Since, α= 0.05, then Zα/2 = 1.96. Thus, 

n = 0.5 x [Zα/2/ E]

= 0.5 x  [1.96/ 0.02]


Thus the researcher should sample at least 4,802 in the research study.