Sample Size Formula, Definition, Solved Examples

Sample Size Formula: In the field of statistics, the sample size represents the number of observations used to measure population estimates within a specific group. Essentially, it signifies the quantity of individual samples utilized in a data analysis. By leveraging the difference between the overall population and the sample, it determines the appropriate sample size. This process involves the selection of a subset from a population to infer the characteristics of the entire group, a practice known as sampling, typically symbolized as 'n'.

Sample Size Formula

When dealing with a smaller sample size, the T distribution is used instead of the normal distribution. This specialized distribution comes into play when the sample size is under 30. Utilizing the t statistic, this test examines the null hypothesis using both one-tailed and two-tailed approaches, particularly when the population variance is unknown and the sample size is small. This adjusted sample size is also referred to as the modified sample size.

Let's modify the given mathematical expressions in a more structured and explanatory form:

Adjusted Sample Size Equation:

A = n / (1 + (n – 1)/P)

Where, A represents the adjusted sample size, n stands for the sample size,  P denotes the population size. For a population of infinite size, the formula is expressed concerning the z-value and error margin:

Sample Size Formula:

n = Z ² p(1 – p)/m ²

Where,  n is the sample size, Z represents the z-value,  p signifies the proportion of the population (typically assumed as 0.5),  m indicates the margin of error.

Sample Size Formula Solved Examples

Example 1: Determine the adjusted sample size for a sample size of 150 from a population of 30,000.

Solution: Given: n=150 ,P=30,000

Using the Sample Size Formula: A = n / (1 + (n – 1)/P)

A=  150 ​  / (1+149/ 30000  ​)

A= 1.004967 150 ​

A≈149.254

Example 2 : Calculate the adjusted sample size for a sample of 50 from a population of 5,000.

Solution: Given: n=50 , P=5,000

Using the Sample Size Formula: A = n / (1 + (n – 1)/P)

A= 50/ (1+ 49/5000) ​ ​

A= 1.0098 50 ​

A≈49.55

Example 3: Determine the population size if the adjusted sample size is 85.6, given a sample size of 90.

Solution: Given: A=85.6 , n=90

Using the formula: A = n / (1 + (n – 1)/P)

85.6=  90/ (1+ 89 ​/P)

1+ 89 / P ​ = 90 ​/ 85.6

89 ​/ P = 90 / 85.6 ​ −1

89/ P ​ ≈1.0491

P≈ 1.0491 89 ​

P≈84.78

The estimated population size is approximately 84.78. These examples demonstrate how to calculate adjusted sample sizes using the formula and also estimate population size based on the provided adjusted sample size and actual sample size.

The sample size in statistics is a crucial factor used to estimate characteristics of a larger population based on observations within a specific group. It determines the quantity of individual samples used in data analysis, facilitating the accurate representation of a population's features through a subset of observations. The process involves distinguishing between the overall population and the sample to determine an appropriate sample size.

When dealing with smaller sample sizes, statistical analysis may utilize the T distribution instead of the normal distribution, especially when the sample size is below 30. This specialized distribution, using the t statistic, becomes valuable for testing hypotheses, particularly in scenarios where the population variance is unknown and the sample size is small. The adjusted sample size, or modified sample size, accounts for these statistical considerations.

The mathematical formulas, such as the Adjusted Sample Size Equation and the Sample Size Calculation Formula, offer structured methods for determining sample sizes based on specific parameters like the sample size, population size, z-value, proportion of the population, and margin of error.

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