Sample Size Formula
The sample size formula helps us find the accurate sample size through the difference between the population and the sample. To recall, the number of observations in a given sample population is known as sample size. Since it not possible to survey the whole population, we take a sample from the population and then conduct a survey or research. The sample size is denoted by “n” or “N”. Here, it is written as “SS”. Let us learn the sample size formula along with a few solved examples.
What Is Sample Size Formula?
The sample size formula is determined in two steps. First, we calculate the sample size for the infinite population and second we adjust the sample size to the required population. The sample size formula can be given as:
Formula 1: Sample size for infinite population
S= Z^{2 }× P^{ }× \(\dfrac{(1P)}{M^2}\)
Formula 2: Adjusted sample size
Adjusted Sample Size = \(\dfrac{(S)}{1 + \dfrac{(S1)}{\text{Population}}}\)
where,
 S = sample size for infinite population
 Z = Z score
 P = population proportion ( Assumed as 50% or 0.5)
 M = Margin of error
Note: Z score is determined based on the confidence level.
Confidence Level: Probability that the value of a parameter falls within a specified range of values. For example, for 95% confidence level Z score is 1.960. The margin of error: It is defined as a small amount that is allowed for in case of miscalculation or change of circumstances. Generally, the margin of error is taken as 5% or 0.05.
Let us see how to use the sample size formula in the following solved examples section.
Solved Examples Using Sample Size Formula

Example 1:Calculate the sample size for a population of 100000. Take confidence level as 95% and margin of error as 5%.
Solution:
To Find: Sample size for 100000 population .
We will calculate the sample size first by calculating it for infinite size and then adjusting it to the required size.
Given: Z = 1.960, P =0.5, M = 0.05Using sample size formula,
S= Z^{2} × P × \(\dfrac{(1P)}{M^2}\)
S= (1.960)^{2 }× 0.5^{ }× \(\dfrac{(10.5)}{0.05^2}\)
= 3.8416 × 0.25 / 0.0025
S = 384.16Answer: The sample size for the infinite population is 384.16

Example 2 : Using the sample size formula, adjust the sample size for the required population in solved example 1.
Solution:
To Find: Adjusted sample size.
Given: Z = 1.960, P =0.5, M = 0.05
Using sample size formula for adjusted sample size ,
Adjusted Sample Size = \(\dfrac{(S)}{1 + \dfrac{(S1)}{\text{Population}}}\)
= \(\dfrac{(384.16)}{1 + \dfrac{(384.16 1)}{100000}}\)
= 382.69 or
= 383 approx.Answer: The required sample size for a population of 100000 is 383.